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Table Of Integrals, Series And Products (7Ed , Elsevier, 2007 Gradshteyn I , Ryzhik 1220S)

Table of Integrals, Series, and Products
Seventh Edition
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Table of Integrals, Series, and Products
Seventh Edition
I.S. Gradshteyn and I.M. Ryzhik
Alan Jeffrey, Editor
University of Newcastle upon Tyne, England
Daniel Zwillinger, Editor
Rensselaer Polytechnic Institute, USA
Translated from Russian by Scripta Technica, Inc.
AMSTERDAM • BOSTON • HEIDELBERG • LONDON
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∞
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ISBN-13: 978-0-12-373637-6
ISBN-10: 0-12-373637-4
PRINTED IN THE UNITED STATES OF AMERICA
07 08
09 10 11 9
8
7
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5
4
3
2 1
Contents
Preface to the Seventh Edition
Acknowledgments
The Order of Presentation of the Formulas
Use of the Tables
Index of Special Functions
Notation
Note on the Bibliographic References
0
Introduction
0.1
0.11
0.12
0.13
0.14
0.15
0.2
0.21
0.22
0.23–0.24
0.25
0.26
0.3
0.30
0.31
0.32
0.33
0.4
0.41
0.42
0.43
0.44
1
xxi
xxiii
xxvii
xxxi
xxxix
xliii
xlvii
Finite Sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Progressions . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Sums of powers of natural numbers . . . . . . . . . . . . . . . .
Sums of reciprocals of natural numbers . . . . . . . . . . . . . .
Sums of products of reciprocals of natural numbers . . . . . . .
Sums of the binomial coeﬃcients . . . . . . . . . . . . . . . . .
Numerical Series and Inﬁnite Products . . . . . . . . . . . . . .
The convergence of numerical series . . . . . . . . . . . . . . .
Convergence tests . . . . . . . . . . . . . . . . . . . . . . . . .
Examples of numerical series . . . . . . . . . . . . . . . . . . .
Inﬁnite products . . . . . . . . . . . . . . . . . . . . . . . . . .
Examples of inﬁnite products . . . . . . . . . . . . . . . . . . .
Functional Series . . . . . . . . . . . . . . . . . . . . . . . . . .
Deﬁnitions and theorems . . . . . . . . . . . . . . . . . . . . . .
Power series . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Fourier series . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Asymptotic series . . . . . . . . . . . . . . . . . . . . . . . . . .
Certain Formulas from Diﬀerential Calculus . . . . . . . . . . . .
Diﬀerentiation of a deﬁnite integral with respect to a parameter .
The nth derivative of a product (Leibniz’s rule) . . . . . . . . . .
The nth derivative of a composite function . . . . . . . . . . . .
Integration by substitution . . . . . . . . . . . . . . . . . . . . .
Elementary Functions
Power of Binomials . . . .
1.11
Power series . . . . . . .
1.12
Series of rational fractions
1.2
The Exponential Function
1.1
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1
1
1
1
3
3
3
6
6
6
8
14
14
15
15
16
19
21
21
21
22
22
23
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25
25
25
26
26
vi
CONTENTS
1.21
1.22
1.23
1.3–1.4
1.30
1.31
1.32
1.33
1.34
1.35
1.36
1.37
1.38
1.39
1.41
1.42
1.43
1.44–1.45
1.46
1.47
1.48
1.49
1.5
1.51
1.52
1.6
1.61
1.62–1.63
1.64
2
Series representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Functional relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Series of exponentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Trigonometric and Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . .
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The basic functional relations . . . . . . . . . . . . . . . . . . . . . . . . . . .
The representation of powers of trigonometric and hyperbolic functions in terms
of functions of multiples of the argument (angle) . . . . . . . . . . . . . . . . .
The representation of trigonometric and hyperbolic functions of multiples of
the argument (angle) in terms of powers of these functions . . . . . . . . . . .
Certain sums of trigonometric and hyperbolic functions . . . . . . . . . . . . .
Sums of powers of trigonometric functions of multiple angles . . . . . . . . . .
Sums of products of trigonometric functions of multiple angles . . . . . . . . .
Sums of tangents of multiple angles . . . . . . . . . . . . . . . . . . . . . . . .
Sums leading to hyperbolic tangents and cotangents . . . . . . . . . . . . . . .
The representation of cosines and sines of multiples of the angle as ﬁnite products
The expansion of trigonometric and hyperbolic functions in power series . . . .
Expansion in series of simple fractions . . . . . . . . . . . . . . . . . . . . . .
Representation in the form of an inﬁnite product . . . . . . . . . . . . . . . . .
Trigonometric (Fourier) series . . . . . . . . . . . . . . . . . . . . . . . . . . .
Series of products of exponential and trigonometric functions . . . . . . . . . .
Series of hyperbolic functions . . . . . . . . . . . . . . . . . . . . . . . . . . .
Lobachevskiy’s “Angle of Parallelism” Π(x) . . . . . . . . . . . . . . . . . . .
The hyperbolic amplitude (the Gudermannian) gd x . . . . . . . . . . . . . . .
The Logarithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Series representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Series of logarithms (cf. 1.431) . . . . . . . . . . . . . . . . . . . . . . . . . .
The Inverse Trigonometric and Hyperbolic Functions . . . . . . . . . . . . . . .
The domain of deﬁnition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Functional relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Series representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Indeﬁnite Integrals of Elementary Functions
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.00
General remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.01
The basic integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.02
General formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1
Rational Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.10
General integration rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.11–2.13
Forms containing the binomial a + bxk . . . . . . . . . . . . . . . . . . . . . .
2.14
Forms containing the binomial 1 ± xn . . . . . . . . . . . . . . . . . . . . . .
2.15
Forms containing pairs of binomials: a + bx and α + βx . . . . . . . . . . . . .
2.16
Forms containing the trinomial a + bxk + cx2k . . . . . . . . . . . . . . . . . .
2.17
Forms containing the quadratic trinomial a + bx + cx2 and powers of x . . . .
2.18
Forms containing the quadratic trinomial a + bx + cx2 and the binomial α + βx
2.2
Algebraic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.20
Introduction . . . . . . . . . . . . . . . . .√. . . . . . . . . . . . . . . . . . .
2.21
Forms containing the binomial a + bxk and x . . . . . . . . . . . . . . . . .
2.0
26
27
27
28
28
28
31
33
36
37
38
39
39
41
42
44
45
46
51
51
51
52
53
53
55
56
56
56
60
63
63
63
64
65
66
66
68
74
78
78
79
81
82
82
83
CONTENTS
2.22–2.23
2.24
2.25
2.26
2.27
2.28
2.29
2.3
2.31
2.32
2.4
2.41–2.43
2.44–2.45
2.46
2.47
2.48
2.5–2.6
2.50
2.51–2.52
2.53–2.54
2.55–2.56
2.57
2.58–2.62
2.63–2.65
2.66
2.67
2.7
2.71
2.72–2.73
2.74
2.8
2.81
2.82
2.83
2.84
2.85
vii
n
Forms containing √
(a + bx)k . . . . . . . . . . . . . . . . . . . . . . . . . .
Forms containing √a + bx and the binomial α + βx . . . . . . . . . . . . . .
Forms containing √a + bx + cx2 . . . . . . . . . . . . . . . . . . . . . . . .
Forms containing √a + bx + cx2 and integral powers of x . . . . . . . . . . .
Forms containing √a + cx2 and integral powers of x . . . . . . . . . . . . . .
Forms containing a + bx + cx2 and ﬁrst- and second-degree polynomials . .
Integrals that can be reduced to elliptic or pseudo-elliptic integrals . . . . . .
The Exponential Function . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Forms containing eax . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The exponential combined with rational functions of x . . . . . . . . . . . . .
Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Powers of sinh x, cosh x, tanh x, and coth x . . . . . . . . . . . . . . . . . .
Rational functions of hyperbolic functions . . . . . . . . . . . . . . . . . . .
Algebraic functions of hyperbolic functions . . . . . . . . . . . . . . . . . . .
Combinations of hyperbolic functions and powers . . . . . . . . . . . . . . .
Combinations of hyperbolic functions, exponentials, and powers . . . . . . . .
Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Powers of trigonometric functions . . . . . . . . . . . . . . . . . . . . . . . .
Sines and cosines of multiple angles and of linear and more complicated functions of the argument . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Rational functions of√the sine and cosine
. . . . . . . . . . . . . . . . . . . .
√
Integrals containing a ± b sin x or a ± b cos x . . . . . . . . . . . . . . . .
Integrals reducible to elliptic and pseudo-elliptic integrals . . . . . . . . . . .
Products of trigonometric functions and powers . . . . . . . . . . . . . . . .
Combinations of trigonometric functions and exponentials . . . . . . . . . . .
Combinations of trigonometric and hyperbolic functions . . . . . . . . . . . .
Logarithms and Inverse-Hyperbolic Functions . . . . . . . . . . . . . . . . . .
The logarithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Combinations of logarithms and algebraic functions . . . . . . . . . . . . . .
Inverse hyperbolic functions . . . . . . . . . . . . . . . . . . . . . . . . . . .
Inverse Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . .
Arcsines and arccosines . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The arcsecant, the arccosecant, the arctangent, and the arccotangent . . . .
Combinations of arcsine or arccosine and algebraic functions . . . . . . . . . .
Combinations of the arcsecant and arccosecant with powers of x . . . . . . .
Combinations of the arctangent and arccotangent with algebraic functions . .
3–4 Deﬁnite Integrals of Elementary Functions
3.0
Introduction . . . . . . . . . . . . . . .
3.01
Theorems of a general nature . . . . . .
3.02
Change of variable in a deﬁnite integral .
3.03
General formulas . . . . . . . . . . . . .
3.04
Improper integrals . . . . . . . . . . . .
3.05
The principal values of improper integrals
3.1–3.2
Power and Algebraic Functions . . . . .
3.11
Rational functions . . . . . . . . . . . .
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84
88
92
94
99
103
104
106
106
106
110
110
125
132
139
148
151
151
151
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161
171
179
184
214
227
231
237
237
238
240
241
241
242
242
244
244
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247
247
247
248
249
251
252
253
253
viii
CONTENTS
3.12
3.13–3.17
3.18
3.19–3.23
3.24–3.27
3.3–3.4
3.31
3.32–3.34
3.35
3.36–3.37
3.38–3.39
3.41–3.44
3.45
3.46–3.48
3.5
3.51
3.52–3.53
3.54
3.55–3.56
3.6–4.1
3.61
3.62
3.63
3.64–3.65
3.66
3.67
3.68
3.69–3.71
3.72–3.74
3.75
3.76–3.77
3.78–3.81
3.82–3.83
3.84
3.85–3.88
3.89–3.91
3.92
3.93
3.94–3.97
3.98–3.99
4.11–4.12
4.13
Products of rational functions and expressions that can be reduced to square
roots of ﬁrst- and second-degree polynomials . . . . . . . . . . . . . . . . . . .
Expressions that can be reduced to square roots of third- and fourth-degree
polynomials and their products with rational functions . . . . . . . . . . . . . .
Expressions that can be reduced to fourth roots of second-degree polynomials
and their products with rational functions . . . . . . . . . . . . . . . . . . . . .
Combinations of powers of x and powers of binomials of the form (α + βx) . .
Powers of x, of binomials of the form α + βxp and of polynomials in x . . . . .
Exponential Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Exponential functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Exponentials of more complicated arguments . . . . . . . . . . . . . . . . . . .
Combinations of exponentials and rational functions . . . . . . . . . . . . . . .
Combinations of exponentials and algebraic functions . . . . . . . . . . . . . .
Combinations of exponentials and arbitrary powers . . . . . . . . . . . . . . . .
Combinations of rational functions of powers and exponentials . . . . . . . . .
Combinations of powers and algebraic functions of exponentials . . . . . . . . .
Combinations of exponentials of more complicated arguments and powers . . .
Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Hyperbolic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Combinations of hyperbolic functions and algebraic functions . . . . . . . . . .
Combinations of hyperbolic functions and exponentials . . . . . . . . . . . . .
Combinations of hyperbolic functions, exponentials, and powers . . . . . . . . .
Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Rational functions of sines and cosines and trigonometric functions of multiple
angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Powers of trigonometric functions . . . . . . . . . . . . . . . . . . . . . . . . .
Powers of trigonometric functions and trigonometric functions of linear functions
Powers and rational functions of trigonometric functions . . . . . . . . . . . . .
Forms containing powers of linear functions of trigonometric functions . . . . .
Square roots of expressions containing trigonometric functions . . . . . . . . .
Various forms of powers of trigonometric functions . . . . . . . . . . . . . . . .
Trigonometric functions of more complicated arguments . . . . . . . . . . . . .
Combinations of trigonometric and rational functions . . . . . . . . . . . . . .
Combinations of trigonometric and algebraic functions . . . . . . . . . . . . . .
Combinations of trigonometric functions and powers . . . . . . . . . . . . . . .
Rational functions of x and of trigonometric functions . . . . . . . . . . . . . .
Powers of trigonometric
with other powers . . . . . . . . .
functions combined
√
2
2
2
Integrals containing 1 − k sin x, 1 − k cos2 x, and similar expressions . .
Trigonometric functions of more complicated arguments combined with powers
Trigonometric functions and exponentials . . . . . . . . . . . . . . . . . . . . .
Trigonometric functions of more complicated arguments combined with exponentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Trigonometric and exponential functions of trigonometric functions . . . . . . .
Combinations involving trigonometric functions, exponentials, and powers . . .
Combinations of trigonometric and hyperbolic functions . . . . . . . . . . . . .
Combinations involving trigonometric and hyperbolic functions and powers . . .
Combinations of trigonometric and hyperbolic functions and exponentials . . . .
254
254
313
315
322
334
334
336
340
344
346
353
363
364
371
371
375
382
386
390
390
395
397
401
405
408
411
415
423
434
436
447
459
472
475
485
493
495
497
509
516
522
CONTENTS
4.14
4.2–4.4
4.21
4.22
4.23
4.24
4.25
4.26–4.27
4.28
4.29–4.32
4.33–4.34
4.35–4.36
4.37
4.38–4.41
4.42–4.43
4.44
4.5
4.51
4.52
4.53–4.54
4.55
4.56
4.57
4.58
4.59
4.6
4.60
4.61
4.62
4.63–4.64
5
ix
Combinations of trigonometric and hyperbolic functions, exponentials, and powers
Logarithmic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Logarithmic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Logarithms of more complicated arguments . . . . . . . . . . . . . . . . . . . .
Combinations of logarithms and rational functions . . . . . . . . . . . . . . . .
Combinations of logarithms and algebraic functions . . . . . . . . . . . . . . .
Combinations of logarithms and powers . . . . . . . . . . . . . . . . . . . . . .
Combinations involving powers of the logarithm and other powers . . . . . . . .
Combinations of rational functions of ln x and powers . . . . . . . . . . . . . .
Combinations of logarithmic functions of more complicated arguments and powers
Combinations of logarithms and exponentials . . . . . . . . . . . . . . . . . . .
Combinations of logarithms, exponentials, and powers . . . . . . . . . . . . . .
Combinations of logarithms and hyperbolic functions . . . . . . . . . . . . . . .
Logarithms and trigonometric functions . . . . . . . . . . . . . . . . . . . . . .
Combinations of logarithms, trigonometric functions, and powers . . . . . . . .
Combinations of logarithms, trigonometric functions, and exponentials . . . . .
Inverse Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . .
Inverse trigonometric functions . . . . . . . . . . . . . . . . . . . . . . . . . .
Combinations of arcsines, arccosines, and powers . . . . . . . . . . . . . . . . .
Combinations of arctangents, arccotangents, and powers . . . . . . . . . . . . .
Combinations of inverse trigonometric functions and exponentials . . . . . . . .
A combination of the arctangent and a hyperbolic function . . . . . . . . . . .
Combinations of inverse and direct trigonometric functions . . . . . . . . . . .
A combination involving an inverse and a direct trigonometric function and a
power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Combinations of inverse trigonometric functions and logarithms . . . . . . . . .
Multiple Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Change of variables in multiple integrals . . . . . . . . . . . . . . . . . . . . .
Change of the order of integration and change of variables . . . . . . . . . . .
Double and triple integrals with constant limits . . . . . . . . . . . . . . . . . .
Multiple integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Indeﬁnite Integrals of Special Functions
Elliptic Integrals and Functions . . . . . . . . . . . . . . . .
Complete elliptic integrals . . . . . . . . . . . . . . . . . . .
5.11
5.12
Elliptic integrals . . . . . . . . . . . . . . . . . . . . . . . .
5.13
Jacobian elliptic functions . . . . . . . . . . . . . . . . . . .
5.14
Weierstrass elliptic functions . . . . . . . . . . . . . . . . . .
5.2
The Exponential Integral Function . . . . . . . . . . . . . .
5.21
The exponential integral function . . . . . . . . . . . . . . .
5.22
Combinations of the exponential integral function and powers
5.23
Combinations of the exponential integral and the exponential
5.3
The Sine Integral and the Cosine Integral . . . . . . . . . . .
5.4
The Probability Integral and Fresnel Integrals . . . . . . . . .
5.5
Bessel Functions . . . . . . . . . . . . . . . . . . . . . . . .
5.1
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525
527
527
529
535
538
540
542
553
555
571
573
578
581
594
599
599
599
600
601
605
605
605
607
607
607
607
608
610
612
619
619
619
621
623
626
627
627
627
628
628
629
629
x
CONTENTS
6–7 Deﬁnite Integrals of Special Functions
6.1
Elliptic Integrals and Functions . . . . . . . . . . . . . . . . . . . . . . . . .
6.11
Forms containing F (x, k) . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.12
Forms containing E (x, k) . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.13
Integration of elliptic integrals with respect to the modulus . . . . . . . . . .
6.14–6.15
Complete elliptic integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.16
The theta function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.17
Generalized elliptic integrals . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2–6.3
The Exponential Integral Function and Functions Generated by It . . . . . . .
6.21
The logarithm integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.22–6.23
The exponential integral function . . . . . . . . . . . . . . . . . . . . . . . .
6.24–6.26
The sine integral and cosine integral functions . . . . . . . . . . . . . . . . .
6.27
The hyperbolic sine integral and hyperbolic cosine integral functions . . . . .
6.28–6.31
The probability integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.32
Fresnel integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.4
The Gamma Function and Functions Generated by It . . . . . . . . . . . . .
6.41
The gamma function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.42
Combinations of the gamma function, the exponential, and powers . . . . . .
6.43
Combinations of the gamma function and trigonometric functions . . . . . . .
6.44
The logarithm of the gamma function∗ . . . . . . . . . . . . . . . . . . . . .
6.45
The incomplete gamma function . . . . . . . . . . . . . . . . . . . . . . . .
6.46–6.47
The function ψ(x) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.5–6.7
Bessel Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.51
Bessel functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.52
Bessel functions combined with x and x2 . . . . . . . . . . . . . . . . . . . .
6.53–6.54
Combinations of Bessel functions and rational functions . . . . . . . . . . . .
6.55
Combinations of Bessel functions and algebraic functions . . . . . . . . . . .
6.56–6.58
Combinations of Bessel functions and powers . . . . . . . . . . . . . . . . . .
6.59
Combinations of powers and Bessel functions of more complicated arguments
6.61
Combinations of Bessel functions and exponentials . . . . . . . . . . . . . . .
6.62–6.63
Combinations of Bessel functions, exponentials, and powers . . . . . . . . . .
6.64
Combinations of Bessel functions of more complicated arguments, exponentials,
and powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.65
Combinations of Bessel and exponential functions of more complicated arguments and powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.66
Combinations of Bessel, hyperbolic, and exponential functions . . . . . . . . .
6.67–6.68
Combinations of Bessel and trigonometric functions . . . . . . . . . . . . . .
6.69–6.74
Combinations of Bessel and trigonometric functions and powers . . . . . . . .
6.75
Combinations of Bessel, trigonometric, and exponential functions and powers
6.76
Combinations of Bessel, trigonometric, and hyperbolic functions . . . . . . .
6.77
Combinations of Bessel functions and the logarithm, or arctangent . . . . . .
6.78
Combinations of Bessel and other special functions . . . . . . . . . . . . . . .
6.79
Integration of Bessel functions with respect to the order . . . . . . . . . . . .
6.8
Functions Generated by Bessel Functions . . . . . . . . . . . . . . . . . . . .
6.81
Struve functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.82
Combinations of Struve functions, exponentials, and powers . . . . . . . . . .
6.83
Combinations of Struve and trigonometric functions . . . . . . . . . . . . . .
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631
631
631
632
632
632
633
635
636
636
638
639
644
645
649
650
650
652
655
656
657
658
659
659
664
670
674
675
689
694
699
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708
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711
713
717
727
742
747
747
748
749
753
753
754
755
CONTENTS
6.84–6.85
6.86
6.87
6.9
6.91
6.92
6.93
6.94
7.1–7.2
7.11
7.12–7.13
7.14
7.15
7.16
7.17
7.18
7.19
7.21
7.22
7.23
7.24
7.25
7.3–7.4
7.31
7.32
7.325∗
7.33
7.34
7.35
7.36
7.37–7.38
7.39
7.41–7.42
7.5
7.51
7.52
7.53
7.54
7.6
7.61
7.62–7.63
7.64
7.65
xi
Combinations of Struve and Bessel functions . . . . . . . . . . . . . . . . . . .
Lommel functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Thomson functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Mathieu Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Mathieu functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Combinations of Mathieu, hyperbolic, and trigonometric functions . . . . . . .
Combinations of Mathieu and Bessel functions . . . . . . . . . . . . . . . . . .
Relationships between eigenfunctions of the Helmholtz equation in diﬀerent
coordinate systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Associated Legendre Functions . . . . . . . . . . . . . . . . . . . . . . . . . .
Associated Legendre functions . . . . . . . . . . . . . . . . . . . . . . . . . . .
Combinations of associated Legendre functions and powers . . . . . . . . . . .
Combinations of associated Legendre functions, exponentials, and powers . . .
Combinations of associated Legendre and hyperbolic functions . . . . . . . . .
Combinations of associated Legendre functions, powers, and trigonometric
functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A combination of an associated Legendre function and the probability integral .
Combinations of associated Legendre and Bessel functions . . . . . . . . . . . .
Combinations of associated Legendre functions and functions generated by
Bessel functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Integration of associated Legendre functions with respect to the order . . . . .
Combinations of Legendre polynomials, rational functions, and algebraic functions
Combinations of Legendre polynomials and powers . . . . . . . . . . . . . . . .
Combinations of Legendre polynomials and other elementary functions . . . . .
Combinations of Legendre polynomials and Bessel functions . . . . . . . . . . .
Orthogonal Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Combinations of Gegenbauer polynomials Cnν (x) and powers . . . . . . . . . .
Combinations of Gegenbauer polynomials Cnν (x) and elementary functions . . .
Complete System of Orthogonal Step Functions . . . . . . . . . . . . . . . . .
Combinations of the polynomials C νn (x) and Bessel functions; Integration of
Gegenbauer functions with respect to the index . . . . . . . . . . . . . . . . .
Combinations of Chebyshev polynomials and powers . . . . . . . . . . . . . . .
Combinations of Chebyshev polynomials and elementary functions . . . . . . .
Combinations of Chebyshev polynomials and Bessel functions . . . . . . . . . .
Hermite polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Jacobi polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Laguerre polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Hypergeometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Combinations of hypergeometric functions and powers . . . . . . . . . . . . . .
Combinations of hypergeometric functions and exponentials . . . . . . . . . . .
Hypergeometric and trigonometric functions . . . . . . . . . . . . . . . . . . .
Combinations of hypergeometric and Bessel functions . . . . . . . . . . . . . .
Conﬂuent Hypergeometric Functions . . . . . . . . . . . . . . . . . . . . . . .
Combinations of conﬂuent hypergeometric functions and powers . . . . . . . .
Combinations of conﬂuent hypergeometric functions and exponentials . . . . .
Combinations of conﬂuent hypergeometric and trigonometric functions . . . . .
Combinations of conﬂuent hypergeometric functions and Bessel functions . . .
756
760
761
763
763
763
767
767
769
769
770
776
778
779
781
782
787
788
789
791
792
794
795
795
797
798
798
800
802
803
803
806
808
812
812
814
817
817
820
820
822
829
830
xii
CONTENTS
7.66
7.67
7.68
7.69
7.7
7.71
7.72
7.73
7.74
7.75
7.76
7.77
7.8
7.81
7.82
7.83
Combinations of conﬂuent hypergeometric functions, Bessel functions, and powers
Combinations of conﬂuent hypergeometric functions, Bessel functions, exponentials, and powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Combinations of conﬂuent hypergeometric functions and other special functions
Integration of conﬂuent hypergeometric functions with respect to the index . .
Parabolic Cylinder Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Parabolic cylinder functions . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Combinations of parabolic cylinder functions, powers, and exponentials . . . . .
Combinations of parabolic cylinder and hyperbolic functions . . . . . . . . . . .
Combinations of parabolic cylinder and trigonometric functions . . . . . . . . .
Combinations of parabolic cylinder and Bessel functions . . . . . . . . . . . . .
Combinations of parabolic cylinder functions and conﬂuent hypergeometric
functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Integration of a parabolic cylinder function with respect to the index . . . . . .
Meijer’s and MacRobert’s Functions (G and E) . . . . . . . . . . . . . . . . .
Combinations of the functions G and E and the elementary functions . . . . .
Combinations of the functions G and E and Bessel functions . . . . . . . . . .
Combinations of the functions G and E and other special functions . . . . . . .
8–9 Special Functions
8.1
Elliptic Integrals and Functions . . . . . . . . . . . . . . . . . . . . . . . . . .
8.11
Elliptic integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.12
Functional relations between elliptic integrals . . . . . . . . . . . . . . . . . . .
8.13
Elliptic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.14
Jacobian elliptic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.15
Properties of Jacobian elliptic functions and functional relationships between them
8.16
The Weierstrass function ℘(u) . . . . . . . . . . . . . . . . . . . . . . . . . .
8.17
The functions ζ(u) and σ(u) . . . . . . . . . . . . . . . . . . . . . . . . . . .
Theta functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.18–8.19
8.2
The Exponential Integral Function and Functions Generated by It . . . . . . . .
8.21
The exponential integral function Ei(x) . . . . . . . . . . . . . . . . . . . . . .
8.22
The hyperbolic sine integral shi x and the hyperbolic cosine integral chi x . . .
8.23
The sine integral and the cosine integral: si x and ci x . . . . . . . . . . . . . .
8.24
The logarithm integral li(x) . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.25
The probability integral Φ(x), the Fresnel integrals S (x) and C (x), the error
function erf(x), and the complementary error function erfc(x) . . . . . . . . .
8.26
Lobachevskiy’s function L(x) . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.3
Euler’s Integrals of the First and Second Kinds . . . . . . . . . . . . . . . . . .
8.31
The gamma function (Euler’s integral of the second kind): Γ(z) . . . . . . . .
8.32
Representation of the gamma function as series and products . . . . . . . . . .
8.33
Functional relations involving the gamma function . . . . . . . . . . . . . . . .
8.34
The logarithm of the gamma function . . . . . . . . . . . . . . . . . . . . . . .
8.35
The incomplete gamma function . . . . . . . . . . . . . . . . . . . . . . . . .
8.36
The psi function ψ(x) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.37
The function β(x) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.38
The beta function (Euler’s integral of the ﬁrst kind): B(x, y) . . . . . . . . . .
8.39
The incomplete beta function Bx (p, q) . . . . . . . . . . . . . . . . . . . . . .
8.4–8.5
Bessel Functions and Functions Associated with Them . . . . . . . . . . . . . .
831
834
839
841
841
841
842
843
844
845
849
849
850
850
854
856
859
859
859
863
865
866
870
873
876
877
883
883
886
886
887
887
891
892
892
894
895
898
899
902
906
908
910
910
CONTENTS
8.40
8.41
8.42
8.43
8.44
8.45
8.46
8.47–8.48
8.49
8.51–8.52
8.53
8.54
8.55
8.56
8.57
8.58
8.59
8.6
8.60
8.61
8.62
8.63
8.64
8.65
8.66
8.67
8.7–8.8
8.70
8.71
8.72
8.73–8.74
8.75
8.76
8.77
8.78
8.79
8.81
8.82–8.83
8.84
8.85
8.9
8.90
8.91
8.919
8.92
8.93
8.94
xiii
Deﬁnitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Integral representations of the functions Jν (z) and Nν (z) . . . . . . . .
(1)
(2)
Integral representations of the functions Hν (z) and Hν (z) . . . . . .
Integral representations of the functions Iν (z) and Kν (z) . . . . . . . .
Series representation . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Asymptotic expansions of Bessel functions . . . . . . . . . . . . . . . .
Bessel functions of order equal to an integer plus one-half . . . . . . . .
Functional relations . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Diﬀerential equations leading to Bessel functions . . . . . . . . . . . . .
Series of Bessel functions . . . . . . . . . . . . . . . . . . . . . . . . .
Expansion in products of Bessel functions . . . . . . . . . . . . . . . . .
The zeros of Bessel functions . . . . . . . . . . . . . . . . . . . . . . .
Struve functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Thomson functions and their generalizations . . . . . . . . . . . . . . .
Lommel functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Anger and Weber functions Jν (z) and Eν (z) . . . . . . . . . . . . . . .
Neumann’s and Schläﬂi’s polynomials: O n (z) and S n (z) . . . . . . . .
Mathieu Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Mathieu’s equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Periodic Mathieu functions . . . . . . . . . . . . . . . . . . . . . . . .
(2n)
(2n+1)
(2n+1)
(2n+2)
Recursion relations for the coeﬃcients A2r , A2r+1 , B2r+1 , B2r+2
Mathieu functions with a purely imaginary argument . . . . . . . . . . .
Non-periodic solutions of Mathieu’s equation . . . . . . . . . . . . . . .
Mathieu functions for negative q . . . . . . . . . . . . . . . . . . . . .
Representation of Mathieu functions as series of Bessel functions . . . .
The general theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Associated Legendre Functions . . . . . . . . . . . . . . . . . . . . . .
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Integral representations . . . . . . . . . . . . . . . . . . . . . . . . . .
Asymptotic series for large values of |ν| . . . . . . . . . . . . . . . . . .
Functional relations . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Special cases and particular values . . . . . . . . . . . . . . . . . . . .
Derivatives with respect to the order . . . . . . . . . . . . . . . . . . .
Series representation . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The zeros of associated Legendre functions . . . . . . . . . . . . . . . .
Series of associated Legendre functions . . . . . . . . . . . . . . . . . .
Associated Legendre functions with integer indices . . . . . . . . . . . .
Legendre functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Conical functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Toroidal functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Orthogonal Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . .
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Legendre polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . .
Series of products of Legendre and Chebyshev polynomials . . . . . . .
Series of Legendre polynomials . . . . . . . . . . . . . . . . . . . . . .
Gegenbauer polynomials Cnλ (t) . . . . . . . . . . . . . . . . . . . . . .
The Chebyshev polynomials Tn (x) and Un (x) . . . . . . . . . . . . . .
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910
912
914
916
918
920
924
926
931
933
940
941
942
944
945
948
949
950
950
951
951
952
953
953
954
957
958
958
960
962
964
968
969
970
972
972
974
975
980
981
982
982
983
988
988
990
993
xiv
CONTENTS
8.95
8.96
8.97
9.1
9.10
9.11
9.12
9.13
9.14
9.15
9.16
9.17
9.18
9.19
9.2
9.20
9.21
9.22–9.23
9.24–9.25
9.26
9.3
9.30
9.31
9.32
9.33
9.34
9.4
9.41
9.42
9.5
9.51
9.52
9.53
9.54
9.55
9.56
9.6
9.61
9.62
9.63
9.64
9.65
9.7
9.71
9.72
The Hermite polynomials H n (x) . . . . . . . . . . . . . . . . . . . . . . . . .
Jacobi’s polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The Laguerre polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Hypergeometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Deﬁnition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Integral representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Representation of elementary functions in terms of a hypergeometric functions .
Transformation formulas and the analytic continuation of functions deﬁned by
hypergeometric series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A generalized hypergeometric series . . . . . . . . . . . . . . . . . . . . . . . .
The hypergeometric diﬀerential equation . . . . . . . . . . . . . . . . . . . . .
Riemann’s diﬀerential equation . . . . . . . . . . . . . . . . . . . . . . . . . .
Representing the solutions to certain second-order diﬀerential equations using
a Riemann scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Hypergeometric functions of two variables . . . . . . . . . . . . . . . . . . . .
A hypergeometric function of several variables . . . . . . . . . . . . . . . . . .
Conﬂuent Hypergeometric Functions . . . . . . . . . . . . . . . . . . . . . . .
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The functions Φ(α, γ; z) and Ψ(α, γ; z) . . . . . . . . . . . . . . . . . . . . . .
The Whittaker functions Mλ,μ (z) and Wλ,μ (z) . . . . . . . . . . . . . . . . . .
Parabolic cylinder functions Dp (z) . . . . . . . . . . . . . . . . . . . . . . . .
Conﬂuent hypergeometric series of two variables . . . . . . . . . . . . . . . . .
Meijer’s G-Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Deﬁnition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Functional relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A diﬀerential equation for the G-function . . . . . . . . . . . . . . . . . . . . .
Series of G-functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Connections with other special functions . . . . . . . . . . . . . . . . . . . . .
MacRobert’s E-Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Representation by means of multiple integrals . . . . . . . . . . . . . . . . . .
Functional relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Riemann’s Zeta Functions ζ(z, q) and ζ(z), and the Functions Φ(z, s, v) and ξ(s)
Deﬁnition and integral representations . . . . . . . . . . . . . . . . . . . . . .
Representation as a series or as an inﬁnite product . . . . . . . . . . . . . . . .
Functional relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Singular points and zeros . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The Lerch function Φ(z, s, v) . . . . . . . . . . . . . . . . . . . . . . . . . . .
The function ξ (s) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Bernoulli Numbers and Polynomials, Euler Numbers . . . . . . . . . . . . . . .
Bernoulli numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Bernoulli polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Euler numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The functions ν(x), ν(x, α), μ(x, β), μ(x, β, α), and λ(x, y) . . . . . . . . . .
Euler polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Bernoulli numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Euler numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
996
998
1000
1005
1005
1005
1006
1008
1010
1010
1014
1017
1018
1022
1022
1022
1023
1024
1028
1031
1032
1032
1033
1034
1034
1034
1035
1035
1035
1036
1036
1037
1037
1038
1039
1040
1040
1040
1041
1043
1043
1044
1045
1045
1045
CONTENTS
9.73
9.74
xv
Euler’s and Catalan’s constants . . . . . . . . . . . . . . . . . . . . . . . . . .
Stirling numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10 Vector Field Theory
10.1–10.8
Vectors, Vector Operators, and Integral
10.11
Products of vectors . . . . . . . . . .
10.12
Properties of scalar product . . . . . .
10.13
Properties of vector product . . . . . .
10.14
Diﬀerentiation of vectors . . . . . . . .
10.21
Operators grad, div, and curl . . . . .
10.31
Properties of the operator ∇ . . . . .
10.41
Solenoidal ﬁelds . . . . . . . . . . . .
10.51–10.61 Orthogonal curvilinear coordinates . .
10.71–10.72 Vector integral theorems . . . . . . . .
10.81
Integral rate of change theorems . . .
1046
1046
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1049
1049
1049
1049
1049
1050
1050
1051
1052
1052
1055
1057
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1059
1059
1059
1060
1061
12 Integral Inequalities
12.11
Mean Value Theorems . . . . . . . . . . . . . . . . . . . .
12.111
First mean value theorem . . . . . . . . . . . . . . . . . .
12.112
Second mean value theorem . . . . . . . . . . . . . . . . .
12.113
First mean value theorem for inﬁnite integrals . . . . . . .
12.114
Second mean value theorem for inﬁnite integrals . . . . . .
12.21
Diﬀerentiation of Deﬁnite Integral Containing a Parameter
12.211
Diﬀerentiation when limits are ﬁnite . . . . . . . . . . . . .
12.212
Diﬀerentiation when a limit is inﬁnite . . . . . . . . . . . .
12.31
Integral Inequalities . . . . . . . . . . . . . . . . . . . . .
12.311
Cauchy-Schwarz-Buniakowsky inequality for integrals . . .
12.312
Hölder’s inequality for integrals . . . . . . . . . . . . . . .
12.313
Minkowski’s inequality for integrals . . . . . . . . . . . . .
12.314
Chebyshev’s inequality for integrals . . . . . . . . . . . . .
12.315
Young’s inequality for integrals . . . . . . . . . . . . . . .
12.316
Steﬀensen’s inequality for integrals . . . . . . . . . . . . .
12.317
Gram’s inequality for integrals . . . . . . . . . . . . . . . .
12.318
Ostrowski’s inequality for integrals . . . . . . . . . . . . .
12.41
Convexity and Jensen’s Inequality . . . . . . . . . . . . . .
12.411
Jensen’s inequality . . . . . . . . . . . . . . . . . . . . . .
12.412
Carleman’s inequality for integrals . . . . . . . . . . . . . .
12.51
Fourier Series and Related Inequalities . . . . . . . . . . .
12.511
Riemann-Lebesgue lemma . . . . . . . . . . . . . . . . . .
12.512
Dirichlet lemma . . . . . . . . . . . . . . . . . . . . . . .
12.513
Parseval’s theorem for trigonometric Fourier series . . . . .
12.514
Integral representation of the nth partial sum . . . . . . . .
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1063
1063
1063
1063
1063
1064
1064
1064
1064
1064
1064
1064
1065
1065
1065
1065
1065
1066
1066
1066
1066
1066
1067
1067
1067
1067
Theorems
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11 Algebraic Inequalities
11.1–11.3
General Algebraic Inequalities . . . . . . . . . .
11.11
Algebraic inequalities involving real numbers . .
11.21
Algebraic inequalities involving complex numbers
11.31
Inequalities for sets of complex numbers . . . .
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.
xvi
CONTENTS
12.515
12.516
12.517
Generalized Fourier series . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Bessel’s inequality for generalized Fourier series . . . . . . . . . . . . . . . . .
Parseval’s theorem for generalized Fourier series . . . . . . . . . . . . . . . . .
1067
1068
1068
13 Matrices and Related Results
13.11–13.12
Special Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.111
Diagonal matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.112
Identity matrix and null matrix . . . . . . . . . . . . . . . . . . . . . . .
13.113
Reducible and irreducible matrices . . . . . . . . . . . . . . . . . . . . . .
13.114
Equivalent matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.115
Transpose of a matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.116
Adjoint matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.117
Inverse matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.118
Trace of a matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.119
Symmetric matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.120
Skew-symmetric matrix . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.121
Triangular matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.122
Orthogonal matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.123
Hermitian transpose of a matrix . . . . . . . . . . . . . . . . . . . . . . .
13.124
Hermitian matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.125
Unitary matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.126
Eigenvalues and eigenvectors . . . . . . . . . . . . . . . . . . . . . . . .
13.127
Nilpotent matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.128
Idempotent matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.129
Positive deﬁnite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.130
Non-negative deﬁnite . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.131
Diagonally dominant . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.21
Quadratic Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.211
Sylvester’s law of inertia . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.212
Rank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.213
Signature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.214
Positive deﬁnite and semideﬁnite quadratic form . . . . . . . . . . . . . .
13.215
Basic theorems on quadratic forms . . . . . . . . . . . . . . . . . . . . .
13.31
Diﬀerentiation of Matrices . . . . . . . . . . . . . . . . . . . . . . . . . .
13.41
The Matrix Exponential . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.411
Basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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1069
1069
1069
1069
1069
1069
1069
1070
1070
1070
1070
1070
1070
1070
1070
1070
1071
1071
1071
1071
1071
1071
1071
1071
1072
1072
1072
1072
1072
1073
1074
1074
14 Determinants
14.11
14.12
14.13
14.14
14.15*
14.16
14.17
14.18
14.21
14.31
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1075
1075
1075
1075
1076
1076
1076
1077
1077
1077
1078
Expansion of Second- and Third-Order Determinants
Basic Properties . . . . . . . . . . . . . . . . . . . .
Minors and Cofactors of a Determinant . . . . . . . .
Principal Minors . . . . . . . . . . . . . . . . . . . .
Laplace Expansion of a Determinant . . . . . . . . .
Jacobi’s Theorem . . . . . . . . . . . . . . . . . . .
Hadamard’s Theorem . . . . . . . . . . . . . . . . .
Hadamard’s Inequality . . . . . . . . . . . . . . . . .
Cramer’s Rule . . . . . . . . . . . . . . . . . . . . .
Some Special Determinants . . . . . . . . . . . . . .
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CONTENTS
14.311
14.312
14.313
14.314
14.315
14.316
14.317
15 Norms
15.1–15.9
15.11
15.21
15.211
15.212
15.213
15.31
15.311
15.312
15.313
15.41
15.411
15.412
15.413
15.51
15.511
15.512
15.61
15.611
15.612
15.71
15.711
15.712
15.713
15.714
15.715
15.81–15.82
15.811
15.812
15.813
15.814
15.815
15.816
15.817
15.818
15.819
15.820
15.821
15.822
xvii
Vandermonde’s determinant (alternant) . . . . .
Circulants . . . . . . . . . . . . . . . . . . . . .
Jacobian determinant . . . . . . . . . . . . . .
Hessian determinants . . . . . . . . . . . . . .
Wronskian determinants . . . . . . . . . . . . .
Properties . . . . . . . . . . . . . . . . . . . .
Gram-Kowalewski theorem on linear dependence
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1078
1078
1078
1079
1079
1079
1080
Vector Norms . . . . . . . . . . . . . . . . . . . . . . . . . .
General Properties . . . . . . . . . . . . . . . . . . . . . . .
Principal Vector Norms . . . . . . . . . . . . . . . . . . . .
The norm ||x||1 . . . . . . . . . . . . . . . . . . . . . . . .
The norm ||x||2 (Euclidean or L2 norm) . . . . . . . . . . .
The norm ||x||∞ . . . . . . . . . . . . . . . . . . . . . . . .
Matrix Norms . . . . . . . . . . . . . . . . . . . . . . . . .
General properties . . . . . . . . . . . . . . . . . . . . . . .
Induced norms . . . . . . . . . . . . . . . . . . . . . . . . .
Natural norm of unit matrix . . . . . . . . . . . . . . . . . .
Principal Natural Norms . . . . . . . . . . . . . . . . . . . .
Maximum absolute column sum norm . . . . . . . . . . . . .
Spectral norm . . . . . . . . . . . . . . . . . . . . . . . . .
Maximum absolute row sum norm . . . . . . . . . . . . . . .
Spectral Radius of a Square Matrix . . . . . . . . . . . . . .
Inequalities concerning matrix norms and the spectral radius
Deductions from Gerschgorin’s theorem (see 15.814) . . . .
Inequalities Involving Eigenvalues of Matrices . . . . . . . . .
Cayley-Hamilton theorem . . . . . . . . . . . . . . . . . . .
Corollaries . . . . . . . . . . . . . . . . . . . . . . . . . . .
Inequalities for the Characteristic Polynomial . . . . . . . . .
Named and unnamed inequalities . . . . . . . . . . . . . . .
Parodi’s theorem . . . . . . . . . . . . . . . . . . . . . . . .
Corollary of Brauer’s theorem . . . . . . . . . . . . . . . . .
Ballieu’s theorem . . . . . . . . . . . . . . . . . . . . . . . .
Routh-Hurwitz theorem . . . . . . . . . . . . . . . . . . . .
Named Theorems on Eigenvalues . . . . . . . . . . . . . . .
Schur’s inequalities . . . . . . . . . . . . . . . . . . . . . . .
Sturmian separation theorem . . . . . . . . . . . . . . . . .
Poincare’s separation theorem . . . . . . . . . . . . . . . . .
Gerschgorin’s theorem . . . . . . . . . . . . . . . . . . . . .
Brauer’s theorem . . . . . . . . . . . . . . . . . . . . . . . .
Perron’s theorem . . . . . . . . . . . . . . . . . . . . . . . .
Frobenius theorem . . . . . . . . . . . . . . . . . . . . . . .
Perron–Frobenius theorem . . . . . . . . . . . . . . . . . . .
Wielandt’s theorem . . . . . . . . . . . . . . . . . . . . . .
Ostrowski’s theorem . . . . . . . . . . . . . . . . . . . . . .
First theorem due to Lyapunov . . . . . . . . . . . . . . . .
Second theorem due to Lyapunov . . . . . . . . . . . . . . .
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1081
1081
1081
1081
1081
1081
1081
1082
1082
1082
1082
1082
1082
1082
1083
1083
1083
1083
1084
1084
1084
1084
1085
1086
1086
1086
1086
1087
1087
1087
1087
1088
1088
1088
1088
1088
1088
1089
1089
1089
xviii
CONTENTS
15.823
15.91
15.911
15.912
Hermitian matrices and diophantine relations
rational angles due to Calogero and Perelomov
Variational Principles . . . . . . . . . . . . . .
Rayleigh quotient . . . . . . . . . . . . . . . .
Basic theorems . . . . . . . . . . . . . . . . .
involving
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circular functions
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of
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1089
1091
1091
1091
16 Ordinary Diﬀerential Equations
1093
16.1–16.9
Results Relating to the Solution of Ordinary Diﬀerential Equations . . . . . . . 1093
16.11
First-Order Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1093
16.111
Solution of a ﬁrst-order equation . . . . . . . . . . . . . . . . . . . . . . . . . 1093
16.112
Cauchy problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1093
16.113
Approximate solution to an equation . . . . . . . . . . . . . . . . . . . . . . . 1093
16.114
Lipschitz continuity of a function . . . . . . . . . . . . . . . . . . . . . . . . . 1094
16.21
Fundamental Inequalities and Related Results . . . . . . . . . . . . . . . . . . 1094
16.211
Gronwall’s lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1094
16.212
Comparison of approximate solutions of a diﬀerential equation . . . . . . . . . 1094
16.31
First-Order Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1094
16.311
Solution of a system of equations . . . . . . . . . . . . . . . . . . . . . . . . . 1094
16.312
Cauchy problem for a system . . . . . . . . . . . . . . . . . . . . . . . . . . . 1095
16.313
Approximate solution to a system . . . . . . . . . . . . . . . . . . . . . . . . . 1095
16.314
Lipschitz continuity of a vector . . . . . . . . . . . . . . . . . . . . . . . . . . 1095
16.315
Comparison of approximate solutions of a system . . . . . . . . . . . . . . . . 1096
16.316
First-order linear diﬀerential equation . . . . . . . . . . . . . . . . . . . . . . . 1096
16.317
Linear systems of diﬀerential equations . . . . . . . . . . . . . . . . . . . . . . 1096
16.41
Some Special Types of Elementary Diﬀerential Equations . . . . . . . . . . . . 1097
16.411
Variables separable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1097
16.412
Exact diﬀerential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1097
16.413
Conditions for an exact equation . . . . . . . . . . . . . . . . . . . . . . . . . 1097
16.414
Homogeneous diﬀerential equations . . . . . . . . . . . . . . . . . . . . . . . . 1097
16.51
Second-Order Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1098
16.511
Adjoint and self-adjoint equations . . . . . . . . . . . . . . . . . . . . . . . . . 1098
16.512
Abel’s identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1098
16.513
Lagrange identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1099
16.514
The Riccati equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1099
16.515
Solutions of the Riccati equation . . . . . . . . . . . . . . . . . . . . . . . . . 1099
Solution of a second-order linear diﬀerential equation . . . . . . . . . . . . . . 1100
16.516
16.61–16.62
Oscillation and Non-Oscillation Theorems for Second-Order Equations . . . . . 1100
16.611
First basic comparison theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 1100
16.622
Second basic comparison theorem . . . . . . . . . . . . . . . . . . . . . . . . . 1101
16.623
Interlacing of zeros . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1101
16.624
Sturm separation theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1101
16.625
Sturm comparison theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1101
16.626
Szegö’s comparison theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1101
16.627
Picone’s identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1102
16.628
Sturm-Picone theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1102
16.629
Oscillation on the half line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1102
16.71
Two Related Comparison Theorems . . . . . . . . . . . . . . . . . . . . . . . . 1103
16.711
Theorem 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1103
CONTENTS
16.712
16.81–16.82
16.811
16.822
16.823
16.91
16.911
16.912
16.913
16.914
16.92
16.921
16.922
16.923
16.924
16.93
xix
Theorem 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Non-Oscillatory Solutions . . . . . . . . . . . . . . . . . . . . .
Kneser’s non-oscillation theorem . . . . . . . . . . . . . . . . .
Comparison theorem for non-oscillation . . . . . . . . . . . . . .
Necessary and suﬃcient conditions for non-oscillation . . . . . .
Some Growth Estimates for Solutions of Second-Order Equations
Strictly increasing and decreasing solutions . . . . . . . . . . . .
General result on dominant and subdominant solutions . . . . .
Estimate of dominant solution . . . . . . . . . . . . . . . . . . .
A theorem due to Lyapunov . . . . . . . . . . . . . . . . . . . .
Boundedness Theorems . . . . . . . . . . . . . . . . . . . . . .
All solutions of the equation . . . . . . . . . . . . . . . . . . . .
If all solutions of the equation . . . . . . . . . . . . . . . . . . .
If a(x) → ∞ monotonically as x → ∞, then all solutions of . . .
Consider the equation . . . . . . . . . . . . . . . . . . . . . . .
Growth of maxima of |y| . . . . . . . . . . . . . . . . . . . . . .
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1103
1103
1103
1104
1104
1104
1104
1104
1105
1105
1106
1106
1106
1106
1106
1106
17 Fourier, Laplace, and Mellin Transforms
17.1–17.4
Integral Transforms . . . . . . . . . . . . . . . . . . . . . . . . .
17.11
Laplace transform . . . . . . . . . . . . . . . . . . . . . . . . . .
17.12
Basic properties of the Laplace transform . . . . . . . . . . . . . .
17.13
Table of Laplace transform pairs . . . . . . . . . . . . . . . . . .
17.21
Fourier transform . . . . . . . . . . . . . . . . . . . . . . . . . . .
17.22
Basic properties of the Fourier transform . . . . . . . . . . . . . .
17.23
Table of Fourier transform pairs . . . . . . . . . . . . . . . . . . .
17.24
Table of Fourier transform pairs for spherically symmetric functions
17.31
Fourier sine and cosine transforms . . . . . . . . . . . . . . . . . .
17.32
Basic properties of the Fourier sine and cosine transforms . . . . .
17.33
Table of Fourier sine transforms . . . . . . . . . . . . . . . . . . .
17.34
Table of Fourier cosine transforms . . . . . . . . . . . . . . . . . .
17.35
Relationships between transforms . . . . . . . . . . . . . . . . . .
17.41
Mellin transform . . . . . . . . . . . . . . . . . . . . . . . . . . .
17.42
Basic properties of the Mellin transform . . . . . . . . . . . . . .
17.43
Table of Mellin transforms . . . . . . . . . . . . . . . . . . . . . .
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1107
1107
1107
1107
1108
1117
1118
1118
1120
1121
1121
1122
1126
1129
1129
1130
1131
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18 The z-Transform
18.1–18.3
Deﬁnition, Bilateral, and
18.1
Deﬁnitions . . . . . . .
18.2
Bilateral z-transform . .
18.3
Unilateral z-transform .
References
Supplemental references
Index of Functions and Constants
General Index of Concepts
Unilateral z-Transforms
. . . . . . . . . . . . .
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.
.
1135
1135
1135
1136
1138
1141
1145
1151
1161
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Preface to the Seventh Edition
Since the publication in 2000 of the completely reset sixth edition of Gradshteyn and Ryzhik, users of the
reference work have continued to submit corrections, new results that extend the work, and suggestions
for changes that improve the presentation of existing entries. It is a matter of regret to us that the
structure of the book makes it impossible to acknowledge these individual contributions, so, as usual,
the names of the many new contributors have been added to the acknowledgment list at the front of the
book.
This seventh edition contains the corrections received since the publication of the sixth edition in
2000, together with a considerable amount of new material acquired from isolated sources. Following
our previous conventions, an amended entry has a superscript “11” added to its entry reference number,
where the equivalent superscript number for the sixth edition was “10.” Similarly, an asterisk on an
entry’s reference number indicates a new result. When, for technical reasons, an entry in a previous
edition has been removed, to preserve the continuity of numbering between the new and older editions
the subsequent entries have not been renumbered, so the numbering will jump.
We wish to express our gratitude to all who have been in contact with us with the object of improving
and extending the book, and we want to give special thanks to Dr. Victor H. Moll for his interest in
the book and for the many contributions he has made over an extended period of time. We also wish to
acknowledge the contributions made by Dr. Francis J. O’Brien Jr. of the Naval Station in Newport, in
particular for results involving integrands where exponentials are combined with algebraic functions.
Experience over many years has shown that each new edition of Gradshteyn and Ryzhik generates
a fresh supply of suggestions for new entries, and for the improvement of the presentation of existing
entries and errata. In view of this, we do not expect this new edition to be free from errors, so all
users of this reference work who identify errors, or who wish to propose new entries, are invited to
contact the authors, whose email addresses are listed below. Corrections will be posted on the web site
www.az-tec.com/gr/errata.
Alan Jeﬀrey
[email protected]
Daniel Zwillinger
[email protected]
xxi
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Acknowledgments
The publisher and editors would like to take this opportunity to express their gratitude to the following
users of the Table of Integrals, Series, and Products who, either directly or through errata published
in Mathematics of Computation, have generously contributed corrections and addenda to the original
printing.
Dr. A. Abbas
Dr. P. B. Abraham
Dr. Ari Abramson
Dr. Jose Adachi
Dr. R. J. Adler
Dr. N. Agmon
Dr. M. Ahmad
Dr. S. A. Ahmad
Dr. Luis Alvarez-Ruso
Dr. Maarten H P Ambaum
Dr. R. K. Amiet
Dr. L. U. Ancarani
Dr. M. Antoine
Dr. C. R. Appledorn
Dr. D. R. Appleton
Dr. Mitsuhiro Arikawa
Dr. P. Ashoshauvati
Dr. C. L. Axness
Dr. E. Badralexe
Dr. S. B. Bagchi
Dr. L. J. Baker
Dr. R. Ball
Dr. M. P. Barnett
Dr. Florian Baumann
Dr. Norman C. Beaulieu
Dr. Jerome Benoit
Mr. V. Bentley
Dr. Laurent Berger
Dr. M. van den Berg
Dr. N. F. Berk
Dr. C. A. Bertulani
Dr.
Dr.
Dr.
Dr.
Dr.
Dr.
Dr.
Dr.
Dr.
Dr.
Dr.
Dr.
Dr.
Dr.
Dr.
Dr.
Dr.
Dr.
Dr.
Dr.
Dr.
Dr.
Dr.
Dr.
Dr.
Dr.
Dr.
Dr.
Dr.
Dr.
Dr.
J. Betancort-Rijo
P. Bickerstaﬀ
Iwo Bialynicki-Birula
Chris Bidinosti
G. R. Bigg
Ian Bindloss
L. Blanchet
Mike Blaskiewicz
R. D. Blevins
Anders Blom
L. M. Blumberg
R. Blumel
S. E. Bodner
M. Bonsager
George Boros
S. Bosanac
B. Van den Bossche
A. Boström
J. E. Bowcock
T. H. Boyer
K. M. Briggs
D. J. Broadhurst
Chris Van Den Broeck
W. B. Brower
H. N. Browne
Christoph Bruegger
William J. Bruno
Vladimir Bubanja
D. J. Buch
D. J. Bukman
F. M. Burrows
xxiii
Dr. R. Caboz
Dr. T. Calloway
Dr. F. Calogero
Dr. D. Dal Cappello
Dr. David Cardon
Dr. J. A. Carlson Gallos
Dr. B. Carrascal
Dr. A. R. Carr
Dr. S. Carter
Dr. G. Cavalleri
Mr. W. H. L. Cawthorne
Dr. A. Cecchini
Dr. B. Chan
Dr. M. A. Chaudhry
Dr. Sabino Chavez-Cerda
Dr. Julian Cheng
Dr. H. W. Chew
Dr. D. Chin
Dr. Young-seek Chung
Dr. S. Ciccariello
Dr. N. S. Clarke
Dr. R. W. Cleary
Dr. A. Clement
Dr. P. Cochrane
Dr. D. K. Cohoon
Dr. L. Cole
Dr. Filippo Colomo
Dr. J. R. D. Copley
Dr. D. Cox
Dr. J. Cox
Dr. J. W. Criss
xxiv
Dr. A. E. Curzon
Dr. D. Dadyburjor
Dr. D. Dajaputra
Dr. C. Dal Cappello
Dr. P. Daly
Dr. S. Dasgupta
Dr. John Davies
Dr. C. L. Davis
Dr. A. Degasperis
Dr. B. C. Denardo
Dr. R. W. Dent
Dr. E. Deutsch
Dr. D. deVries
Dr. P. Dita
Dr. P. J. de Doelder
Dr. Mischa Dohler
Dr. G. Dôme
Dr. Shi-Hai Dong
Dr. Balazs Dora
Dr. M. R. D’Orsogna
Dr. Adrian A. Dragulescu
Dr. Eduardo Duenez
Mr. Tommi J. Dufva
Dr. E. B. Dussan, V
Dr. C. A. Ebner
Dr. M. van der Ende
Dr. Jonathan Engle
Dr. G. Eng
Dr. E. S. Erck
Dr. Jan Erkelens
Dr. Olivier Espinosa
Dr. G. A. Estévez
Dr. K. Evans
Dr. G. Evendon
Dr. V. I. Fabrikant
Dr. L. A. Falkovsky
Dr. K. Farahmand
Dr. Richard J. Fateman
Dr. G. Fedele
Dr. A. R. Ferchmin
Dr. P. Ferrant
Dr. H. E. Fettis
Dr. W. B. Fichter
Dr. George Fikioris
Mr. J. C. S. S. Filho
Dr. L. Ford
Dr. Nicolao Fornengo
Acknowledgments
Dr. J. France
Dr. B. Frank
Dr. S. Frasier
Dr. Stefan Fredenhagen
Dr. A. J. Freeman
Dr. A. Frink
Dr. Jason M. Gallaspy
Dr. J. A. C. Gallas
Dr. J. A. Carlson Gallas
Dr. G. R. Gamertsfelder
Dr. T. Garavaglia
Dr. Jaime Zaratiegui Garcia
Dr. C. G. Gardner
Dr. D. Garﬁnkle
Dr. P. N. Garner
Dr. F. Gasser
Dr. E. Gath
Dr. P. Gatt
Dr. D. Gay
Dr. M. P. Gelfand
Dr. M. R. Geller
Dr. Ali I. Genc
Dr. Vincent Genot
Dr. M. F. George
Dr. P. Germain
Dr. Ing. Christoph Gierull
Dr. S. P. Gill
Dr. Federico Girosi
Dr. E. A. Gislason
Dr. M. I. Glasser
Dr. P. A. Glendinning
Dr. L. I. Goldﬁscher
Dr. Denis Golosov
Dr. I. J. Good
Dr. J. Good
Mr. L. Gorin
Dr. Martin Götz
Dr. R. Govindaraj
Dr. M. De Grauf
Dr. L. Green
Mr. Leslie O. Green
Dr. R. Greenwell
Dr. K. D. Grimsley
Dr. Albert Groenenboom
Dr. V. Gudmundsson
Dr. J. Guillera
Dr. K. Gunn
Dr. D. L. Gunter
Dr. Julio C. Gutiérrez-Vega
Dr. Roger Haagmans
Dr. H. van Haeringen
Dr. B. Haﬁzi
Dr. Bahman Haﬁzi
Dr. T. Hagfors
Dr. M. J. Haggerty
Dr. Timo Hakulinen
Dr. Einar Halvorsen
Dr. S. E. Hammel
Dr. E. Hansen
Dr. Wes Harker
Dr. T. Harrett
Dr. D. O. Harris
Dr. Frank Harris
Mr. Mazen D. Hasna
Dr. Joel G. Heinrich
Dr. Sten Herlitz
Dr. Chris Herzog
Dr. A. Higuchi
Dr. R. E. Hise
Dr. Henrik Holm
Dr. Helmut Hölzler
Dr. N. Holte
Dr. R. W. Hopper
Dr. P. N. Houle
Dr. C. J. Howard
Dr. J. H. Hubbell
Dr. J. R. Hull
Dr. W. Humphries
Dr. Jean-Marc Huré
Dr. Ben Yu-Kuang Hu
Dr. Y. Iksbe
Dr. Philip Ingenhoven
Mr. L. Iossif
Dr. Sean A. Irvine
Dr. Óttar Ísberg
Dr. Cyril-Daniel Iskander
Dr. S. A. Jackson
Dr. John David Jackson
Dr. Francois Jaclot
Dr. B. Jacobs
Dr. E. C. James
Dr. B. Jancovici
Dr. D. J. Jeﬀrey
Dr. H. J. Jensen
Acknowledgments
Dr.
Dr.
Dr.
Dr.
Dr.
Dr.
Dr.
Dr.
Dr.
Dr.
Dr.
Dr.
Dr.
Dr.
Dr.
Dr.
Dr.
Dr.
Dr.
Dr.
Dr.
Dr.
Dr.
Dr.
Dr.
Dr.
Dr.
Dr.
Dr.
Dr.
Dr.
Dr.
Dr.
Dr.
Dr.
Dr.
Dr.
Dr.
Dr.
Dr.
Dr.
Dr.
Dr.
Dr.
Dr.
Dr.
Dr.
Edwin F. Johnson
I. R. Johnson
Steven Johnson
I. Johnstone
Y. P. Joshi
Jae-Hun Jung
Damir Juric
Florian Kaempfer
S. Kanmani
Z. Kapal
Dave Kasper
M. Kaufman
B. Kay
Avinash Khare
Ilki Kim
Youngsun Kim
S. Klama
L. Klingen
C. Knessl
M. J. Knight
Mel Knight
Yannis Kohninos
D. Koks
L. P. Kok
K. S. Kölbig
Y. Komninos
D. D. Konowalow
Z. Kopal
I. Kostyukov
R. A. Krajcik
Vincent Krakoviack
Stefan Kramer
Tobias Kramer
Hermann Krebs
J. W. Krozel
E. D. Krupnikov
Kun-Lin Kuo
E. A. Kuraev
Konstantinos Kyritsis
Velimir Labinac
A. D. J. Lambert
A. Lambert
A. Larraza
K. D. Lee
M. Howard Lee
M. K. Lee
P. A. Lee
xxv
Dr. Todd Lee
Dr. J. Legg
Dr. Armando Lemus
Dr. S. L. Levie
Dr. D. Levi
Dr. Michael Lexa
Dr. Kuo Kan Liang
Dr. B. Linet
Dr. M. A. Lisa
Dr. Donald Livesay
Dr. H. Li
Dr. Georg Lohoefer
Dr. I. M. Longman
Dr. D. Long
Dr. Sylvie Lorthois
Dr. Y. L. Luke
Dr. W. Lukosz
Dr. T. Lundgren
Dr. E. A. Luraev
Dr. R. Lynch
Dr. R. Mahurin
Dr. R. Mallier
Dr. G. A. Mamon
Dr. A. Mangiarotti
Dr. I. Manning
Dr. J. Marmur
Dr. A. Martin
Sr. Yuzo Maruyama
Dr. David J. Masiello
Dr. Richard Marthar
Dr. H. A. Mavromatis
Dr. M. Mazzoni
Dr. K. B. Ma
Dr. P. McCullagh
Dr. J. H. McDonnell
Dr. J. R. McGregor
Dr. Kim McInturﬀ
Dr. N. McKinney
Dr. David McA McKirdy
Dr. Rami Mehrem
Dr. W. N. Mei
Dr. Angelo Melino
Mr. José Ricardo Mendes
Dr. Andy Mennim
Dr. J. P. Meunier
Dr. Gerard P. Michon
Dr. D. F. R. Mildner
Dr. D. L. Miller
Dr. Steve Miller
Dr. P. C. D. Milly
Dr. S. P. Mitra
Dr. K. Miura
Dr. N. Mohankumar
Dr. M. Moll
Dr. Victor H. Moll
Dr. D. Monowalow
Mr. Tony Montagnese
Dr. Jim Morehead
Dr. J. Morice
Dr. W. Mueck
Dr. C. Muhlhausen
Dr. S. Mukherjee
Dr. R. R. Müller
Dr. Pablo Parmezani Munhoz
Dr. Paul Nanninga
Dr. A. Natarajan
Dr. Stefan Neumeier
Dr. C. T. Nguyen
Dr. A. C. Nicol
Dr. M. M. Nieto
Dr. P. Noerdlinger
Dr. A. N. Norris
Dr. K. H. Norwich
Dr. A. H. Nuttall
Dr. Frank O’Brien
Dr. R. P. O’Keeﬀe
Dr. A. Ojo
Dr. P. Olsson
Dr. M. Ortner
Dr. S. Ostlund
Dr. J. Overduin
Dr. J. Pachner
Dr. John D. Paden
Mr. Robert A. Padgug
Dr. D. Papadopoulos
Dr. F. J. Papp
Mr. Man Sik Park
Dr. Jong-Do Park
Dr. B. Patterson
Dr. R. F. Pawula
Dr. D. W. Peaceman
Dr. D. Pelat
Dr. L. Peliti
Dr. Y. P. Pellegrini
xxvi
Dr.
Dr.
Dr.
Dr.
Dr.
Dr.
Dr.
Dr.
Dr.
Dr.
Dr.
Dr.
Dr.
Dr.
Dr.
Dr.
Dr.
Dr.
Dr.
Dr.
Dr.
Dr.
Dr.
Dr.
Dr.
Dr.
Dr.
Dr.
Dr.
Dr.
Dr.
Dr.
Dr.
Dr.
Dr.
Dr.
Dr.
Dr.
Dr.
Dr.
Dr.
Dr.
Dr.
Dr.
Dr.
Dr.
Dr.
Acknowledgments
G. J. Pert
Nicola Pessina
J. B. Peterson
Rickard Petersson
Andrew Plumb
Dror Porat
E. A. Power
E. Predazzi
William S. Price
Paul Radmore
F. Raynal
X. R. Resende
J. M. Riedler
Thomas Richard
E. Ringel
T. M. Roberts
N. I. Robinson
P. A. Robinson
D. M. Rosenblum
R. A. Rosthal
J. R. Roth
Klaus Rottbrand
D. Roy
E. Royer
D. Rudermann
Sanjib Sabhapandit
C. T. Sachradja
J. Sadiku
A. Sadiq
Motohiko Saitoh
Naoki Saito
A. Salim
J. H. Samson
Miguel A. Sanchis-Lozano
J. A. Sanders
M. A. F. Sanjun
P. Sarquiz
Avadh Saxena
Vito Scarola
O. Schärpf
A. Scherzinger
B. Schizer
Martin Schmid
J. Scholes
Mel Schopper
H. J. Schulz
G. J. Sears
Dr. Kazuhiko Seki
Dr. B. Seshadri
Dr. A. Shapiro
Dr. Masaki Shigemori
Dr. J. S. Sheng
Dr. Kenneth Ing Shing
Dr. Tomohiro Shirai
Dr. S. Shlomo
Dr. D. Siegel
Dr. Matthew Stapleton
Dr. Steven H. Simon
Dr. Ashok Kumar Singal
Dr. C. Smith
Dr. G. C. C. Smith
Dr. Stefan Llewellyn Smith
Dr. S. Smith
Dr. G. Solt
Dr. J. Sondow
Dr. A. Sørenssen
Dr. Marcus Spradlin
Dr. Andrzej Staruszkiewicz
Dr. Philip C. L. Stephenson
Dr. Edgardo Stockmeyer
Dr. J. C. Straton
Mr. H. Suraweera
Dr. N. F. Svaiter
Dr. V. Svaiter
Dr. R. Szmytkowski
Dr. S. Tabachnik
Dr. Erik Talvila
Dr. G. Tanaka
Dr. C. Tanguy
Dr. G. K. Tannahill
Dr. B. T. Tan
Dr. C. Tavard
Dr. Gonçalo Tavares
Dr. Aba Teleki
Dr. Arash Dahi Taleghani
Dr. D. Temperley
Dr. A. J. Tervoort
Dr. Theodoros Theodoulidis
Dr. D. J. Thomas
Dr. Michael Thorwart
Dr. S. T. Thynell
Dr. D. C. Torney
Dr. R. Tough
Dr. B. F. Treadway
Dr. Ming Tsai
Dr. N. Turkkan
Dr. Sandeep Tyagi
Dr. J. J. Tyson
Dr. S. Uehara
Dr. M. Vadacchino
Dr. O. T. Valls
Dr. D. Vandeth
Mr. Andras Vanyolos
Dr. D. Veitch
Mr. Jose Lopez Vicario
Dr. K. Vogel
Dr. J. M. M. J. Vogels
Dr. Alexis De Vos
Dr. Stuart Walsh
Dr. Reinhold Wannemacher
Dr. S. Wanzura
Dr. J. Ward
Dr. S. I. Warshaw
Dr. R. Weber
Dr. Wei Qian
Dr. D. H. Werner
Dr. E. Wetzel
Dr. Robert Whittaker
Dr. D. T. Wilton
Dr. C. Wiuf
Dr. K. T. Wong
Mr. J. N. Wright
Dr. J. D. Wright
Dr. D. Wright
Dr. D. Wu
Dr. Michel Daoud Yacoub
Dr. Yu S. Yakovlev
Dr. H.-C. Yang
Dr. J. J. Yang
Dr. Z. J. Yang
Dr. J. J. Wang
Dr. Peter Widerin
Mr. Chun Kin Au Yeung
Dr. Kazuya Yuasa
Dr. S. P. Yukon
Dr. B. Zhang
Dr. Y. C. Zhang
Dr. Y. Zhao
Dr. Ralf Zimmer
The Order of Presentation of the
Formulas
The question of the most expedient order in which to give the formulas, in particular, in what division
to include particular formulas such as the deﬁnite integrals, turned out to be quite complicated. The
thought naturally occurs to set up an order analogous to that of a dictionary. However, it is almost
impossible to create such a system for the formulas of integral calculus. Indeed, in an arbitrary formula
of the form
b
f (x) dx = A
a
one may make a large number of substitutions of the form x = ϕ(t) and thus obtain a number of
“synonyms” of the given formula. We must point out that the table of deﬁnite integrals by Bierens
de Haan and the earlier editions of the present reference both sin in the plethora of such “synonyms”
and formulas of complicated form. In the present edition, we have tried to keep only the simplest of the
“synonym” formulas. Basically, we judged the simplicity of a formula from the standpoint of the simplicity
of the arguments of the “outer” functions that appear in the integrand. Where possible, we have replaced
a complicated formula with a simpler one. Sometimes, several complicated formulas were thereby reduced
to a single, simpler one. We then kept only the simplest formula. As a result of such substitutions, we
sometimes obtained an integral that could be evaluated by use of the formulas of Chapter Two and the
Newton–Leibniz formula, or to an integral of the form
a
f (x) dx,
−a
where f (x) is an odd function. In such cases, the complicated integrals have been omitted.
Let us give an example using the expression
π/4
0
p−1
(cot x − 1)
sin2 x
ln tan x dx = −
π
cosec pπ.
p
By making the natural substitution u = cot x − 1, we obtain
∞
π
up−1 ln(1 + u) du = cosec pπ.
p
0
Integrals similar to formula (0.1) are omitted in this new edition. Instead, we have formula (0.2).
xxvii
(0.1)
(0.2)
xxviii
The Order of Presentation of the Formulas
As a second example, let us take
π/2
ln (tanp x + cotp x) ln tan x dx = 0.
I=
0
The substitution u = tan x yields
∞
I=
0
ln (up + u−p ) ln u
du.
1 + u2
If we now set υ = ln u, we obtain
∞
∞
pυ
υeυ
ln (2 cosh pυ)
−pυ
dυ.
dυ
=
ln
e
+
e
υ
I=
2υ
2 cosh υ
−∞ 1 + e
−∞
The integrand is odd, and, consequently, the integral is equal to 0.
Thus, before looking for an integral in the tables, the user should simplify as much as possible the
arguments (the “inner” functions) of the functions in the integrand.
The functions are ordered as follows: First we have the elementary functions:
1.
2.
3.
4.
5.
6.
7.
The function f (x) = x.
The exponential function.
The hyperbolic functions.
The trigonometric functions.
The logarithmic function.
The inverse hyperbolic functions. (These are replaced with the corresponding logarithms in the
formulas containing deﬁnite integrals.)
The inverse trigonometric functions.
Then follow the special functions:
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
Elliptic integrals.
Elliptic functions.
The logarithm integral, the exponential integral, the sine integral, and the cosine integral functions.
Probability integrals and Fresnel’s integrals.
The gamma function and related functions.
Bessel functions.
Mathieu functions.
Legendre functions.
Orthogonal polynomials.
Hypergeometric functions.
Degenerate hypergeometric functions.
Parabolic cylinder functions.
Meijer’s and MacRobert’s functions.
Riemann’s zeta function.
The integrals are arranged in order of outer function according to the above scheme: the farther down
in the list a function occurs, (i.e., the more complex it is) the later will the corresponding formula appear
The Order of Presentation of the Formulas
xxix
in the tables. Suppose that several expressions have the same outer function. For example, consider
sin ex , sin x, sin ln x. Here, the outer function is the sine function in all three cases. Such expressions are
then arranged in order of the inner function. In the present work, these functions are therefore arranged
in the following order: sin x, sin ex , sin ln x.
Our list does not include polynomials, rational functions, powers, or other algebraic functions. An
algebraic function that is included in tables of deﬁnite integrals can usually be reduced to a ﬁnite combination of roots of rational power. Therefore, for classifying our formulas, we can conditionally treat a
power function as a generalization of an algebraic and, consequently, of a rational function.∗ We shall
distinguish between all these functions and those listed above, and we shall treat them as operators.
Thus, in the expression sin2 ex , we shall think of the squaring operator as applied to the outer function,
sin x+cos x
namely, the sine. In the expression sin
x−cos x , we shall think of the rational operator as applied to the
trigonometric functions sine and cosine. We shall arrange the operators according to the following order:
1.
2.
3.
4.
Polynomials (listed in order of their degree).
Rational operators.
Algebraic operators (expressions of the form Ap/q , where q and p are rational, and q > 0; these are
listed according to the size of q).
Power operators.
Expressions with the same outer and inner functions are arranged in the order of complexity of the
operators. For example, the following functions [whose outer functions are all trigonometric, and whose
inner functions are all f (x) = x] are arranged in the order shown:
sin x,
sin x cos x,
1
= cosec x,
sin x
sin x
= tan x,
cos x
sin x + cos x
,
sin x − cos x
sinm x,
sinm x cos x.
Furthermore, if two outer functions ϕ1 (x) and ϕ2 (x), where ϕ1 (x) is more complex than ϕ2 (x), appear
in an integrand and if any of the operations mentioned are performed on them, the corresponding integral
will appear [in the order determined by the position of ϕ2 (x) in the list] after all integrals containing
only the function ϕ1 (x). Thus, following the trigonometric functions are the trigonometric and power
functions [that is, ϕ2 (x) = x]. Then come
• combinations of trigonometric and exponential functions,
• combinations of trigonometric functions, exponential functions, and powers, etc.,
• combinations of trigonometric and hyperbolic functions, etc.
Integrals containing two functions ϕ1 (x) and ϕ2 (x) are located in the division and order corresponding
to the more complicated function of the two. However, if the positions of several integrals coincide
because they contain the same complicated function, these integrals are put in the position deﬁned by
the complexity of the second function.
To these rules of a general nature, we need to add certain particular considerations that will be easily
1
understood from the tables. For example, according to the above remarks, the function e x comes after
1
1
ex as regards complexity, but ln x and ln are equally complex since ln = − ln x. In the section on
x
x
“powers and algebraic functions,” polynomials, rational functions, and powers of powers are formed from
power functions of the form (a + bx)n and (α + βx)ν .
∗ For any natural number n, the involution (a + bx)n of the binomial a + bx is a polynomial. If n is a negative integer,
(a + bx)n is a rational function. If n is irrational, the function (a + bx)n is not even an algebraic function.
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Use of the Tables∗
For the eﬀective use of the tables contained in this book, it is necessary that the user should ﬁrst become
familiar with the classiﬁcation system for integrals devised by the authors Ryzhik and Gradshteyn. This
classiﬁcation is described in detail in the section entitled The Order of Presentation of the Formulas (see
page xxvii) and essentially involves the separation of the integrand into inner and outer functions. The
principal function involved in the integrand is called the outer function, and its argument, which is itself
usually another function, is called the inner function. Thus, if the integrand comprised the expression
ln sin x, the outer function would be the logarithmic function while its argument, the inner function,
would be the trigonometric function sin x. The desired integral would then be found in the section
dealing with logarithmic functions, its position within that section being determined by the position of
the inner function (here a trigonometric function) in Gradshteyn and Ryzhik’s list of functional forms.
It is inevitable that some duplication of symbols will occur within such a large collection of integrals,
and this happens most frequently in the ﬁrst part of the book dealing with algebraic and trigonometric
integrands. The symbols most frequently involved are α, β, γ, δ, t, u, z, zk , and Δ. The expressions
associated with these symbols are used consistently within each section and are deﬁned at the start of
each new section in which they occur. Consequently, reference should be made to the beginning of the
section being used in order to verify the meaning of the substitutions involved.
Integrals of algebraic functions are expressed as combinations of roots with rational power indices,
and deﬁnite integrals of such functions are frequently expressed in terms of the Legendre elliptic integrals
F (φ, k), E (φ, k) and Π(φ, n, k), respectively, of the ﬁrst, second, and third kinds.
The four inverse hyperbolic functions arcsinh z, arccosh z, arctanh z, and arccoth z are introduced
through the deﬁnitions
1
arcsinh(iz)
i
1
arccos z = arccosh(z)
i
1
arctan z = arctanh(iz)
i
arccot z = i arccoth(iz)
arcsin z =
∗ Prepared
by Alan Jeﬀrey for the English language edition.
xxxi
xxxii
Use of the Tables
or
1
arcsin(iz)
i
arccosh z = i arccos z
1
arctanh z = arctan(iz)
i
1
arccoth z = arccot(−iz)
i
arcsinh z =
The numerical constants C and G which often appear in the deﬁnite integrals denote Euler’s constant
and Catalan’s constant, respectively. Euler’s constant C is deﬁned by the limit
s
1
− ln s = 0.577215 . . . .
C = lim
s→∞
m
m=1
On occasion, other writers denote Euler’s constant by the symbol γ, but this is also often used instead
to denote the constant
γ = eC = 1.781072 . . . .
Catalan’s constant G is related to the complete elliptic integral
π/2
dx
K ≡ K(k) ≡
0
1 − k 2 sin2 x
by the expression
∞
(−1)m
1 1
K dk =
= 0.915965 . . . .
G=
2 0
(2m + 1)2
m=0
Since the notations and deﬁnitions for higher transcendental functions that are used by diﬀerent
authors are by no means uniform, it is advisable to check the deﬁnitions of the functions that occur in
these tables. This can be done by identifying the required function by symbol and name in the Index of
Special Functions and Notation on page xxxix, and by then referring to the deﬁning formula or section
number listed there. We now present a brief discussion of some of the most commonly used alternative
notations and deﬁnitions for higher transcendental functions.
Bernoulli and Euler Polynomials and Numbers
Extensive use is made throughout the book of the Bernoulli and Euler numbers Bn and En that are
deﬁned in terms of the Bernoulli and Euler polynomials of order n, B n (x) and E n (x), respectively. These
polynomials are deﬁned by the generating functions
∞
tn
text
=
for |t| < 2π
B n (x)
t
e − 1 n=0
n!
and
∞
2ext
tn
=
for |t| < π.
E n (x)
t
e + 1 n=0
n!
The Bernoulli numbers are always denoted by Bn and are deﬁned by the relation
Bn = Bn (0)
for n = 0, 1, . . . ,
when
B0 = 1,
1
B1 = − ,
2
B2 =
1
,
6
B4 = −
1
,....
30
Use of the Tables
xxxiii
The Euler numbers En are deﬁned by setting
1
for n = 0, 1, . . .
2
The En are all integral, and E0 = 1, E2 = −1, E4 = 5, E6 = −61, . . . .
An alternative deﬁnition of Bernoulli numbers, which we shall denote by the symbol Bn∗ , uses the
same generating function but identiﬁes the Bn∗ diﬀerently in the following manner:
1
t2
t4
t
= 1 − t + B1∗ − B2∗ + . . . .
t
e −1
2
2!
4!
This deﬁnition then gives rise to the alternative set of Bernoulli numbers
E n = 2n E n
B1∗ = 1/6, B2∗ = 1/30, B3∗ = 1/42, B4∗ = 1/30, B5∗ = 5/66,
B6∗ = 691/2730, B7∗ = 7/6, B8∗ = 3617/510, . . . .
These diﬀerences in notation must also be taken into account when using the following relationships
that exist between the Bernoulli and Euler polynomials:
B n (x) =
n
1 n
Bn−k E k (2x)
k
2n
n = 0, 1, . . .
k=0
E n−1 (x) =
2n
n
E n−1 (x) =
x 2
B n (x) − 2n B n
n
2
or
and
Bn
x+1
2
− Bn
x
2
n = 1, 2, . . .
n−2
n −1 n n−k
2
− 1 Bn−k B n (x)
n = 2, 3, . . .
k
2
k=0
There are also alternative deﬁnitions of the Euler polynomial of order n, and it should be noted that
some authors, using a modiﬁcation of the third expression above, call
2
x B n (x) − 2n B n
n+1
2
the Euler polynomial of order n.
E n−2 (x) = 2
Elliptic Functions and Elliptic Integrals
The following notations are often used in connection with the inverse elliptic functions sn u, cn u, and
dn u:
1
sn u
sn u
sc u =
cn u
sn u
sd u =
dn u
ns u =
1
cn u
cn u
cs u =
sn u
cn u
cd u =
dn u
nc u =
1
dn u
dn u
ds u =
sn u
dn u
dc u =
cn u
nd u =
xxxiv
Use of the Tables
The elliptic integral of the third kind is deﬁned by Gradshteyn and Ryzhik to be
ϕ
da
2
Π ϕ, n , k =
2
2
0
1 − n sin a
1 − k 2 sin2 a
−∞ < n2 < ∞
sin ϕ
dx
=
2
2
(1 − n x ) (1 − x2 ) (1 − k 2 x2 )
0
The Jacobi Zeta Function and Theta Functions
The Jacobi zeta function zn(u, k), frequently written Z(u), is deﬁned by the relation
u
E
E
2
zn(u, k) = Z(u) =
dn υ −
dυ = E(u) − u.
K
K
0
This is related to the theta functions by the relationship
∂
zn(u, k) =
ln Θ(u)
∂u
giving
(i).
(ii).
(iii).
(iv).
πu
π ϑ1 2K
zn(u, k) =
2K ϑ πu
1
2K
πu
π ϑ2 2K
zn(u, k) =
2K ϑ πu
2
2K
πu
π ϑ3 2K
zn(u, k) =
2K ϑ πu
3
2K
πu
π ϑ4 2K
zn(u, k) =
2K ϑ πu
4
2K
−
cn u dn u
sn u
−
dn u sn u
cn u
− k2
sn u cn u
dn u
Many diﬀerent notations for the theta function are in current use. The most common variants are the
replacement of the argument u by the argument u/π and, occasionally, a permutation of the identiﬁcation
of the functions ϑ1 to ϑ4 with the function ϑ4 replaced by ϑ.
The Factorial (Gamma) Function
In older reference texts, the gamma function Γ(z), deﬁned by the Euler integral
∞
Γ(z) =
tz−1 e−t dt,
0
is sometimes expressed in the alternative notation
Γ(1 + z) = z! = Π(z).
On occasions, the related derivative of the logarithmic factorial function Ψ(z) is used where
(z!)
d (ln z!)
=
= Ψ(z).
dz
z!
Use of the Tables
xxxv
This function satisﬁes the recurrence relation
Ψ(z) = Ψ(z − 1) +
and is deﬁned by the series
Ψ(z) = −C +
∞
n=0
1
z−1
1
1
−
n+1 z+n
.
The derivative Ψ (z) satisﬁes the recurrence relation
Ψ (z + 1) = Ψ (z) −
and is deﬁned by the series
Ψ (z) =
1
z2
∞
1
.
(z
+
n)2
n=0
Exponential and Related Integrals
The exponential integrals E n (z) have been deﬁned by Schloemilch using the integral
∞
E n (z) =
e−zt t−n dt
(n = 0, 1, . . . , Re z > 0) .
1
They should not be confused with the Euler polynomials already mentioned. The function E 1 (z) is
related to the exponential integral Ei(z) through the expressions
∞
E 1 (z) = − Ei(−z) =
e−t t−1 dt
z
and
z
dt
= Ei (ln z)
[z > 1] .
ln
t
0
The functions E n (z) satisfy the recurrence relations
1 −z
e − z E n−1 (z)
E n (z) =
[n > 1]
n−1
and
E n (z) = − E n−1 (z)
with
li(z) =
E 0 (z) = e−z /z.
The function E n (z) has the asymptotic expansion
e−z
3π
n n(n + 1) n(n + 1)(n + 2)
−
+
·
·
·
|arg
z|
<
E n (z) ∼
1− +
z
z
z2
z3
2
while for large n,
n
e−x
n(n − 2x) n 6x2 − 8nx + n2
1+
+
+
+ R(n, x) ,
E n (x) =
x+n
(x + n)2
(x + n)4
(x + n)6
where
1
−0.36n−4 ≤ R(n, x) ≤ 1 +
[x > 0] .
n−4
x+n−1
The sine and cosine integrals si(x) and ci(x) are related to the functions Si(x) and Ci(x) by the
integrals
x
π
sin t
Si(x) =
dt = si(x) +
t
2
0
and
xxxvi
Use of the Tables
(cos t − 1)
dt.
t
0
The hyperbolic sine and cosine integrals shi(x) and chi(x) are deﬁned by the relations
x
sinh t
shi(x) =
dt
t
0
and
x
(cosh t − 1)
chi(x) = C + ln x +
dt.
t
0
Some authors write
x
(1 − cos t)
Cin(x) =
dt
t
0
so that
Cin(x) = − Ci(x) + ln x + C.
The error function erf(x) is deﬁned by the relation
x
2
2
erf(x) = Φ(x) = √
e−t dt,
π 0
and the complementary error function erfc(x) is related to the error function erfc(x) and to Φ(x) by the
expression
erfc(x) = 1 − erf(x).
The Fresnel integrals S (x) and C (x) are deﬁned by Gradshteyn and Ryzhik as
x
2
S (x) = √
sin t2 dt
2π 0
and
x
2
C (x) = √
cos t2 dt.
2π 0
Other deﬁnitions that are in use are
x
x
πt2
πt2
dt,
C 1 (x) =
dt,
S 1 (x) =
sin
cos
2
2
0
0
and
x
x
sin t
cos t
1
1
√ dt,
√ dt.
S 2 (x) = √
C 2 (x) = √
t
t
2π 0
2π 0
These are related by the expressions
2
= S 2 x2
S (x) = S 1 x
π
and
2
= C 2 x2
C (x) = C 1 x
π
x
Ci(x) = C + ln x +
Hermite and Chebyshev Orthogonal Polynomials
The Hermite polynomials H n (x) are related to the Hermite polynomials He n (x) by the relations
x
He n (x) = 2−n/2 H n √
2
and
√
H n (x) = 2n/2 He n x 2 .
Use of the Tables
xxxvii
These functions satisfy the diﬀerential equations
dHn
d2 H n
+ 2n H n = 0
− 2x
2
dx
dx
and
d2 He n
d He n
+ n He n = 0.
−x
dx2
dx
They obey the recurrence relations
H n+1 = 2x H n −2n H n−1
and
He n+1 = x He n −n He n−1 .
The ﬁrst six orthogonal polynomials He n are
He 0 = 1, He 1 = x, He 2 = x2 − 1, He 3 = x3 − 3x, He 4 = x4 − 6x2 + 3, He 5 = x5 − 10x3 + 15x.
Sometimes the Chebyshev polynomial U n (x) of the second kind is deﬁned as a solution of the equation
d2 y
dy
+ n(n + 2)y = 0.
1 − x2
− 3x
dx2
dx
Bessel Functions
A variety of diﬀerent notations for Bessel functions are in use. Some common ones involve the replacement
of Y n (z) by N n (z) and the introduction of the symbol
Λn (z) =
1
z
2
−n
Γ(n + 1) J n (z).
In the book by Gray, Mathews, and MacRobert, the symbol Y n (z) is used to denote
1
π Y n (z) +
2
(ln 2 − C) J n (z) while Neumann uses the symbol Y (n) (z) for the identical quantity.
(2)
The Hankel functions H (1)
ν (z) and H ν (z) are sometimes denoted by Hsν (z) and Hiν (z), and some
1
authors write Gν (z) =
πi H (1)
ν (z).
2
The Neumann polynomial O n (t) is a polynomial of degree n + 1 in 1/t, with O 0 (t) = 1/t. The
polynomials O n (t) are deﬁned by the generating function
∞
1
= J 0 (z) O 0 (t) + 2
J k (z) O k (t),
t−z
k=1
giving
O n (t) =
[n/2]
1 n(n − k − 1)!
4
k!
where 12 n signiﬁes the integral part of
polynomials:
k=0
1
2 n.
2
t
n−2k+1
for n = 1, 2, . . . ,
The following relationship holds between three successive
(n − 1) O n+1 (t) + (n + 1) O n−1 (t) −
2 n2 − 1
nπ
2n
O n (t) =
sin2
.
t
t
2
xxxviii
Use of the Tables
The Airy functions Ai(z) and Bi(z) are independent solutions of the equation
d2 u
− zu = 0.
dz 2
The solutions can be represented in terms of Bessel functions by the expressions
2 3/2
2 3/2
2 3/2
1√
1 z
Ai(z) =
z
z
K 1/3
z
z I −1/3
− I 1/3
=
3
3
3
π 3
3
2 3/2
2 3/2
1√
Ai(−z) =
z
z
z J 1/3
+ J −1/3
3
3
3
and by
Bi(z) =
Bi(−z) =
z
3
I −1/3
2 3/2
z
3
+ I 1/3
2 3/2
z
3
z
3
J −1/3
2 3/2
z
3
− J 1/3
2 3/2
z
3
,
.
Parabolic Cylinder Functions and Whittaker Functions
The diﬀerential equation
d2 y 2
+ az + bz + c y = 0
dz 2
has associated with it the two equations
1 2
1 2
d2 y
d2 y
z + a y = 0 and 2 −
z + a y = 0,
+
2
dz
4
dz
4
the solutions of which are parabolic cylinder functions. The ﬁrst equation can be derived from the second
by replacing z by zeiπ/4 and a by −ia.
The solutions of the equation
1 2
d2 y
z +a y =0
−
dz 2
4
are sometimes written U (a, z) and V (a, z). These solutions are related to Whittaker’s function D p (z) by
the expressions
U (a, z) = D −a− 12 (z)
and
1
1
D −a− 12 (−z) + (sin πa) D −a− 12 (z) .
V (a, z) = Γ
+a
π
2
Mathieu Functions
There are several accepted notations for Mathieu functions and for their associated parameters. The
deﬁning equation used by Gradshteyn and Ryzhik is
d2 y + a − 2k 2 cos 2z y = 0
with k 2 = q.
dz 2
Diﬀerent notations involve the replacement of a and q in this equation by h and θ, λ and h2 , and
√
b and c = 2 q, respectively. The periodic solutions sen (z, q) and cen (z, q) and the modiﬁed periodic
solutions Sen (z, q) and Cen (z, q) are suitably altered and, sometimes, re-normalized. A description of
these relationships together with the normalizing factors is contained in: Tables Relating to Mathieu
Functions. National Bureau of Standards, Columbia University Press, New York, 1951.
Index of Special Functions
Notation
β(x)
Γ(z)
γ(a, x), Γ(a, x)
Δ(n − k)
ξ(s)
λ(x, y)
μ(x, β), μ(x, β, α)
ν(x), ν(x, α)
Π(x)
Π(ϕ, n, k)
ζ(u)
ζ(z,
q),
ζ(z)
πu πu , Θ1 (u) = ϑ⎫
Θ(u) ⎧
= ϑ4 2K
3 2K
⎪
⎨ ϑ0 (υ | τ ) = ϑ4 (υ | τ ), ⎪
⎬
ϑ1 (υ | τ ), ϑ2 (υ | τ ),
⎪
⎪
⎩
⎭
ϑ3 (υ | τ )
σ(u)
Φ(x)
Φ(z, s, υ)
Φ(a,
c;
x)
= 1 F 1 (α; γ;
⎧
⎫ x)
Φ
(α,
β,
γ,
x,
y)
⎪
⎪
⎨ 1
⎬
Φ2 (β, β , γ, x, y)
⎪
⎪
⎩
⎭
Φ3 (β, γ, x, y)
ψ(x)
℘(u)
am(u, k)
Bn
B n (x)
B(x, y)
Bx (p, q)
bei(z), ber(z)
Name of the function and the number of
the formula containing its deﬁnition
Lobachevskiy’s angle of parallelism
Elliptic integral of the third kind
Weierstrass zeta function
Riemann’s zeta functions
Jacobian theta function
8.37
8.31–8.33
8.35
18.1
9.56
9.640
9.640
9.640
1.48
8.11
8.17
9.51–9.54
8.191–8.196
Elliptic theta functions
8.18, 8.19
Weierstrass sigma function
See probability integral
Lerch function
Conﬂuent hypergeometric function
8.17
8.25
9.55
9.21
Degenerate hypergeometric series in two
variables
9.26
Euler psi function
Weierstrass elliptic function
Amplitude (of an elliptic function)
Bernoulli numbers
Bernoulli polynomials
Beta functions
Incomplete beta functions
Thomson functions
8.36
8.16
8.141
9.61, 9.71
9.620
8.38
8.39
8.56
Gamma function
Incomplete gamma functions
Unit integer pulse function
continued on next page
xxxix
xl
Index of Special Functions
continued from previous page
Notation
C
C (x)
C ν (a)
C λn (t)
C λn (x)
ce2n (z, q),
ce2n+1 (z, q)
Ce2n (z, q),
Ce2n+1 (z, q)
chi(x)
ci(x)
cn(u)
D(k) ≡ D
D(ϕ, k)
D n (z), D p (z)
dn u
e1 , e2 , e3
En
E
(ϕ,
k) E(k) = E
E(k ) = E
E(p; ar : q; s : x)
Eν (z)
Ei(z)
erf(x)
erfc(x) = 1 − erf(x)
F (ϕ, k)
(α
,
.
.
.
, αp ; β1 , . . . , βq ; z)
F
p q
1
(α,
β;
γ;
z) = F (α, β; γ; z)
F
2 1
(α;
γ;
z) = Φ(α, γ; z)
F
1 1
FΛ (α :β1 , . . . , βn ;
γ1 , . . . . . . , γn : z1 , . . . , zn )
F1 , F2 , F3 , F4
fen (z, q), Fen (z, q) . . .
Feyn (z, q), Fekn (z, q) . . .
G
g2 , g3
gd x
gen (z, q), Gen (z, q)
Geyn (z, q), Gekn (z, q)
!
! ,... ,ap
x ! ab11,...
G m,n
p,q
,bq
Name of the function and the number of
the formula containing its deﬁnition
Euler constant
9.73, 8.367
Fresnel cosine integral
8.25
Young functions
3.76
Gegenbauer polynomials
8.93
Gegenbauer functions
8.932 1
Periodic Mathieu functions (Mathieu
8.61
functions of the ﬁrst kind)
Associated (modiﬁed) Mathieu functions of
8.63
the ﬁrst kind
Hyperbolic cosine integral function
8.22
Cosine integral
8.23
Cosine amplitude
8.14
Elliptic integral
8.112
Elliptic integral
8.111
Parabolic cylinder functions
9.24–9.25
Delta amplitude
8.14
(used with the Weierstrass function)
8.162
Euler numbers
9.63, 9.72
Elliptic integral of the second kind
8.11–8.12
Complete elliptic integral of the second
8.11-8.12
kind
MacRobert’s function
9.4
Weber function
8.58
Exponential integral function
8.21
Error function
8.25
Complementary error function
8.25
Elliptic integral of the ﬁrst kind
8.11–8.12
Generalized hypergeometric series
9.14
Gauss hypergeometric function
9.10–9.13
Degenerate hypergeometric function
9.21
Hypergeometric function of several
9.19
variables
Hypergeometric functions of two variables
9.18
Other nonperiodic solutions of Mathieu’s
8.64, 8.663
equation
Catalan constant
9.73
Invariants of the ℘(u)-function
8.161
Gudermannian
1.49
Other nonperiodic solutions of Mathieu’s
equation
8.64, 8.663
Meijer’s functions
9.3
continued on next page
Index of Special Functions
xli
continued from previous page
Notation
h(n)
heiν (z), herν (z)
H (1)
ν (z),
H (2)
ν (z)
πu H (u) = ϑ1 2K πu
H 1 (u) = ϑ2 2K
H n (z)
Hν (z)
I ν (z)
I x (p, q)
J ν (z)
Jν (z)
kν (x)
K(k) = K, K(k ) = K
K ν (z)
kei(z), ker(z)
L(x)
Lν (z)
Lα
n (z)
li(x)
M λ,μ (z)
O n (x)
P μν (z),
P μν (x)
P ν (z), P ν (x) ⎫
⎧
⎨a b c
⎬
P α β γ δ
⎩ ⎭
α β γ
P (α,β)
(x)
n
Q μν (z),
Q μν (x)
Q ν (z), Q ν (x)
S (x)
S n (x)
s μ,ν (z), S μ,ν (z)
se2n+1 (z, q), se2n+2 (z, q)
Se2n+1 (z, q),
Se2n+2 (z, q)
shi(x)
si(x)
sn u
T n (x)
U n (x)
Name of the function and the number of
the formula containing its deﬁnition
Unit integer function
18.1
Thomson functions
8.56
Hankel functions of the ﬁrst and second
8.405, 8.42
kinds
Theta function
8.192
Theta function
8.192
Hermite polynomials
8.95
Struve functions
8.55
Bessel functions of an imaginary argument
8.406, 8.43
Normalized incomplete beta function
8.39
Bessel function
8.402, 8.41
Anger function
8.58
Bateman’s function
9.210 3
Complete elliptic integral of the ﬁrst kind
8.11–8.12
Bessel functions of imaginary argument
8.407, 8.43
Thomson functions
8.56
Lobachevskiy’s function
8.26
Modiﬁed Struve function
8.55
Laguerre polynomials
8.97
Logarithm integral
8.24
Whittaker functions
9.22, 9.23
Neumann’s polynomials
8.59
Associated Legendre functions of the ﬁrst
8.7, 8.8
kind
Legendre functions and polynomials
8.82, 8.83, 8.91
Riemann’s diﬀerential equation
9.160
Jacobi’s polynomials
Associated Legendre functions of the
second kind
Legendre functions of the second kind
Fresnel sine integral
Schläﬂi’s polynomials
Lommel functions
Periodic Mathieu functions
Mathieu functions of an imaginary
argument
Hyperbolic sine integral
Sine integral
Sine amplitude
Chebyshev polynomial of the 1st kind
Chebyshev polynomials of the 2nd kind
8.96
8.7, 8.8
8.82, 8.83
8.25
8.59
8.57
8.61
8.63
8.22
8.23
8.14
8.94
8.94
continued on next page
xlii
Index of Special Functions
continued from previous page
Notation
U ν (w, z), V ν (w, z)
W λ,μ (z)
Y ν (z)
Z ν (z)
Zν (z)
Name of the function and the number of
the formula containing its deﬁnition
Lommel functions of two variables
8.578
Whittaker functions
9.22, 9.23
Neumann functions
8.403, 8.41
Bessel functions
8.401
Bessel functions
Notation
Symbol
Meaning
The integral part of the real number x (also denoted by [x])
x
(b+)
a
(b−)
a
"
C
PV
Contour integrals; the path of integration starting at the point a extends
to the point b (along a straight line unless there is an indication to the
contrary), encircles the point b along a small circle in the positive
(negative) direction, and returns to the point a, proceeding along the
original path in the opposite direction.
Line integral along the curve C
"
Principal value integral
z = x − iy
The complex conjugate of z = x + iy
n!
= 1 · 2 · 3 . . . n,
(2n + 1)!!
= 1 · 3 . . . (2n + 1). (double factorial notation)
(2n)!!
= 2 · 4 . . . (2n). (double factorial notation)
0!! = 1 and (−1)!! = 1
(cf. 3.372 for n = 0)
00 = 1
(cf. 0.112 and 0.113 for q = 0)
p!
p(p − 1) . . . (p − n + 1)
=
,
1 · 2...n
n!(p − n)!
[n = 1, 2, . . . , p ≥ n]
=
p
n
x
n
(a)n
n
k=m
n
#
,
0! = 1
p
0
= 1,
p
n
=
p!
n!(p − n)!
= x(x − 1) . . . (x − n + 1)/n! [n = 0, 1, . . . ]
= a(a + 1) . . . (a + n − 1) =
(Pochhammer symbol)
n
= um + um+1 + . . . + un . If n < m, we deﬁne
uk = 0
uk
,
Γ(a+n)
Γ(a)
m,n
$
k=m
Summation over all integral values of n excluding n = 0, and summation
over all integral values of n and m excluding m = n = 0, respectively.
#
$
An empty
has value 0, and an empty
has value 1
continued on next page
xliii
xliv
Notation
continued from previous page
Symbol
1 i=j
δij =
0 i=j
Meaning
Kronecker delta
τ
Theta function parameter (cf. 8.18)
× and ∧
Vector product (cf. 10.11)
·
Scalar product (cf. 10.11)
∇ or “del”
Vector operator (cf. 10.21)
∇2
Laplacian (cf. 10.31)
∼
Asymptotically equal to
arg z
The argument of the complex number z = x + iy
curl or rot
Vector operator (cf. 10.21)
div
Vector operator (divergence) (cf. 10.21)
F
Fourier transform (cf. 17.21)
Fc
Fourier cosine transform (cf. 17.31)
Fs
Fourier sine transform (cf. 17.31)
grad
Vector operator (gradient) (cf. 10.21)
hi and gij
Metric coeﬃcients (cf. 10.51)
H
Hermitian transpose of a vector or matrix (cf. 13.123)
0 x<0
H(x) =
1 x≥0
Heaviside step function
Im z ≡ y
The imaginary part of the complex number z = x + iy
k
The letter k (when not used as an index of summation) denotes a number
in the interval [0, 1]. This notation is used in√integrals that lead to elliptic
integrals. In such a connection, the number 1 − k 2 is denoted by k .
L
Laplace transform (cf. 17.11)
M
Mellin transform (cf. 17.41)
N
The natural numbers (0, 1, 2, . . . )
O(f (z))
The order of the function f (z). Suppose that the point z approaches z0 .
If there exists an M > 0 such that |g(z)| ≤ M |f (z)| in some suﬃciently
small neighborhood of the point z0 , we write g(z) = O(f (z)).
continued on next page
Notation
xlv
continued from previous page
Symbol
Meaning
q
The nome, a theta function parameter (cf. 8.18)
R
The real numbers
R(x)
A rational function
Re z ≡ x
The real part of the complex number z = x + iy
Snm
Stirling number of the ﬁrst kind (cd. 9.74)
Sm
n
Stirling number of the second kind (cd. 9.74)
⎧
⎪
⎨+1
sign x =
0
⎪
⎩
−1
x>0
x=0
x<0
The sign (signum) of the real number x
T
Transpose of a vector or matrix (cf. 13.115)
Z
The integers (0, ±1, ±2, . . . )
Zb
Bilateral z transform (cf. 18.1)
Zu
Unilateral z transform (cf. 18.1)
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Note on the Bibliographic References
The letters and numbers following equations refer to the sources used by Russian editors. The key to the
letters will be found preceding each entry in the Bibliography beginning on page 1141. Roman numerals
indicate the volume number of a multivolume work. Numbers without parentheses indicate page numbers,
numbers in single parentheses refer to equation numbers in the original sources.
Some formulas were changed from their form in the source material. In such cases, the letter a appears
at the end of the bibliographic references.
As an example, we may use the reference to equation 3.354–5:
ET I 118 (1) a
The key on page 1141 indicates that the book referred to is:
Erdélyi, A. et al., Tables of Integral Transforms.
The Roman numeral denotes volume one of the work; 118 is the page on which the formula will be
found; (1) refers to the number of the formula in this source; and the a indicates that the expression
appearing in the source diﬀers in some respect from the formula in this book.
In several cases, the editors have used Russian editions of works published in other languages. Under
such circumstances, because the pagination and numbering of equations may be altered, we have referred
the reader only to the original sources and dispensed with page and equation numbers.
xlvii
This page intentionally left blank
0 Introduction
0.1 Finite Sums
0.11 Progressions
0.111 Arithmetic progression.
n−1
n
n
(a + kr) = [2a + (n − 1)r] = (a + l)
2
2
[l = a + (n − 1)r is the last term]
k=0
0.112 Geometric progression.
n
a (q n − 1)
aq k−1 =
q−1
[q = 1]
k=1
0.113 Arithmetic-geometric progression.
n−1
rq 1 − q n−1
a − [a + (n − 1)r]q n
+
(a + kr)q k =
1−q
(1 − q)2
k=0
0.1148
n−1
k 2 xk =
−n + 2n − 1 x
2
k=1
n+2
[q = 1, n > 1]
2
n+1
+ 2n − 2n − 1 x
− n2 xn + x2 + x
(1 − x)3
JO (5)
0.12 Sums of powers of natural numbers
0.121
n
k=1
1.
n
k=1
2.
3.
n
nq
1 q
nq+1
1 q
1 q
+
+
B2 nq−1 +
B4 nq−3 +
B6 nq−5 + · · ·
q+1
2
2 1
4 3
6 5
nq+1
nq
qnq−1
q(q − 1)(q − 2) q−3 q(q − 1)(q − 2)(q − 3)(q − 4) q−5
=
+
+
−
n
n
+
− ···
q+1
2
12
720
30, 240
[last term contains either n or n2 ] CE 332
kq =
k=
n(n + 1)
2
CE 333
n(n + 1)(2n + 1)
6
k=1
2
n
n(n + 1)
3
k =
2
k2 =
CE 333
CE 333
k=1
1
2
Finite Sums
4.
5.
6.
7.
n
k=1
n
k=1
n
k=1
n
k4 =
1
n(n + 1)(2n + 1)(3n2 + 3n − 1)
30
CE 333
k5 =
1 2
n (n + 1)2 (2n2 + 2n − 1)
12
CE 333
k6 =
1
n(n + 1)(2n + 1)(3n4 + 6n3 − 3n + 1)
42
CE 333
k7 =
1 2
n (n + 1)2 (3n4 + 6n3 − n2 − 4n + 2)
24
CE 333
k=1
0.122
0.122
n
2q q+1 1
n
−
q+1
2
(2k − 1)q =
k=1
q q−1
1
2 B2 nq−1 −
1
4
q q−3 3
2
2 − 1 B4 nq−3 − · · ·
3
[last term contains either n or n2 .]
1.
2.
3.
4.11
5.10
6.10
n
k=1
n
k=1
n
k=1
n
k=1
n
k=1
n
(2k − 1) = n2
1
n(4n2 − 1)
3
(2k − 1)2 =
(2k − 1)3 = n2 (2n2 − 1)
(mk − 1) =
n
(mk − 1)2 =
1
n[m2 (n + 1)(2n + 1) − 6m(n + 1) + 6]
6
(mk − 1)3 =
1
n[m3 n(n + 1)2 − 2m2 (n + 1)(2n + 1) + 6m(n + 1) − 4]
4
k(k + 1)2 =
k=1
0.124
1.
2.10
1
n(n + 1)(n + 2)(3n + 5)
12
q
1
k n2 − k 2 = q(q + 1) 2n2 − q 2 − q
4
k=1
n
k(k + 1)3 =
k=1
0.125
0.126
JO (32b)
n
[m(n + 1) − 2]
2
k=1
0.123
JO (32a)
n
k=1
n
k=0
1
n(n + 1) 12n3 + 63n2 + 107n + 58
60
k! · k = (n + 1)! − 1
(n + k)!
=
k!(n − k)!
[q = 1, 2, . . .]
e
K 1
π n+ 2
AD (188.1)
1
2
WA 94
0.142
Sums of the binomial coeﬃcients
3
0.13 Sums of reciprocals of natural numbers
∞
n
1
1
Ak
= C + ln n +
−
,
k
2n
n(n + 1) . . . (n + k − 1)
0.13111
k=1
JO (59), AD (1876)
k=2
where
1 1
x(1 − x)(2 − x)(3 − x) · · · (k − 1 − x) dx
k 0
1
1
A2 =
,
A3 =
12
12
19
9
A4 =
,
A5 =
,
120
20
3
n
2 − 1 B4
1
B2
1
7
0.132
= (C + ln n) + ln 2 + 2 +
+ ...
2k − 1
2
8n
64n4
k=1
n
3
2n + 1
1
= −
0.133
k2 − 1
4 2n(n + 1)
Ak =
JO (71a)a
JO (184f)
k=2
0.14 Sums of products of reciprocals of natural numbers
1.
n
k=1
2.
n
k=1
3.
4.
n
1
=
[p + (k − 1)q](p + kq)
p(p + nq)
GI III (64)a
n(2p + nq + q)
1
=
[p + (k − 1)q](p + kq)[p + (k + 1)q]
2p(p + q)(p + nq)[p + (n + 1)q]
GI III (65)a
n
1
[p + (k − 1)q](p + kq) . . . [p + (k + l)q]
k=1
1
1
1
−
=
(l + 1)q p(p + q) . . . (p + lq) (p + nq)[p + (n + 1)q] . . . [p + (n + l)q]
n
k=1
1
1
=
[1 + (k − 1)q][1 + (k − l)q + p]
p
%
n
k=1
1
−
1 + (k − 1)q
n
k=1
1
1 + (k − 1)q + p
&
AD (1856)a
GI III (66)a
0.142
n
k2 + k − 1
k=1
(k + 2)!
=
1
n+1
−
2 (n + 2)!
JO (157)
0.15 Sums of the binomial coeﬃcients
Notation: n is a natural number.
1.
m
k=0
2.
1+
n+k
n
=
n+m+1
n+1
n
n
+
+ . . . = 2n−1
2
4
KR 64 (70.1)
KR 62 (58.1)
4
Finite Sums
3.
4.
n
n
n
+
+
+ . . . = 2n−1
1
3
5
m
n−1
n
= (−1)m
(−1)k
m
k
0.152
KR 62 (58.1)
[n ≥ 1]
KR 64 (70.2)
k=0
0.152
1.
2.
3.
n
n
n
+
+
+ ... =
0
3
6
n
n
n
+
+
+ ... =
1
4
7
nπ
1 n
2 + 2 cos
3
3
1
(n − 2)π
2n + 2 cos
3
3
KR 62 (59.1)
KR 62 (59.2)
(n − 4)π
3
n
n
n
1
+
+
+ ... =
3
2
5
8
2n + 2 cos
n
0
n
1
n
2
n
3
2n−1 + 2 2 cos
KR 62 (59.3)
0.153
1.
2.
3.
4.
n
4
n
+
5
n
+
6
n
+
7
n
8
n
+
9
n
+
10
n
+
11
+
+
1
2
1
+ ... =
2
1
+ ... =
2
1
+ ... =
2
+ ... =
nπ
4
n
nπ
2n−1 + 2 2 sin
4
n
nπ
2n−1 − 2 2 cos
4
n
nπ
2n−1 − 2 2 sin
4
n
KR 63 (60.1)
KR 63 (60.2)
KR 63 (60.3)
KR 63 (60.4)
0.154
1.
n
n
= 2n−1 (n + 2)
k
(k + 1)
k=0
2.
n
(−1)k+1 k
k=1
3.
N
n
n
N
KR 63 (66.2)
(−1)k
n n
k = (−1)n n!
k
[n ≥ 0;
00 ≡ 1]
(−1)k
n
(α + k)n = (−1)n n!
k
[n ≥ 0;
00 ≡ 1]
(−1)k
N
k
[N ≥ n ≥ 1,
k=0
6.
[n ≥ 2]
N
k
k=0
5.
KR 63 (66.1)
(−1)k
k=0
4.
n
=0
k
[n ≥ 0]
k=0
k n−1 = 0
(α + k)n−1 = 0
[N ≥ n ≥ 1;
00 ≡ 1]
00 ≡ 1 N, n ∈ N + ]
0.155
1.
n
(−1)k+1
k=1
k+1
n
n
=
k
n+1
KR 63 (67)
0.159
2.
Sums of the binomial coeﬃcients
n
1
k+1
k=0
3.
n
αk+1
k+1
k=0
4.
n
2n+1 − 1
=
k
n+1
KR 63 (68.1)
n
(α + 1)n+1 − 1
=
k
n+1
KR 63 (68.2)
n
(−1)k+1
n
n
1
=
k
m
m=1
k
k=1
5
KR 64 (69)
0.156
1.
p
n
k
k=0
2.
n−p
n
k
k=0
m
p−k
=
n
p+k
=
n+m
p
[m is a natural number]
(2n)!
(n − p)!(n + p)!
KR 64 (71.1)
KR 64 (71.2)
0.157
1.
n
n
k
2
2n
n
=
k=0
2.
2n
2n
k
(−1)k
k=0
3.
2n+1
n
k
k=1
0.15810
1.
2.
3.
4.
0.15910
1.
n
k
n 2k
2
=
2n
n
= (−1)n
2n + 1
k
(−1)k
k=0
4.
2
KR 64 (72.1)
2
KR 64 (72.2)
=0
KR 64 (72.3)
(2n − 1)!
KR 64 (72.4)
2
[(n − 1)!]
−2
k+1
k=1
2n − k
n−k
2n − k − 1
n−k−1
− 2k+1
k=1
2n − k
n−k
2n − k − 1
n−k−1
− 2k+1
k=1
2n − k
n−k
2n − k − 1
n−k−1
− 2k+1
k=1
2n − k
n−k
2n − k − 1
n−k−1
n 2k
n 2k
n 2k
n k=0
2n
n−k
−
2n
n−k−1
2n
n
k = 4n −
k 2 = 4n −
2n
n
3 · 4n
k 3 = (6n + 13)4n − 18n
k 4 = (32n2 + 104n)
1 n
k=
4 −
2
2n
n
2n
n
2n
n
− (60n + 75)4n
6
Numerical Series and Inﬁnite Products
2.
3.
n −
k=0
2n
n−k
2n
n−k−1
−
k=0
2n
n−k
2n
n−k−1
2n
k
αk +
n k2 =
1
(2n + 1)
2
k3 =
(3n + 2) n 1
·4 −
4
2
2n
n
0.160
− 4n
2n
n
0.16010
1.
2n
k=n+1
2.
n
(−1)r
r=0
n
r
1
2
2n
n
αn +
n−1
(1 + α)2n−1 (1 − α) 2
k=0
2k
k
(3n + 1)
α
(1 + α)2
k
=
1
(1 + α)2n
2
Γ (r + b)
B (n + a − b, b)
=
Γ (r + a)
Γ (a − b)
0.2 Numerical Series and Inﬁnite Products
0.21 The convergence of numerical series
The series
∞
0.211
uk = u1 + u2 + u3 + . . .
k=1
is said to converge absolutely if the series
∞
0.212
|uk | = |u1 | + |u2 | + |u3 | + · · · ,
k=1
composed of the absolute values of its terms converges. If the series 0.211 converges and the series 0.212
diverges, the series 0.211 is said to converge conditionally. Every absolutely convergent series converges.
0.22 Convergence tests
Suppose that
lim |uk |
k→∞
1/k
=q
If q < 1, the series 0.211 converges absolutely. On the other hand, if q > 1, the series 0.211 diverges.
(Cauchy)
0.222 Suppose that
!
!
! uk+1 !
!=q
lim !!
k→∞ uk !
!
!
! uk+1 !
!
Here, if q < 1, the series 0.211 converges absolutely. If q > 1, the series 0.211 diverges. If !!
uk !
approaches 1 but remains greater than unity, then the series 0.211 diverges. (d’Alembert)
0.223
Suppose that
!
!
! uk !
!
!
lim k !
−1 =q
k→∞
uk+1 !
Here, if q > 1, the series 0.211 converges absolutely. If q < 1, the series 0.211 diverges. (Raabe)
0.229
Convergence tests
7
0.224
Suppose that f (x) is a positive decreasing function and that
ek f ek
lim
=q
k→∞ f (k)
#∞
for natural k. If q < 1, the series k=1 f (k) converges. If q > 1, this series diverges. (Ermakov)
0.225 Suppose that
!
!
! uk !
q
|vk |
!
!
! uk+1 ! = 1 + k + k p ,
where p > 1 and the |vk | are bounded, that is, the |vk | are all less than some M , which is independent
of k. Here, if q > 1, the series 0.211 converges absolutely. If q ≤ 1, this series diverges. (Gauss)
0.226 Suppose that a function f (x) deﬁned for x ≥ q ≥ 1 is continuous, positive, and decreasing. Under
these conditions, the series
∞
f (k)
k=1
converges or diverges accordingly as the integral
∞
f (x) dx
q
converges or diverges (the Cauchy integral test).
0.227 Suppose that all terms of a sequence u1 , u2 , . . . , un are positive. In such a case, the series
1.
∞
(−1)k+1 uk = u1 − u2 + u3 − . . .
k=1
is called an alternating series.
If the terms of an alternating series decrease monotonically in absolute value and approach zero,
that is, if
2.
3.
uk+1 < uk and lim uk = 0,
k→∞
the series 0.227 1 converges. Here, the remainder of the series is
!
!∞
∞
n
!
!
!
!
(−1)k−n+1 uk = ! (−1)k+1 uk −
(−1)k+1 uk < un+1 !
!
!
k=n+1
0.228
1.
k=1
(Leibniz)
k=1
If the series
∞
vk = v1 + v2 + . . . + vk + . . .
k=1
2.
converges and the numbers uk form a monotonic bounded sequence, that is, if |uk | < M for some
number M and for all k, the series
∞
uk vk = u1 v1 + u2 v2 + . . . + uk vk + . . .
FI II 354
k=1
converges. (Abel)
0.229 If the partial sums of the series 0.228 1 are bounded and if the numbers uk constitute a monotonic
sequence that approaches zero, that is, if
8
Numerical Series and Inﬁnite Products
!
!
n
!
!
!
!
vk ! < M
!
!
!
[n = 1, 2, . . .]
0.231
and lim uk = 0,
k=1
FI II 355
k→∞
then the series 0.228 2 converges (Dirichlet).
0.23–0.24 Examples of numerical series
0.231
Progressions
∞
1.
aq k =
k=0
∞
2.
a
1−q
(a + kr)q k =
k=0
0.232
1.
∞
∞
1
= ln 2
k
(−1)k+1
π
1
1
=1−2
=
2k − 1
(4k − 1)(4k + 1)
4
3.
(cf. 0.113)
(cf. 1.511)
k=1
∞
ka
k=1
[|q| < 1]
∞
k=1
∗
rq
a
+
1 − q (1 − q)2
(−1)k+1
k=1
2.
[|q| < 1]
(cf. 1.643)
⎤
⎡
a
i
1 (−1)j (a + 1)!(i − j)a ⎦
1
⎣
=
bk
(b − 1)a+1 i=1 ba−i j=0
j!(a + 1 − j)!
[a = 1, 2, 3, . . . ,
0.233
1.
∞
1
1
1
= 1 + p + p + . . . = ζ(p)
kp
2
3
b = 1]
[Re p > 1]
WH
[Re p > 0]
WH
k=1
2.
∞
(−1)k+1
k=1
3.10
∞
1
22n−1 π 2n
|B2n |,
=
k 2n
(2n)!
k=1
4.
∞
(−1)k+1
k=1
5.
∞
k=1
6.
1
= (1 − 21−p ) ζ(p)
kp
∞
k=1
∞
1
π2
=
k2
6
1
(22n−1 − 1)π 2n
|B2n |
=
k 2n
(2n)!
1
(22n − 1)π 2n
|B2n |
=
2n
(2k − 1)
2 · (2n)!
(−1)k+1
FI II 721
k=1
1
π 2n+1
|E2n |
= 2n+2
2n+1
(2k − 1)
2
(2n)!
JO (165)
JO (184b)
JO (184d)
0.236
0.234
1.
2.
3.
4.
5.
6.
7.
8.6
9.∗
Examples of numerical series
∞
k=1
∞
k=1
∞
k=0
∞
k=1
∞
k=1
∞
k=1
∞
(−1)k+1
1
π2
=
2
k
12
9
EU
1
π2
=
2
(2k − 1)
8
EU
(−1)k
=G
(2k + 1)2
FI II 482
(−1)k+1
π3
=
(2k − 1)3
32
EU
1
π4
=
4
(2k − 1)
96
EU
(−1)k+1
5π 5
=
(2k − 1)5
1536
EU
(−1)k+1
k=1
∞
k
π2
− ln 2
=
(k + 1)2
12
1
= 2 − 2 ln 2
k(2k + 1)
k=1
∞
√
Γ n + 12
= π ln 4
2
n Γ (n)
n=1
0.235 Sn =
∞
k=1
S1 =
(4k 2
1
,
2
1
n
− 1)
S2 =
π2 − 8
,
16
S3 =
32 − 3π 2
,
64
S4 =
π 4 + 30π 2 − 384
768
JO (186)
0.236
1.
2.
3.
4.
5.
∞
k=1
∞
k=1
∞
k=1
∞
1
= 2 ln 2 − 1
k (4k 2 − 1)
BR 51a
3
1
= (ln 3 − 1)
2
k (9k − 1)
2
BR 51a
3
1
= −3 + ln 3 + 2 ln 2
k (36k 2 − 1)
2
k
=
2
(4k 2 − 1)
k=1
∞
k=1
1
k (4k 2
2
− 1)
1
8
=
BR 52, AD (6913.3)
BR 52
3
− 2 ln 2
2
BR 52
10
6.
Numerical Series and Inﬁnite Products
∞
2
k=1
7.6
12k 2 − 1
∞
k=1
k (4k 2 − 1)
0.237
= 2 ln 2
AD (6917.3), BR 52
1
π2
− 2 ln 2
=4−
2
k(2k + 1)
4
0.237
1.
∞
k=1
2.
∞
k=1
3.
∞
k=2
4.
1
1
=
(2k − 1)(2k + 1)
2
1 π
1
= −
(4k − 1)(4k + 1)
2
8
3
1
=
(k − 1)(k + 1)
4
∞
k=1,k=m
5.
AD (6917.2), BR 52
∞
k=1,k=m
[cf. 0.133],
3
1
=− 2
(m + k)(m − k)
4m
[m is an integer]
AD (6916.1)
3
(−1)k−1
=
(m − k)(m + k)
4m2
[m is an even number]
AD (6916.2)
0.238
1.
∞
k=1
2.
∞
k=1
3.
1
1
= ln 2 −
(2k − 1)2k(2k + 1)
2
1
(−1)k+1
= (1 − ln 2)
(2k − 1)2k(2k + 1)
2
∞
1 1
π
1
= − ln 3 + √
(3k + 1)(3k + 2)(3k + 3)(3k + 4)
6 4
12 3
k=0
GI III (93)
GI III (94)a
GI III (95)
0.239
1.11
∞
(−1)k+1
1
1
=
3k − 2
3
π
√ + ln 2
3
GI III (85), BR∗ 161 (1)
(−1)k+1
1
1
=
3k − 1
3
π
√ − ln 2
3
BR∗ 161 (1)
(−1)k+1
,
√
1 +
1
= √ π + 2 ln
2+1
4k − 3
4 2
k=1
2.7
∞
k=1
3.
∞
k=1
4.
∞
k=1
k+3 1
π 1
(−1)[ 2 ] = + ln 2
k
4
2
BR∗ 161 (1)
GI III (87)
0.241
Examples of numerical series
∞
5.
k+3
(−1)[ 2 ]
k=1
∞
6.
k+5
(−1)[ 3 ]
k=1
∞
7.
k=1
11
1
π
= √
2k − 1
2 2
1
5π
=
2k − 1
12
GI III (88)
1
π √
1
= −
2+1
(8k − 1)(8k + 1)
2 16
0.241
∞
1
= ln 2
2k k
1.
JO (172g)
k=1
∞
2.
k=1
3.11
1
2k k 2
∞
2n
n
n=0
4.10
5.10
6.
i 2j
i 2j
j=1
7.10
i 2j
j=1
8.
10
i 2j
j=1
9.10
2n
j=n+1
10.10
1
π2
2
− (ln 2)
12 2
pn = √
JO (174)
1
1 − 4p
p
∞
pn
ln(1 − x)
π2
−
dx
=
2
n
6
x
1
n=1
j=1
10
=
i
k=0
2i − j
i−j
2i − j
i−j
2i − j
i−j
2i − j
i−j
2n
j
i+k
k
kj +
− 2j+1
−2
j+1
− 2j+1
−2
1
2
j+1
2n
n
2i−k = 4i
1
4
[0 ≤ p ≤ 1]
2i − (j + 1)
i − (j + 1)
2i − (j + 1)
i − (j + 1)
2i − (j + 1)
i − (j + 1)
2i − (j + 1)
i − (j + 1)
kn +
0≤p<
j = 4i −
2i
i
+ n
= 0,
m
j 2 = 4i
2i
i
− 3 · 4i
+ n
= 0,
m
j 3 = (6i + 13)4i − 18i
,
m<0
2i
i
+ n
= 0,
m
j 4 = 32i2 + 104i
n−1
(1 + k)2n−1 (1 − k) 2
i=0
2i
i
2i
i
,
m<0
,
m<0
− (60i + 75)4i
k
(1 + k)2
i
=
1
(1 + k)2n
2
12
Numerical Series and Inﬁnite Products
11.10
i
i+k
h
k=0
12.
10
i
k=0
13.10
i
k=0
14.10
i
k=0
0.242
∞
1.
∞
k=1
k = (i + 1)4i − (2i + 1)
1 i
=
4 +
2
2i
k
k=
2i
k
k2 = (2i + 1)i4i−1 −
(−1)k
2i
i
2i
i
2i
k
k=0
0.243
i−k
0.242
i i
4
2
i2
2
2i
i
1
n2
=
n2k
n2 + 1
[n > 1]
1
1
1
=
[p + (k − 1)q](p + kq) . . . [p + (k + l)q]
(l + 1)q p(p + q) . . . (p + lq)
(see also 0.141 3)
2.7
3.
1 i tp−1 (1 − t)t
x
=
dt
[p + (k − 1)q][p + (k − 1)q + 1][p + (k − 1)q + 2] . . . [p + (k − 1)q + l] l! 0 1 − xtq
k=1
p > 0, x2 < 1
BR∗ 161 (2), AD (6.704)
∞
∞
k=0
k−1
1
(2k + 1)3
(2k + 1)πx
1
(2k + 1)π
tanh
+ x tanh
x
2
2x
0.244
1.
∞
k=1
2.
∞
1
1
=
(k + p)(k + q)
q−p
(−1)k+1
k=1
3.10
∞
k=1
1
=
p + (k − 1)q
1
0
1
0
1.
[p > −1,
tp−1
dt
1 + tq
[p > 0,
q
1
1
1
=
(k + p)(k + q)
q − p m=p+1 m
∞
1
= e = 2.71828 . . .
k!
k=0
2.11
∞
(−1)k
k=0
k!
=
π3
16
xp − xq
dx
1−x
Summations of reciprocals of factorials
0.245
=
1
≈ 0.1839397 . . .
2e
q > −1,
q > 0]
[q > p > −1,
p = q]
GI III (90)
BR∗ 161 (1)
p and q integers]
0.249
3.
4.
5.
6.
7.
8.
Examples of numerical series
∞
k=1
∞
k=1
∞
k=0
∞
k=0
∞
k=0
∞
k=0
0.246
1.
2.
3.
4.
5.
6.
∞
k=0
∞
k=0
∞
k=0
∞
k=0
∞
k=0
0.247
0.248
1
k
= = 0.36787 . . .
(2k + 1)!
e
k
=1
(k + 1)!
1
1
=
(2k)!
2
∞
k=1
∞
k=1
e+
1
1
=
(2k + 1)!
2
1
(k!)
2
e−
1
e
= 1.17520 . . .
= I 0 (2) = 2.27958530 . . .
1
= I 1 (2) = 1.590636855 . . .
k!(k + 1)!
1
= I n (2)
k!(k + n)!
(−1)k
(k!)
= J 0 (2) = 0.22389078 . . .
(−1)k
= J 1 (2) = 0.57672481 . . .
k!(k + 1)!
(−1)k
= J n (2)
k!(k + n)!
1
k!
=
(n + k − 1)!
(n − 2) · (n − 1)!
kn
= Sn ,
k!
∞
(k + 1)3
k=0
= 1.54308 . . .
(−1)k−1
= sin 1 = 0.84147 . . .
(2k − 1)!
S1 = e,
S5 = 52e,
0.2497
1
e
(−1)k
= cos 1 = 0.54030 . . .
(2k)!
2
k=0
∞
13
k!
= 15e
S2 = 2e,
S6 = 203e,
S3 = 5e,
S7 = 877e,
S4 = 15e
S8 = 4140e
14
Numerical Series and Inﬁnite Products
0.250
0.25 Inﬁnite products
0.250
Suppose that a sequence of numbers a1 , a2 , . . . , ak , . . . is given. If the limit lim
n→∞
n
-
(1 + ak )
k=1
exists, whether ﬁnite or inﬁnite (but of deﬁnite sign), this limit is called the value of the inﬁnite product
∞
(1 + ak ), and we write
k=1
1.
lim
n
-
n→∞
(1 + ak ) =
k=1
∞
-
(1 + ak )
k=1
If an inﬁnite product has a ﬁnite nonzero value, it is said to converge. Otherwise, the inﬁnite product is
said to diverge. We assume that no ak is equal to −1.
FI II 400
0.251 For the inﬁnite product 0.250 1. to converge, it is necessary that lim ak = 0.
FI II 403
k→∞
0.252 If ak > 0 or ak < 0 for all values of the index k starting with some#
particular value, then, for the
∞
product 0.250 1 to converge, it is necessary and suﬃcient that the series k=1 ak converge.
∞
∞
0.253 The product
(1 + ak ) is said to converge absolutely if the product
(1 + |ak |) converges.
k=1
k=1
FI II 403
0.254
0.255
Absolute convergence of an inﬁnite product implies its convergence.
∞
∞
(1 + ak ) converges absolutely if, and only if, the series
ak converges abThe product
k=1
k=1
solutely.
FI II 406
0.26 Examples of inﬁnite products
0.261
∞
-
1+
(−1)k+1
2k − 1
1−
1
k2
1−
1
(2k)2
1−
1
(2k + 1)2
k=1
=
√
2
EU
0.262
1.
∞
k=2
2.
∞
k=1
3.
∞
k=1
=
1
2
=
FI II 401
2
π
=
FI II 401
π
4
FI II 401
0.263
1.
e=
2
·
1
2.∗
e=
2
1
4
3
1/2
1/2
6·8
5·7
22
1·3
1/3
1/4
10 · 12 · 14 · 16
9 · 11 · 13 · 15
23 · 4
1 · 33
1/4
1/8
24 · 44
1 · 36 · 5
...
1/5
···
0.303
3.
∗
Deﬁnitions and theorems
π
=
2
1
2
1/2
22
1·3
1/4
23 · 4
1 · 33
1/8
24 · 44
1 · 36 · 5
1/16
···
where the nth factor is the (n + 1)th root of the product
0.264
1.
eC =
2.∗
eC =
15
$n
k=0 (k
+ 1)(−1)
k+1
( nk ) .
√
k
e
1
k=1 1 +
k
∞
-
2
1
1/2
FI II 402
22
1·3
1/3
23 · 4
1 · 33
1/4
24 · 44
1 · 36 · 5
1/5
···
$n
k+1 n
where the nth factor is the (n + 1)th root of the product k=0 (k + 1)(−1) ( k ) . Here C is the
Euler constant, denoted in other works by γ.
/
.
.
0
0
2
1
1 1 1 11 1 1 1 1
0.265
=
·
+
·
+
+
...
FI II 402
π
2
2 2 2
2 2 2 2 2
∞
k
1
1 + x2 =
[0 < x < 1]
0.2668
FI II 401
1−x
k=0
0.3 Functional Series
0.30 Deﬁnitions and theorems
0.301
1.
The series
∞
fk (x),
k=1
the terms of which are functions, is called a functional series. The set of values of the independent
variable x for which the series 0.301 1 converges constitutes what is called the region of convergence of
that series.
0.302 A series that converges for all values of x in a region M is said to converge uniformly in that
region if, for every ε ≥ 0, there exists a number N such that, for n > N , the inequality
!
!
∞
! !
!
!
fk (x)! < ε
!
!
!
k=n+1
holds for all x in M .
0.303
If the terms of the functional series 0.301 1 satisfy the inequalities:
|fk (x)| < uk (k = 1, 2, 3, . . .) ,
throughout the region M , where the uk are the terms of some convergent numerical series
∞
uk = u1 + u2 + . . . + uk + . . . ,
k=1
the series 0.301 1 converges uniformly in M . (Weierstrass)
16
Functional Series
0.304
0.304 Suppose that the series 0.301 1 converges uniformly in a region M and that a set of functions
gk (x) constitutes (for each x) a monotonic sequence, and that these functions are uniformly bounded;
that is, suppose that a number L exists such that the inequalities
1.
|gn (x)| ≤ L
2.
hold for all n and x. Then, the series
∞
fk (x)gk (x)
k=1
converges uniformly in the region M . (Abel)
FI II 451
0.305 Suppose that the partial sums of the series 0.301 1 are uniformly bounded; that is, suppose that,
for some L and for all n and x in M , the inequalities
! n
!
!
!
!
!
fk (x)! < L
!
!
!
k=1
hold. Suppose also that for each x the functions gn (x) constitute a monotonic sequence that approaches
zero uniformly in the region M . Then, the series 0.304 2 converges uniformly in the region M . (Dirichlet)
FI II 451
6
0.306 If the functions fk (x) (for k = 1, 2, 3, . . . ) are integrable on the interval [a, b] and if the series
0.301 1 made up of these functions converges uniformly on that interval, this series may be integrated
termwise; that is,
b ∞
∞ b
fk (x) dx =
fk (x) dx
[a ≤ x ≤ b]
FI II 459
a
k=1
k=1
a
0.307 Suppose that the functions fk (x) (for k = 1, 2, 3, . . . ) have continuous derivatives
#∞ fk (x) on the
interval [a, b]. If the series 0.301 1 converges on this interval and if the series
k=1 fk (x) of these
derivatives converges uniformly, the series 0.301 1 may be diﬀerentiated termwise; that is,
∞
∞
fk (x) =
fk (x)
FI II 460
k=1
k=1
0.31 Power series
0.311
1.
A functional series of the form
∞
ak (x − ξ)k = a0 + a1 (x − ξ) + a2 (x − ξ)2 + . . .
k=0
is called a power series. The following is true of any power series: if it is not everywhere convergent, the
region of convergence is a circle with its center at the point ξ and a radius equal to R; at every interior
point of this circle, the power series 0.311 1 converges absolutely, and outside this circle, it diverges. This
circle is called the circle of convergence, and its radius is called the radius of convergence. If the series
converges at all points of the complex plane, we say that the radius of convergence is inﬁnite (R = +∞).
0.315
0.312
is,
Power series
17
Power series may be integrated and diﬀerentiated termwise inside the circle of convergence; that
x
ξ
∞
ak (x − ξ)
k
k=0
d
dx
∞
∞
ak
(x − ξ)k+1 ,
k+1
dx =
k=0
∞
ak (x − ξ)k
=
k=0
kak (x − ξ)k−1 .
k=1
The radius of convergence of a series that is obtained from termwise integration or diﬀerentiation of
another power series coincides with the radius of convergence of the original series.
Operations on power series
0.313
Division of power series.
∞
k=0
∞
bk xk
=
ak xk
∞
1 ck xk ,
a0
k=0
k=0
where
1
cn−k ak − bn = 0,
a0
n
cn +
k=1
or
⎡
k=0
where
c0 =
a0
a1
a2
..
.
0
a0
a1
..
.
···
⎢
⎢
(−1) ⎢
⎢
cn =
⎢
..
an0 ⎢
.
⎢
⎣an−1 b0 − a0 bn−1 an−2 an−3 · · ·
an b 0 − a0 b n
an−1 an−2 · · ·
Power series raised to powers.
n
∞
∞
k
ak x
=
ck xk ,
n
0.314
a1 b 0 − a0 b 1
a2 b 0 − a0 b 2
a3 b 0 − a0 b 3
..
.
an0 ,
cm
m
1 =
(kn − m + k)ak cm−k
ma0
⎤
0
0⎥
⎥
0⎥
⎥
⎥
⎥
⎥
a0 ⎦
a1
AD (6360)
k=0
for m ≥ 1
[n is a natural number]
AD (6361)
k=1
0.315
The substitution of one series into another.
∞
∞
∞
bk y k =
ck xk
y=
ak xk ;
k=1
c 1 = a1 b 1 ,
k=1
c2 = a2 b1 + a21 b2 ,
k=1
c3 = a3 b1 + 2a1 a2 b2 + a31 b3 ,
c4 = a4 b1 + a22 b2 + 2a1 a3 b2 + 3a21 a2 b3 + a41 b4 ,
...
AD (6362)
18
0.316
Functional Series
Multiplication of power series
∞
∞
∞
ak xk
bk xk =
ck xk
k=0
k=0
cn =
k=0
0.316
n
ak bn−k
FI II 372
k=0
Taylor series
0.317 If a function f (x) has derivatives of all orders throughout a neighborhood of a point ξ, then we
may write the series
1.
(x − ξ) (x − ξ)2 (x − ξ)3 f (ξ) +
f (ξ) +
f (ξ) + . . . ,
1!
2!
3!
which is known as the Taylor series of the function f (x).
f (ξ) +
The Taylor series converges to the function f (x) if the remainder
2.
Rn (x) = f (x) − f (ξ) −
n
(x − ξ)k
k=1
k!
f (k) (ξ)
approaches zero as n → ∞.
The following are diﬀerent forms for the remainder of a Taylor series:
3.
Rn (x) =
4.
Rn (x) =
5.
(x − ξ)n+1 (n+1)
f
(ξ + θ(x − ξ))
(n + 1)!
[0 < θ < 1]
(Lagrange)
(x − ξ)n+1
(1 − θ)n f (n+1) (ξ + θ(x − ξ))
[0 < θ < 1]
n!
ψ(x − ξ) − ψ(0) (x − ξ)n (1 − θ)n (n+1)
f
Rn (x) = (ξ + θ(x − ξ))
ψ [(x − ξ)(1 − θ)]
n!
[0 < θ < 1] ,
(Cauchy)
(Schlömilch)
where ψ(x) is an arbitrary function satisfying the following two conditions: (1) It and its derivative
ψ (x) are continuous in the interval (0, x − ξ); and (2) the derivative ψ (x) does not change sign
in that interval. If we set ψ(x) = xp+1 , we obtain the following form for the remainder:
(x − ξ)n+1 (1 − θ)n−p−1 (n+1)
f
(ξ + θ(x − ξ))
(p + 1)n!
1 x (n+1)
Rn (x) =
f
(t)(x − t)n dt
n! ξ
Rn (x) =
6.
0.318
1.11
0 < θ < 1]
(Rouché)
Other forms in which a Taylor series may be written:
f (a + x) =
∞
xk
k=0
2.
[0 < p ≤ n;
∞
xk
k=0
k!
k!
f (k) (a) = f (a) +
f (k) (0) = f (0) +
x x2
f (a) + f (a) + . . .
1!
2!
x x2
f (0) + f (0) + . . .
1!
2!
(Maclaurin series)
0.323
0.319
Fourier series
19
The Taylor series of functions of several variables:
∂f (ξ, η)
∂f (ξ, η)
+ (y − η)
∂x
∂y
2
2
∂
f
(ξ,
η)
∂ 2 f (ξ, η)
2
2 ∂ f (ξ, η)
+ (y − η)
+ 2(x − ξ)(y − η)
+ ...
(x − ξ)
∂x2
∂x ∂y
∂y 2
f (x, y) = f (ξ, η) + (x − ξ)
+
1
2!
0.32 Fourier series
0.320 Suppose that f (x) is a periodic function of period 2l and that it is absolutely integrable (possibly
improperly) over the interval (−l, l). The following trigonometric series is called the Fourier series of
f (x):
∞
1.
a0 kπx
kπx
+
+ bk sin
,
ak cos
2
l
l
k=1
2.
3.11
the coeﬃcients of which (the Fourier coeﬃcients) are given by the formulas
1 α+2l
1 l
kπt
kπt
ak =
dt =
dt (k = 0, 1, 2, . . .)
f (t) cos
f (t) cos
l −l
l
l α
l
1 α+2l
1 l
kπt
kπt
dt =
dt (k = 1, 2, . . .)
bk =
f (t) sin
f (t) sin
l −l
l
l α
l
Convergence tests
0.321
The Fourier series of a function f (x) at a point x0 converges to the number
f (x0 + 0) + f (x0 − 0)
,
2
if, for some h > 0, the integral
h
|f (x0 + t) + f (x0 − t) − f (x0 + 0) − f (x0 − 0)|
dt
t
0
exists. Here, it is assumed that the function f (x) either is continuous at the point x0 or has a discontinuity
of the ﬁrst kind (a saltus) at that point and that both one-sided limits f (x0 + 0) and f (x0 − 0) exist.
FI III 524
(Dini)
0.322 The Fourier series of a periodic function f (x) that satisﬁes the Dirichlet conditions on the interval
[a, b] converges at every point x0 to the value 12 [f (x0 + 0) + f (x0 − 0)]. (Dirichlet)
We say that a function f (x) satisﬁes the Dirichlet conditions on the interval [a, b] if it is bounded on
that interval and if the interval [a, b] can be partitioned into a ﬁnite number of subintervals inside each
of which the function f (x) is continuous and monotonic.
0.323 The Fourier series of a function f (x) at a point x0 converges to 12 [f (x0 + 0) + f (x0 − 0)] if f (x) is
FI III 528
of bounded variation in some interval (x0 − h, x0 + h) with center at x0 . (Jordan–Dirichlet)
The deﬁnition of a function of bounded variation. Suppose that a function f (x) is deﬁned on some
interval [a, b], where z < b. Let us partition this interval in an arbitrary manner into subintervals with
the dividing points
a = x0 < x1 < x2 < . . . < xn−1 < xn = b
and let us form the sum
20
Functional Series
n
0.324
|f (xk ) − f (xk−1 )|
k=1
Diﬀerent partitions of the interval [a, b] (that is, diﬀerent choices of points of division xi ) yield, generally
speaking, diﬀerent sums. If the set of these sums is bounded above, we say that the function f (x) is of
bounded variation on the interval [a, b]. The least upper bound of these sums is called the total variation
of the function f (x) on the interval [a, b].
0.324 Suppose that a function f (x) is piecewise-continuous on the interval [a, b] and that in each interval
of continuity it has a piecewise-continuous derivative. Then, at every point x0 of the interval [a, b], the
Fourier series of the function f (x) converges to 12 [f (x0 + 0) + f (x0 − 0)].
0.325 A function f (x) deﬁned in the interval (0, l) can be expanded in a cosine series of the form
∞
1.
a0 kπx
+
,
ak cos
2
l
k=1
where
2.
0.326
1.
ak =
2
l
l
f (t) cos
0
A function f (x) deﬁned in the interval (0, l) can be expanded in a sine series of the form
∞
bk sin
k=1
where
2.
kπt
dt
l
bk =
2
l
kπx
,
l
l
f (t) sin
0
kπt
dt
l
The convergence tests for the series 0.325 1 and 0.326 1 are analogous to the convergence tests for the
series 0.320 1 (see 0.321–0.324).
0.327 The Fourier coeﬃcients ak and bk (given by formulas 0.320 2 and 0.320 3) of an absolutely
integrable function approach zero as k → ∞.
If a function f (x) is square-integrable on the interval (−l, l), the equation of closure is satisﬁed:
∞
1 l 2
a20 2
+
ak + b2k =
f (x) dx
(A. M. Lyapunov)
FI III 705
2
l −l
k=1
0.328 Suppose that f (x) and ϕ(x) are two functions that are square-integrable on the interval (−l, l)
and that ak , bk and αk , βk are their Fourier coeﬃcients. For such functions, the generalized equation of
closure (Parseval’s equation) holds:
∞
1 l
a0 α0 +
(ak αk + bk βk ) =
f (x)ϕ(x) dx
FI III 709
2
l −l
k=1
For examples of Fourier series, see 1.44 and 1.45.
0.411
Diﬀerentiation of a deﬁnite integral with respect to a parameter
21
0.33 Asymptotic series
0.330 Included in the collection of all divergent series is the broad class of series known as asymptotic
or semiconvergent series. Despite the fact that these series diverge, the values of the functions that they
represent can be calculated with a high degree of accuracy if we take the sum of a suitable number of
terms of such series. In the case of alternating asymptotic series, we obtain greatest accuracy if we break
oﬀ the series in question at whatever term is of lowest absolute value. In this case, the error (in absolute
value) does not exceed the absolute value of the ﬁrst of the discarded terms (cf. 0.227 3).
Asymptotic series have many properties that are analogous to the properties of convergent series, and,
for that reason, they play a signiﬁcant role in analysis.
The asymptotic expansion of a function is denoted as follows:
∞
An z −n
f (z) ∼
n=0
∞
An
This is the deﬁnition of an asymptotic expansion. The divergent series
is called the asymptotic
zn
n=0
expansion of a function f (z) in a given region of values of arg z if the expression Rn (z) = z n [f (z) − Sn (z)],
n
Ak
, satisﬁes the condition lim Rn (z) = 0 for ﬁxed n.
FI II 820
where Sn (z) =
zk
|z|→∞
k=0
A divergent series that represents the asymptotic expansion of some function is called an asymptotic
series.
0.331 Properties of asymptotic series
1.
The operations of addition, subtraction, multiplication, and raising to a power can be performed
on asymptotic series just as on absolutely convergent series. The series obtained as a result of
these operations will also be asymptotic.
2.
One asymptotic series can be divided by another, provided that the ﬁrst term A0 of the divisor
is not equal to zero. The series obtained as a result of division will also be asymptotic.
FI II 823-825
3.
An asymptotic series can be integrated termwise, and the resultant series will also be asymptotic.
FI II 824
In contrast, diﬀerentiation of an asymptotic series is, in general, not permissible.
4.
A single asymptotic expansion can represent diﬀerent functions. On the other hand, a given
function can be expanded in an asymptotic series in only one manner.
0.4 Certain Formulas from Diﬀerential Calculus
0.41 Diﬀerentiation of a deﬁnite integral with respect to a parameter
0.410
0.411
1.
2.
ϕ(a)
dψ(a)
d
d ϕ(a)
dϕ(a)
− f (ψ(a), a)
+
f (x, a) dx
f (x, a) dx = f (ϕ(a), a)
da ψ(a)
da
da
da
ψ(a)
In particular,
d a
f (x) dx = f (a)
da b
d a
f (x) dx = −f (b)
db b
FI II 680
22
Certain Formulas from Diﬀerential Calculus
0.430
0.42 The nth derivative of a product (Leibniz’s rule)
Suppose that u and v are n-times-diﬀerentiable functions of x. Then,
dn v
n du dn−1 v
n d2 u dn−2 v
n d3 u dn−3 v
dn u
dn (uv)
=u n +
+
+
+ ··· + v n
n
n−1
2
n−2
3
n−3
dx
dx
dx
1 dx dx
2 dx dx
3 dx dx
or, symbolically,
dn (uv)
= (u + v)(n)
dxn
FI I 272
0.43 The nth derivative of a composite function
0.430
1.
If f (x) = F (y)and y = ϕ(x), then
dn
U1 U2 U3 Un (n)
F (y) +
F (y) +
F (y) + . . . +
F (y),
f (x) =
dxn
1!
2!
3!
n!
where
Uk =
n
dn k
k dn k−1 k(k − 1) 2 dn k−2
k−1
k−1 d y
y
y
y
−
y
+
y
−
.
.
.
+
(−1)
ky
dxn
1! dxn
2!
dxn
dxn
y (l)
y y dm F y dn
n!
f
(x)
=
·
·
·
,
dxn
i!j!h! . . . k! dy m 1!
2!
3!
l!
#
Here, the symbol
indicates summation over all solutions in non-negative integers of the equation i + 2j + 3h + . . . + lk = n and m = i + j + h + . . . + k.
i
2.
AD (7361) GO
j
k
h
0.431
1.
(−1)n
dn
F
dxn
1
x
1
1 (n) 1
n−1 n
F
+ 2n−1 F (n−1)
x2n
x
x
1!
x
(n − 1)(n − 2) n(n − 1) (n−2) 1
F
+
+ ...
x2n−2
2!
x
=
AD (7362.1)
⎡
2.
(−1)n
a
dn a
1 a
ex = n ex ⎣
n
dx
x
x
n
+ (n − 1)
+ (n − 1)(n − 2)(n − 3)
n
3
n
1
a
x
a
x
n−1
n−3
+ (n − 1)(n − 2)
⎤
n
2
a
x
n−2
+ . . .⎦
AD (7362.2)
0.432
1.
n(n − 1)
dn 2 (2x)n−2 F (n−1) x2
F x = (2x)n F (n) x2 +
n
dx
1!
n(n − 1)(n − 2)(n − 3)
+
(2x)n−4 F (n−2) x2 +
2!
n(n − 1)(n − 2)(n − 3)(n − 4)(n − 5)
+
(2x)n−6 F (n−3) x2 + . . .
3!
AD (7363.1)
0.440
Integration by substitution
23
⎡
n(n − 1) n(n − 1)(n − 2)(n − 3)
d ax
+
e = (2ax)n eax ⎣1 +
2
dxn
1! (4ax2 )
2! (4ax2 )
⎤
n(n − 1)(n − 2)(n − 3)(n − 4)(n − 5)
+
+ · · ·⎦
3
3! (4ax2 )
n
2.
2
2
AD (7363.2)
p p(p − 1)(p − 2) . . . (p − n + 1)(2ax)
d 1 + ax2 =
n−p
dxn
(1 + ax2 )
n(n − 1) 1 + ax2
n(n − 1)(n − 2)(n − 3)
× 1+
+
2
1!(p − n + 1) 4ax
2!(p − n + 1)(p − n + 2)
n
3.
n
1 + ax2
4ax2
2
+ ... ,
AD (7363.3)
4.
5.
m− 12
(2m − 1)!!
dm−1 sin (m arccos x)
1 − x2
= (−1)m−1
m−1
dx
m
n+1
a
a
n+1
∂n
(−1)n n
(−1)k
=
n!
2
2
2
2
2k
∂a
a +b
a +b
AD (7363.4)
b
a
0≤2k≤n+1
6.
0.433
1.
(−1)n
∂n
∂an
b
2
a + b2
= n!
a
2
a + b2
n+1
(−1)k
0≤2k≤n
n+1
2k + 1
b
a
2k
(3.944.12)
2k+1
(3.944.11)
√
√
√
dn √ F (n) ( x) n(n − 1) F (n−1) ( x) (n + 1)n(n − 1)(n − 2) F (n−2) ( x)
F
x
=
−
+
√
√ n+1
√ n+2 − . . .
n
dxn
1!
2!
(2 x)
(2 x)
(2 x)
AD (7364.1)
√ 2n−1
dn (2n − 1)!! a
1
√
1+a x
=
a2 −
n
n
dx
2
x
x
2
n
n
y
n−p
dn p
n
1
n
1
p−1 d y
p−2 d
y
y
y =p
−
+
− ...
n
dxn
dxn
dxn
1 p−1
2 p−2
dn y 3 n
n
1 dn y
dn
n
1 dn y 2
ln y =
−
+
x − ...
1 1 · y dxn
dxn
dxn
2 2 · y 2 dxn
n−1
2.
0.434
0.435
AD (7364.2)
AD (737.1)
AD (737.2)
0.44 Integration by substitution
0.44011 Let f (g(x)) and g(x) be continuous in [a, b]. Further, let g (x) exist and be continuous there.
b
g(b)
Then
f [g(x)]g (x) dx =
f (u) du
a
g(a)
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1 Elementary Functions
1.1 Power of Binomials
1.11 Power series
∞
q
q(q − 1) 2
q(q − 1) . . . (q − k + 1) k
xk
x + ··· +
x + ··· =
k
2!
k!
k=0
If q is neither a natural number nor zero, the series converges absolutely for |x| < 1 and diverges for
|x| > 1. For x = 1, the series converges for q > −1 and diverges for q ≤ −1. For x = 1, the series
converges absolutely for q > 0. For x = −1, it converges absolutely for q > 0 and diverges for q < 0. If
FI II 425
q = n is a natural number, the series 1.110 is reduced to the ﬁnite sum 1.111.
n
n
xk an−k
1.111 (a + x)n =
k
1.110 (1 + x)q = 1 + qx +
k=0
1.112
1.
(1 + x)−1 = 1 − x + x2 − x3 + · · · =
∞
(−1)k−1 xk−1
k=1
(see also 1.121 2)
2.
(1 + x)−2 = 1 − 2x + 3x2 − 4x3 + · · · =
∞
(−1)k−1 kxk−1
k=1
3.11
4.
1·1 2 1·1·3 3 1·1·3·5 4
1
x +
x −
x + ...
(1 + x)1/2 = 1 + x −
2
2·4
2·4·6
2·4·6·8
1·3 2 1·3·5 3
1
(1 + x)−1/2 = 1 − x +
x −
x + ...
2
2·4
2·4·6
∞
x
=
kxk
(1 − x)2
1.113
x2 < 1
k=1
1.114
1.
√
q
q
q
1+ 1+x =2 1+
1!
x
q(q − 3)
+
4
2!
x
4
2
q(q − 4)(q − 5) x 3
+
+ ...
4
2 3!
x < 1, q is a real number
AD (6351.1)
25
26
2.
The Exponential Function
x+
1 + x2
q
1.121
∞
q 2 q 2 − 22 q 2 − 42 . . . q 2 − (2k)2 x2k+2
=1+
(2k + 2)!
k=0
2
∞ 2
2
q −1
q − 32 . . . q 2 − (2k − 1)2 2k+1
+qx + q
x
(2k + 1)!
k=1
2
x < 1, q is a real number
AD(6351.2)
1.12 Series of rational fractions
1.121
∞
1.
∞
2k−1 x2
x2
x
=
k−1 =
2
1−x
1+x
1 − x2k
k−1
k=1
k−1
x2 < 1
AD (6350.3)
k=1
∞
2.
2k−1
1
=
x−1
x2k−1 + 1
x2 > 1
AD (6350.3)
k=1
1.2 The Exponential Function
1.21 Series representation
1.211
1.11
ex =
∞
xk
k=0
2.
ax =
k!
∞
k
(x ln a)
k!
k=0
3.
e−x =
2
∞
(−1)k
k=0
4.∗
ex = lim
n→∞
1.212 ex (1 + x) =
1+
x2k
k!
x
n
n
∞
xk (k + 1)
k=0
1.213
x
x
=1− +
ex − 1
2
x
1.214 ee = e 1 + x +
k!
∞
B2k x2k
k=1
2
(2k)!
5x3
15x4
2x
+
+
+ ...
2!
3!
4!
[x < 2π]
FI II 520
AD (6460.3)
1.215
3x4
8x5
3x6
56x7
x2
−
−
−
+
+ ...
2!
4!
5!
6!
7!
4x4
31x6
x2
+
−
+ ...
=e 1−
2!
4!
6!
1.
esin x = 1 + x +
AD (6460.4)
2.
ecos x
AD (6460.5)
1.232
Series of exponentials
etan x = 1 + x +
3.
27
3x3
9x4
37x5
x2
+
+
+
+ ...
2!
3!
4!
5!
AD (6460.6)
1.216
2x3
5x4
x2
+
+
+ ...
2!
3!
4!
x3
7x4
x2
−
−
+ ...
=1+x+
2!
3!
4!
1.
earcsin x = 1 + x +
AD (6460.7)
2.
earctan x
AD (6460.8)
1.217
1.
π
∞
1
eπx + e−πx
=
x
πx
−πx
2
e −e
x + k2
(cf. 1.421 3)
AD (6707.1)
(cf. 1.422 3)
AD (6707.2)
k=−∞
∞
(−1)k
2π
=
x
πx
−πx
e −e
x2 + k 2
2.
k=−∞
1.22 Functional relations
1.221
1.
ax = ex ln a
1
aloga x = a logx a = x
2.
1.222
1.
ex = cosh x + sinh x
2.
eix = cos x + i sin x
1.223 eax − ebx = (a − b)x exp
∞ 1
(a − b)2 x2
(a + b)x
1+
2
2k 2 π 2
k=1
1.23 Series of exponentials
1.231
∞
akx =
k=0
1
1 − ax
[a > 1 and x < 0 or 0 < a < 1 and x > 0]
1.232
1.
∞
(−1)k e−2kx
[x > 0]
(−1)k e−(2k+1)x
[x > 0]
tanh x = 1 + 2
k=1
2.
sech x = 2
∞
k=0
3.
cosech x = 2
∞
e−(2k+1)x
k=0
%
4.
∗
∞
cos2n x
sin x = exp −
2n
n=1
[x > 0]
&
[0 ≤ x ≤ π]
MO 216
28
Trigonometric and Hyperbolic Functions
1.311
1.3–1.4 Trigonometric and Hyperbolic Functions
1.30 Introduction
The trigonometric and hyperbolic sines are related by the identities
sinh x =
1
sin(ix),
i
sin x =
1
sinh(ix).
i
The trigonometric and hyperbolic cosines are related by the identities
cosh x = cos(ix),
cos x = cosh(ix).
Because of this duality, every relation involving trigonometric functions has its formal counterpart involving the corresponding hyperbolic functions, and vice versa. In many (though not all) cases, both pairs of
relationships are meaningful.
The idea of matching the relationships is carried out in the list of formulas given below. However, not
all the meaningful “pairs” are included in the list.
1.31 The basic functional relations
1.311
1 ix
e − e−ix
2i
= −i sinh(ix)
1.
sin x =
2.
sinh x =
3.
cos x =
4.
cosh x =
5.
6.
7.
8.
1 x
e − e−x
2
= −i sin(ix)
1 ix
e + e−ix
2
= cosh(ix)
1 x
e + e−x
2
= cos(ix)
1
sin x
= tanh(ix)
cos x
i
1
sinh x
= tan(ix)
tanh x =
cosh x
i
cos x
1
cot x =
=
= i coth(ix)
sin x
tan x
1
cosh x
=
= i cot (ix)
coth x =
sinh x
tanh x
tan x =
1.312
1.
cos2 x + sin2 x = 1
1.314
2.
The basic functional relations
cosh2 x − sinh2 x = 1
1.313
1.
sin (x ± y) = sin x cos y ± sin y cos x
2.
sinh (x ± y) = sinh x cosh y ± sinh y cosh x
3.
sin (x ± iy) = sin x cosh y ± i sinh y cos x
4.
sinh (x ± iy) = sinh x cos y ± i sin y cosh x
5.
cos (x ± y) = cos x cos y ∓ sin x sin y
6.
cosh (x ± y) = cosh x cosh y ± sinh x sinh y
7.
cos (x ± iy) = cos x cosh y ∓ i sin x sinh y
8.
cosh (x ± iy) = cosh x cos y ± i sinh x sin y
tan x ± tan y
tan (x ± y) =
1 ∓ tan x tan y
tanh x ± tanh y
tanh (x ± y) =
1 ± tanh x tanh y
tan x ± i tanh y
tan (x ± iy) =
1 ∓ i tan x tanh y
tanh x ± i tan y
tanh (x ± iy) =
1 ± i tanh x tan y
9.
10.
11.
12.
1.314
1.
2.
3.
4.
5.
6.
7.
8.
9.∗
1
1
(x ± y) cos (x ∓ y)
2
2
1
1
sinh x ± sinh y = 2 sinh (x ± y) cosh (x ∓ y)
2
2
1
1
cos x + cos y = 2 cos (x + y) cos (x − y)
2
2
1
1
cosh x + cosh y = 2 cosh (x + y) cosh (x − y)
2
2
1
1
cos x − cos y = 2 sin (x + y) sin (y − x)
2
2
1
1
cosh x − cosh y = 2 sinh (x + y) sinh (x − y)
2
2
sin (x ± y)
tan x ± tan y =
cos x cos y
sinh (x ± y)
tanh x ± tanh y =
cosh x cosh y
1
π
1
π
sin x ± cos y = ±2 sin (x + y) ±
sin (x − y) ±
4
4
2
2
1
π
1
π
= ±2 cos (x + y) ∓
cos (x − y) ∓
2
4 2
4
1
π
1
π
= 2 sin (x ± y) ±
cos (x ∓ y) ∓
2
4
2
4
sin x ± sin y = 2 sin
29
30
Trigonometric and Hyperbolic Functions
.
10.∗
a sin x ± b cos x = a 1 +
11.∗
±a sin x + b cos x = b
1+
2
b
a
sin x ± arctan
b
a
[a = 0]
a
b
2
+
a ,
cos x ∓ arctan
b
[b = 0]
2
r
r
1
a sin x ± b cos y = q 1 +
sin (x ± y) + arctan
q
2
q
1
1
[q = 0]
q = (a + b) cos (x ∓ y) , r = (a − b) sin (x ∓ y)
2
2
s
s 2
1
a cos x + b cos y = t 1 +
cos (x ∓ y) + arctan
[t = 0]
t
2
t
.
2
t
t
1
cos (x ∓ y) − arctan
= −s 1 +
[s = 0]
s
2
s
1
1
s = (a − b) sin (x ± y) , t = (a + b) cos (x ± y)
2
2
.
12.∗
13.∗
1.315
1.
sin2 x − sin2 y = sin(x + y) sin(x − y) = cos2 y − cos2 x
2.
sinh2 x − sinh2 y = sinh(x + y) sinh(x − y) = cosh2 x − cosh2 y
3.
cos2 x − sin2 y = cos(x + y) cos(x − y) = cos2 y − sin2 x
4.
sinh2 x + cosh2 y = cosh(x + y) cosh(x − y) = cosh2 x + sinh2 y
1.316
1.
2.
1.317
1.
2.
3.
4.
5.
1.315
n
(cos x + i sin x) = cos nx + i sin nx
n
(cosh x + sinh x) = sinh nx + cosh nx
x
1
sin = ±
(1 − cos x)
2
2
x
1
sinh = ±
(cosh x − 1)
2
2
x
1
cos = ±
(1 + cos x)
2
2
x
1
cosh =
(cosh x + 1)
2
2
1 − cos x
sin x
x
=
tan =
2
sin x
1 + cos x
[n is an integer]
[n is an integer]
1.321
6.
Trigonometric and hyperbolic functions: expansion in multiple angles
tanh
31
cosh x − 1
sinh x
x
=
=
2
sinh x
cosh x + 1
The signs in front of the radical in formulas 1.317 1, 1.317 2, and 1.317 3 are taken so as to agree
with the signs of the left-hand members. The sign of the left hand members depends in turn on the value
of x.
1.32 The representation of powers of trigonometric and hyperbolic functions in terms
of functions of multiples of the argument (angle)
1.320
1.
2.
n−1
2n
2n
n−k
sin
(−1)
2
cos 2(n − k)x +
k
n
k=0
n−1
2n
2n
(−1)n 2n
n−k
sinh x = 2n
(−1)
2
cosh 2(n − k)x +
k
n
2
2n
1
x = 2n
2
KR 56 (10, 2)
k=0
3.
sin2n−1 x =
1
22n−2
n−1
(−1)n+k−1
k=0
2n − 1
k
sin(2n − 2k − 1)x
2n − 1
(−1)n−1 (−1)n+k−1
sinh(2n − 2k − 1)x
k
22n−2
k=0
n−1
2n
2n
1
2n
cos x = 2n
2
cos 2(n − k)x +
k
n
2
k=0
n−1
2n
2n
1
2n
cosh x = 2n
2
cosh 2(n − k)x +
k
n
2
KR 56 (10, 4)
n−1
4.
5.
6.
sinh2n−1 x =
KR 56 (10, 1)
k=0
7.
8.
cos2n−1 x =
cosh
2n−1
n−1
1
22n−2
x=
k=0
1
22n−2
n−1
k=0
2n − 1
k
2n − 1
k
cos(2n − 2k − 1)x
cosh(2n − 2k − 1)x
Special cases
1.321
1.
2.
3.
4.
1
(− cos 2x + 1)
2
1
sin3 x = (− sin 3x + 3 sin x)
4
1
sin4 x = (cos 4x − 4 cos 2x + 3)
8
1
sin5 x =
(sin 5x − 5 sin 3x + 10 sin x)
16
sin2 x =
KR 56 (10, 3)
32
5.
6.
Trigonometric and Hyperbolic Functions
1
(− cos 6x + 6 cos 4x − 15 cos 2x + 10)
32
1
sin7 x =
(− sin 7x + 7 sin 5x − 21 sin 3x + 35 sin x)
64
sin6 x =
1.322
1.
sinh2 x =
2.
sinh3 x =
3.
sinh4 x =
4.
sinh5 x =
5.
sinh6 x =
6.
sinh7 x =
1
(cosh 2x − 1)
2
1
(sinh 3x − 3 sinh x)
4
1
(cosh 4x − 4 cosh 2x + 3)
8
1
(sinh 5x − 5 sinh 3x + 10 sinh x)
16
1
(cosh 6x − 6 cosh 4x + 15 cosh 2x + 10)
32
1
(sinh 7x − 7 sinh 5x + 21 sinh 3x + 35 sinh x)
64
1.323
1.
cos2 x =
2.
cos3 x =
3.
cos4 x =
4.
cos5 x =
5.
cos6 x =
6.
cos7 x =
1
(cos 2x + 1)
2
1
(cos 3x + 3 cos x)
4
1
(cos 4x + 4 cos 2x + 3)
8
1
(cos 5x + 5 cos 3x + 10 cos x)
16
1
(cos 6x + 6 cos 4x + 15 cos 2x + 10)
32
1
(cos 7x + 7 cos 5x + 21 cos 3x + 35 cos x)
64
1.324
1.
cosh2 x =
2.
cosh3 x =
3.
cosh4 x =
4.
cosh5 x =
5.
cosh6 x =
6.
cosh7 x =
1
(cosh 2x + 1)
2
1
(cosh 3x + 3 cosh x)
4
1
(cosh 4x + 4 cosh 2x + 3)
8
1
(cosh 5x + 5 cosh 3x + 10 cosh x)
16
1
(cosh 6x + 6 cosh 4x + 15 cosh 2x + 10)
32
1
(cosh 7x + 7 cosh 5x + 21 cosh 3x + 35 cosh x)
64
1.322
1.332
Trigonometric and hyperbolic functions: expansion in powers
33
1.33 The representation of trigonometric and hyperbolic functions of multiples of
the argument (angle) in terms of powers of these functions
1.331
1.7
n
n
sin nx = n cosn−1 x sin x −
cosn−3 x sin3 x +
cosn−5 x sin5 x − . . . ;
3
5
⎧
⎨
n − 2 n−3
2
cosn−3 x
= sin x 2n−1 cosn−1 x −
⎩
1
⎫
⎬
n − 3 n−5
n − 4 n−7
+
2
cosn−5 x −
2
cosn−7 x + . . .
⎭
2
3
AD (3.175)
[(n+1)/2]
2.
sinh nx = x
n
2k − 1
sinh2k−2 x coshn−2k+1 x
k=1
[(n−1)/2]
= sinh x
(−1)k
k=0
3.
n − k − 1 n−2k−1
coshn−2k−1 x
2
k
n
n
cosn−2 x sin2 x +
cosn−4 x sin4 x − . . . ;
2
4
n
n n − 3 n−5
= 2n−1 cosn x − 2n−3 cosn−2 x +
cosn−4 x
2
1
2
1
n n − 4 n−7
−
cosn−6 x + . . .
2
3
2
cos nx = cosn x −
AD (3.175)
[n/2]
4.3
cosh nx =
k=0
n
sinh2k x coshn−2k x
2k
[n/2]
=2
n−1
n
cosh x + n
k=1
1.332
1.
(−1)k
1
k
n − k − 1 n−2k−1
coshn−2k x
2
k−1
2
4n − 22 4n2 − 42
4n2 − 22
3
5
sin 2nx = 2n cos x sin x −
sin x +
sin x − . . .
3!
5!
2n − 2 2n−3 2n−3
2
= (−1)n−1 cos x 22n−1 sin2n−1 x −
sin
x
1!
(2n − 3)(2n − 4) 2n−5 2n−5
2
sin
x
+
2!
(2n − 4)(2n − 5)(2n − 6) 2n−7 2n−7
2
−
sin
x + ...
3!
AD (3.171)
AD (3.173)
34
2.
3.
Trigonometric and Hyperbolic Functions
1.333
(2n − 1)2 − 12
sin3 x
sin(2n − 1)x = (2n − 1) sin x −
3!
(2n − 1)2 − 12 (2n − 1)2 − 32
5
sin x − . . .
AD (3.172)
+
5!
2n − 1 2n−4 2n−3
2
= (−1)n−1 22n−2 sin2n−1 x −
sin
x
1!
(2n − 1)(2n − 4) 2n−6 2n−5
2
sin
x
+
2!
(2n − 1)(2n − 5)(2n − 6) 2n−8 2n−7
2
−
sin
x + ...
AD (3.174)
3!
4n2 4n2 − 22
4n2 4n2 − 2 4n2 − 42
4n2
sin2 x +
sin4 x −
sin6 x + . . .
cos 2nx = 1 −
2!
4!
6!
AD (3.171)
= (−1)n
2n 2n−3 2n−2
2
sin
x
22n−1 sin2n x −
1!
2n(2n − 3) 2n−5 2n−4
2n(2n − 4)(2n − 5) 2n−7 2n−6
2
2
+
sin
x−
sin
x + ...
2!
3!
AD (3.173)a
4.
(2n − 1)2 − 12
sin2 x
cos(2n − 1)x = cos x 1 −
2!
(2n − 1)2 − 12 (2n − 1)2 − 32
4
sin x − . . .
+
4!
2n − 3 2n−4 2n−4
2
= (−1)n−1 cos x 22n−2 sin2n−2 x −
sin
x
1!
(2n − 4)(2n − 5) 2n−6 2n−6
2
sin
x
+
2!
(2n − 5)(2n − 6)(2n − 7) 2n−8 2n−8
2
−
sin
x + ...
3!
AD (3.172)
AD (3.174)
By using the formulas and values of 1.30, we can write formulas for sinh 2nx, sinh(2n − 1)x, cosh 2nx,
and cosh(2n − 1)x that are analogous to those of 1.332, just as was done in the formulas in 1.331.
Special cases
1.333
1.
sin 2x = 2 sin x cos x
2.
sin 3x = 3 sin x − 4 sin3 x
sin 4x = cos x 4 sin x − 8 sin3 x
3.
4.
5.
sin 5x = 5 sin x − 20 sin3 x + 16 sin5 x
sin 6x = cos x 6 sin x − 32 sin3 x + 32 sin5 x
1.337
6.
Trigonometric and hyperbolic functions: expansion in powers
sin 7x = 7 sin x − 56 sin3 x + 112 sin5 x − 64 sin7 x
1.334
1.
sinh 2x = 2 sinh x cosh x
2.
sinh 3x = 3 sinh x + 4 sinh3 x
sinh 4x = cosh x 4 sinh x + 8 sinh3 x
3.11
5.11
sinh 5x = 5 sinh x + 20 sinh3 x + 16 sinh5 x
sinh 6x = cosh x 6 sinh x + 32 sinh3 x + 32 sinh5 x
6.
sinh 7x = 7 sinh x + 56 sinh3 x + 112 sinh5 x + 64 sinh7 x
4.
1.335
1.
cos 2x = 2 cos2 x − 1
2.
cos 3x = 4 cos3 x − 3 cos x
3.
cos 4x = 8 cos4 x − 8 cos2 x + 1
4.
cos 5x = 16 cos5 x − 20 cos3 x + 5 cos x
5.
cos 6x = 32 cos6 x − 48 cos4 x + 18 cos2 x − 1
6.
cos 7x = 64 cos7 x − 112 cos5 x + 56 cos3 x − 7 cos x
1.336
1.
cosh 2x = 2 cosh2 x − 1
2.
cosh 3x = 4 cosh3 x − 3 cosh x
3.
cosh 4x = 8 cosh4 x − 8 cosh2 x + 1
4.
cosh 5x = 16 cosh5 x − 20 cosh3 x + 5 cosh x
5.
cosh 6x = 32 cosh6 x − 48 cosh4 x + 18 cosh2 x − 1
6.
cosh 7x = 64 cosh7 x − 112 cosh5 x + 56 cosh3 x − 7 cosh x
1.337
1.∗
2.∗
3.∗
4.∗
5.∗
6.∗
cos 3x
cos3 x
cos 4x
cos4 x
cos 5x
cos5 x
cos 6x
cos6 x
sin 3x
cos3 x
sin 4x
cos4 x
= 1 − 3 tan2 x
= 1 − 6 tan2 x + tan4 x
= 1 − 10 tan2 x + 5 tan4 x
= 1 − 15 tan2 x + 15 tan4 x − tan6 x
= 3 tan x − tan3 x
= 4 tan x − 4 tan3 x
35
36
7.∗
8.∗
9.∗
10.∗
11.∗
12.∗
13.∗
14.∗
15.∗
16.∗
Trigonometric and Hyperbolic Functions
1.341
sin 5x
= 5 tan x − 10 tan3 x + tan5 x
cos5 x
sin 6x
= 6 tan x − 20 tan3 x + 6 tan5 x
cos6 x
cos 3x
= cot3 x − 3 cot x
sin3 x
cos 4x
= cot4 x − 6 cot2 x + 1
sin4 x
cos 5x
= cot5 x − 10 cot3 x + 5 cot x
sin5 x
cos 6x
= cot6 x − 15 cot4 x + 15 cot2 x − 1
sin6 x
sin 3x
= 3 cot2 x − 1
sin3 x
sin 4x
= 4 cot3 x − 4 cot x
sin4 x
sin 5x
= 5 cot4 x − 10 cot2 x + 1
sin5 x
sin 6x
= 6 cot5 x − 20 cot3 x + 6 cot x
sin6 x
1.34 Certain sums of trigonometric and hyperbolic functions
1.341
1.
n−1
sin(x + ky) = sin x +
k=0
2.
n−1
ny
y
n−1
y sin
cosec
2
2
2
sinh(x + ky) = sinh x +
k=0
3.
n−1
cos(x + ky) = cos x +
k=0
4.
n−1
k=0
5.
ny 1
n−1
y sinh
2
2 sinh y
2
ny
y
n−1
y sin
cosec
2
2
2
cosh(x + ky) = cosh x +
2n−1
(−1)k cos(x + ky) = sin x +
n−1
(−1)k sin(x + ky) = sin x +
k=0
AD (361.9)
ny 1
n−1
y sinh
2
2 sinh y
2
k=0
6.
AD (361.8)
y
2n − 1
y sin ny sec
2
2
n(y + π)
y
n−1
(y + π) sin
sec
2
2
2
JO (202)
AD (202a)
1.351
Sums of powers of trigonometric functions of multiple angles
37
Special cases
1.342
1.
2.10
n
k=1
n
sin kx = sin
nx
x
n+1
x sin
cosec
2
2
2
AD (361.1)
nx
x
n+1
x sin
cosec + 1
2
2
2
sin n + 12 x
nx
n+1
x
1
= cos
sin
x cosec =
1+
2
2
2
2
sin x2
cos kx = cos
k=0
AD (361.2)
3.
4.
n
k=1
n
sin(2k − 1)x = sin2 nx cosec x
1
sin 2nx cosec x
2
cos(2k − 1)x =
k=1
1.343
1.
2.
3.
1 (−1)n cos 2n+1
2 x
(−1) cos kx = − +
x
2
2 cos
k=1
2
n
sin 2nx
(−1)k+1 sin(2k − 1)x = (−1)n+1
2 cos x
n
k=1
n
k
cos(4k − 3)x +
k=1
AD (361.7)
n
JO (207)
AD (361.11)
AD (361.10)
sin(4k − 1)x = sin 2nx (cos 2nx + sin 2nx) (cos x + sin x) cosec 2x
k=1
JO (208)
1.344
1.
n−1
k=1
2.
n−1
k=1
3.
n−1
k=0
sin
π
πk
= cot
n
2n
√
nπ
nπ
n
2πk 2
=
1 + cos
− sin
sin
n
2
2
2
√
nπ
nπ
n
2πk 2
=
1 + cos
+ sin
cos
n
2
2
2
AD (361.19)
AD (361.18)
AD (361.17)
1.35 Sums of powers of trigonometric functions of multiple angles
1.351
1.
n
k=1
1
[(2n + 1) sin x − sin(2n + 1)x] cosec x
4
n cos(n + 1)x sin nx
= −
2
2 sin x
sin2 kx =
AD (361.3)
38
2.
Trigonometric and Hyperbolic Functions
n
k=1
1.352
n−1 1
+ cos nx sin(n + 1)x cosec x
2
2
n cos(n + 1)x sin nx
= +
2
2 sin x
cos2 kx =
AD (361.4)a
3.
n
sin3 kx =
n+1
nx
x 1
3(n + 1)x
3nx
3x
3
sin
x sin
cosec − sin
sin
cosec
4
2
2
2 4
2
2
2
JO (210)
cos3 kx =
n+1
nx
x 1
3(n + 1)
3nx
3x
3
cos
x sin
cosec + cos
x sin
cosec
4
2
2
2 4
2
2
2
JO (211)a
sin4 kx =
1
[3n − 4 cos(n + 1)x sin nx cosec x + cos 2(n + 1)x sin 2nx cosec 2x]
8
JO (212)
cos4 kx =
1
[3n + 4 cos(n + 1)x sin nx cosec x + cos 2(n + 1)x sin 2nx cosec 2x]
8
JO (213)
k=1
4.
n
k=1
5.
n
k=1
6.
n
k=1
1.352
1.11
2.11
n cos 2n−1
sin nx
2 x
−
2 sin x2
4 sin2 x2
k=1
n−1
n sin 2n−1
1 − cos nx
2 x
k cos kx =
−
2 sin x2
4 sin2 x2
n−1
k sin kx =
AD (361.5)
AD (361.6)
k=1
1.353
1.
n−1
pk sin kx =
k=1
2.
n−1
p sin x − pn sin nx + pn+1 sin(n − 1)x
1 − 2p cos x + p2
pk sinh kx =
k=1
3.
n−1
k=0
4.
n−1
k=0
pk cos kx =
p sinh x − pn sinh nx + pn+1 sinh(n − 1)x
1 − 2p cosh x + p2
1 − p cos x − pn cos nx + pn+1 cos(n − 1)x
1 − 2p cos x + p2
pk cosh kx =
AD (361.12)a
1 − p cosh x − pn cosh nx + pn+1 cosh(n − 1)x
1 − 2p cosh x + p2
AD (361.13)a¡
JO (396)
1.36 Sums of products of trigonometric functions of multiple angles
1.361
1.
n
sin kx sin(k + 1)x =
1
[(n + 1) sin 2x − sin 2(n + 1)x] cosec x
4
JO (214)
sin kx sin(k + 2)x =
1
n
cos 2x − cos(n + 3)x sin nx cosec x
2
2
JO (216)
k=1
2.
n
k=1
1.381
3.
Sums leading to hyperbolic tangents and cotangents
2
n
39
n(x + 2y)
x + 2y
n+1
x sin
cosec
2
2
2
n(2y − x)
2y − x
n+1
x sin
cosec
− sin ny −
2
2
2
sin kx cos(2k − 1)y = sin ny +
k=1
JO (217)
1.362
1.
n
x
2k
2
1
x
sec k
2k
2
2
2k sin2
k=1
2.
n
k=1
= 2n sin
2
x
2n
= cosec2 x −
− sin2 x
AD (361.15)
2
1
x
cosec n
2n
2
AD (361.14)
1.37 Sums of tangents of multiple angles
1.371
1.
n
1
x
1
x
tan k = n cot n − 2 cot 2x
k
2
2
2
2
AD (361.16)
n
1
x
22n+2 − 1
1
2 x
tan
=
+ 4 cot2 2x − 2n cot2 n
22k
2k
3 · 22n−1
2
2
AD (361.20)
k=0
2.
k=0
1.38 Sums leading to hyperbolic tangents and cotangents
1.381
⎞
⎛
⎟
1
⎟
⎠
2k + 1
2
π
n sin
4n
= tanh (2nx)
2
tanh x
1+
2k + 1
π
tan2
4n
⎞
⎛
⎜
tanh ⎜
⎝x
1.
n−1
k=0
⎜
tanh ⎜
⎝x
2.
n−1
k=1
1
kπ
2n
2
tanh x
1+
kπ
tan2
2n
n sin2
JO (402)a
⎟
⎟
⎠
= coth (2nx) −
1
(tanh x + coth x)
2n
JO (403)
40
Trigonometric and Hyperbolic Functions
⎞
⎛
⎟
⎟
⎠
2k
+
1
2
π
(2n + 1) sin
tanh x
2(2n + 1)
= tanh (2n + 1) x −
2n + 1
tanh2 x
1+
2k + 1
π
tan2
2(2n + 1)
⎞
⎛
⎜
tanh ⎜
⎝x
3.
n−1
k=0
⎜
tanh ⎜
⎝x
4.
1.382
2
⎟
⎟
⎠
2
kπ
2(2n + 1)
tanh2 x
1+
kπ
tan2
(2n + 1)
n
k=1
JO (404)
(2n + 1) sin2
= coth (2n + 1) x −
coth x
2n + 1
JO (405)
1.382
1.
n−1
k=0
2.
n−1
k=1
3.
n−1
k=0
4.
n
k=1
1
⎛
2k + 1
π
⎜ sin
4n
⎜
⎝
sinh x
⎞ = 2n tanh (nx)
2
1
x ⎟
⎟
tanh
2
2 ⎠
1
⎛
kπ
2
⎜ sin 2n
⎜
⎝ sinh x
⎛
2
⎜ sin
⎜
⎝
⎛
+
+
⎞ = 2n coth (nx) − 2 coth x
JO (407)
1
x ⎟
⎟
tanh
2
2 ⎠
1
2k + 1
π
2(2n + 1)
sinh x
kπ
2
⎜ sin 2n + 1
⎜
⎝
sinh x
JO (406)
1
⎞ = (2n + 1) tanh
+
1
x ⎟
⎟
tanh
2
2 ⎠
⎞ = (2n + 1) coth
+
x ⎟
1
⎟
tanh
2
2 ⎠
(2n + 1)x
2
(2n + 1)x
2
− tanh
− coth
x
2
x
2
JO (408)
JO (409)
1.395
Representing sines and cosines as ﬁnite products
41
1.39 The representation of cosines and sines of multiples of the angle as ﬁnite
products
1.391
1.
sin nx = n sin x cos x
n−2
2
k=1
2.
3.
4.
n
2
-
⎛
⎛
⎞
2
sin x ⎟
⎜
⎝1 −
kπ ⎠
sin2
n
⎞
sin2 x
⎟
⎠
(2k − 1)π
2
k=1
sin
2n
⎛
⎞
n−1
2
2
sin x ⎟
⎜
sin nx = n sin x
⎝1 −
kπ ⎠
k=1
sin2
n
⎛
⎞
n−1
2
2
- ⎜
sin x
⎟
cos nx = cos x
⎝1 −
⎠
(2k − 1)π
2
k=1
sin
2n
cos nx =
⎜
⎝1 −
[n is even]
JO (568)
[n is even]
JO (569)
[n is odd]
JO (570)
[n is odd]
JO (571)a
1.392
1.
sin nx = 2n−1
n−1
-
sin x +
kπ
n
JO (548)
sin x +
2k − 1
π
2n
JO (549)
k=0
2.
cos nx = 2n−1
n
k=1
1.393
1.
n−1
-
cos x +
k=0
1
2k
π = n−1 cos nx
n
2
n
1 = n−1 (−1) 2 − cos nx
2
[n odd]
[n even]
JO (543)
2.11
n−1
-
n−1
sin x +
k=0
(−1) 2
2k
π =
sin nx
n
2n−1
n
(−1) 2
= n−1 (1 − cos nx)
2
[n odd]
[n even]
JO (544)
1.394
n−1
k=0
2kπ
x − 2xy cos α +
n
2
+y
2
= x2n − 2xn y n cos nα + y 2n
1.395
1.
cos nx − cos ny = 2n−1
n−1
k=0
cos x − cos y +
2kπ
n
JO (573)
JO (573)
42
2.
Trigonometric and Hyperbolic Functions
cosh nx − cos ny = 2
n−1
n−1
k=0
2kπ
cosh x − cos y +
n
1.396
JO (538)
1.396
1.
n−1
-
kπ
+1
n
x2 − 2x cos
2kπ
+1
2n + 1
=
x2n+1 − 1
x−1
KR 58 (28.2)
x2 + 2x cos
2kπ
+1
2n + 1
=
x2n+1 − 1
x+1
KR 58 (28.3)
x2 − 2x cos
(2k + 1)π
+1
2n
= x2n + 1
KR 58 (28.4)
k=1
2.
n
k=1
3.
n
k=1
4.
n−1
-
x2n − 1
x2 − 1
x2 − 2x cos
k=0
=
KR 58 (28.1)
1.41 The expansion of trigonometric and hyperbolic functions in power series
1.411
1.
sin x =
∞
(−1)k
k=0
2.
sinh x =
x2k+1
(2k + 1)!
∞
x2k+1
(2k + 1)!
k=0
3.
cos x =
∞
(−1)k
k=0
4.
x2k
(2k)!
∞
x2k
cosh x =
(2k)!
k=0
∞
22k 22k − 1
|B2k |x2k−1
tan x =
(2k)!
6.11
π2
x2 <
4
k=1
∞
22k 22k − 1
2x5
17 7
x3
+
−
x + ··· =
B2k x2k−1
tanh x = x −
3
15
315
(2k)!
k=1
π2
2
x <
4
7.
1 22k |B2k | 2k−1
x
cot x = −
x
(2k)!
5.
∞
x2 < π 2
FI II 523
FI II 523a
k=1
∞
8.
coth x =
2x5
1 22k B2k 2k−1
1 x x3
+ −
+
− ··· = +
x
x 3
45 945
x
(2k)!
k=1
2
x < π2
FI II 522a
1.414
9.
Trigonometric and hyperbolic functions: power series expansion
sec x =
∞
|E2k |
k=0
(2k)!
π2
x2 <
4
x2k
43
CE 330a
∞
10.
sech x = 1 −
E2k
5x4
61x6
x2
+
−
+ ··· = 1 +
x2k
2
24
720
(2k)!
k=1
11.
∞
1 2 22k−1 − 1 |B2k |x2k−1
cosec x = +
x
(2k)!
π2
x2 <
4
x2 < π 2
CE 330
CE 329a
k=1
12.
∞
7x3
31x5
1 2 22k−1 − 1 B2k 2k−1
1 1
−
+ ··· = −
x
cosech x = − x +
x 6
360 15120
x
(2k)!
k=1
2
x < π2
JO (418)
1.412
1.
sin2 x =
∞
(−1)k+1
k=1
2.
cos x = 1 −
2
∞
22k−1 x2k
(2k)!
(−1)k+1
k=1
JO (452)a
22k−1 x2k
(2k)!
JO (443)
∞
3.
4.
32k+1 − 3 2k+1
1
x
(−1)k+1
4
(2k + 1)!
k=1
2k
∞
+ 3 x2k
1
3
k 3
cos x =
(−1)
4
(2k)!
sin3 x =
JO (452a)a
JO (443a)
k=0
1.413
1.
sinh x = cosec x
∞
(−1)k+1
k=1
2.
cosh x = sec x + sec x
∞
(−1)k
k=1
3.
sinh x = sec x
∞
(−1)[k/2]
k=1
4.
cosh x = cosec x
∞
k=1
1.414
1.
22k−1 x4k−2
(4k − 1)!
22k x4k
(4k)!
2k−1 x2k−1
(2k − 1)!
(−1)[(k−1)/2]
2k−1 x2k−1
(2k − 1)!
2
∞
,
+
n + 02 n2 + 22 . . . n2 + (2k)2 2k+2
x
cos n ln x + 1 + x2 = 1 −
(−1)k
(2k + 2)!
k=0
2
x <1
JO (508)
JO (507)
JO (510)
JO (509)
AD (6456.1)
44
Trigonometric and Hyperbolic Functions
1.421
2
2
2
2k+1
∞
,
+
2
2
2
n
n
.
.
.
n
x
+
1
+
3
+
(2k
−
1)
k+1
sin n ln x + 1 + x2 = nx − n
(−1)
(2k + 1)!
k=1
2
x <1
AD (6456.2)
2.
Power series for ln sin x, ln cos x, and ln tan x see 1.518.
1.42 Expansion in series of simple fractions
1.421
∞
1.
tan
πx
4x 1
=
2
π
(2k − 1)2 − x2
BR* (191), AD (6495.1)
k=1
∞
2.10
tanh
4x 1
πx
=
2
π
(2k − 1)2 + x2
k=1
3.
cot πx =
∞
∞
2x x 1
1
1
1
+
+
=
πx
π
x2 − k 2
πx π
k(x − k)
k=1
AD (6495.2), JO (450a)
k=−∞
k=0
∞
4.
2x 1
1
+
coth πx =
πx
π
x2 + k 2
(cf. 1.217 1)
k=1
∞
5.
tan2
πx
2(2k − 1)2 − x2
= x2
2
2
2
2
(12 − x2 ) (32 − x2 ) . . . [(2k − 1)2 − x2 ]
JO (450)
k=1
1.422
1.
∞
πx
4
2k − 1
sec
=
(−1)k+1
2
π
(2k − 1)2 − x2
k=1
2.
sec2
∞
4 πx
= 2
2
π
k=1
1
1
+
(2k − 1 − x)2
(2k − 1 + x)2
AD (6495.3)a
JO (451)a
∞
3.
cosec πx =
2x (−1)k
1
+
πx
π
x2 − k 2
(see also 1.217 2)
AD (6495.4)a
k=1
∞
∞
1
1 1
2 x2 + k 2
=
+
2
π2
(x − k)2
π 2 x2
π2
(x2 − k 2 )
k=−∞
k=1
4.
cosec2 πx =
5.
(−1)k+1
1 + x cosec x
1
= 2−
2
2x
x
(x2 − k 2 π 2 )
JO (446)
∞
JO (449)
k=1
6.
cosec πx =
∞
2 (−1)k
π
x2 − k 2
JO (450b)
k=−∞
∞
1.423
π
π
1 π
1
π2
+
cot
− =
cosec2
2
2
4m
m 4m
m 2
(1 − k 2 m2 )
k=1
JO (477)
1.439
Representation in the form of an inﬁnite product
45
1.43 Representation in the form of an inﬁnite product
1.431
1.
sin x = x
∞
-
x2
k2 π2
1−
k=1
2.
sinh x = x
∞
-
1+
k=1
3.
cos x =
∞
-
1−
k=0
4.
cosh x =
∞
-
x2
k2 π2
EU
4x2
(2k + 1)2 π 2
1+
k=0
EU
EU
4x2
(2k + 1)2 π 2
EU
1.432
1.
11
2.
x2
cos x − cos y = 2 1 − 2
y
cosh x − cos y = 2 1 +
x2
y2
∞
y sin
2
2
k=1
sin2
1−
x2
(2kπ + y)
2
∞
y x2
1+
2
(2kπ + y)2
k=1
1−
x2
(2kπ − y)2
1+
x2
(2kπ − y)2
∞ πx
πx
(−1)k x
− sin
=
1.433 cos
1+
4
4
2k − 1
k=1
%
&
∞
2 2
π + 2x
1
2
2
1.434 cos x = (π + 2x)
1−
4
2kπ
k=1
∞
x+a
sin π(x + a)
x
x
=
1.435
1−
1+
sin πa
a
k−a
k+a
%k=1
&
∞
2
x
sin2 πx
=
1−
1.436 1 −
k−a
sin2 πa k=−∞
%
&
∞
2
2x
sin 3x
=−
1.437
1−
sin x
x + kπ
k=−∞
%
&
∞
2
x
cosh x − cos a
=
1.438
1+
1 − cos a
2kπ + a
AD (653.2)
AD (653.1)
BR* 189
MO 216
MO 216
MO 216
MO 216
MO 216
k=−∞
1.439
1.
sin x = x
∞
k=1
2.
cos
x
2k
∞ x
sin x
4
=
1 − sin2 k
x
3
3
k=1
[|x| < 1]
AD (615), MO 216
MO 216
46
Trigonometric and Hyperbolic Functions
1.441
1.44–1.45 Trigonometric (Fourier) series
1.441
1.
∞
sin kx
k=1
2.
∞
cos kx
k=1
3.
k
k
π−x
2
1
= − ln [2 (1 − cos x)]
2
∞
(−1)k−1 sin kx
∞
=
k
k=1
4.
=
(−1)k−1
k=1
x
2
x
cos kx
= ln 2 cos
k
2
[0 < x < 2π]
FI III 539
[0 < x < 2π]
FI III 530a, AD (6814)
[−π < x < π]
FI III 542
[−π < x < π]
FI III 550
FI III 541
1.442
1.11
∞
sin(2k − 1)x
k=1
2.
2k − 1
∞
cos(2k − 1)x
k=1
2k − 1
=
π
sign x
4
[−π < x < π]
=
1
x
ln cot
2
2
[0 < x < π]
BR* 168, JO (266), GI III(195)
3.
∞
(−1)k−1
1
π x
sin(2k − 1)x
= ln tan
+
2k − 1
2
4
2
+ π
π,
− <x<
2
2
(−1)k−1
cos(2k − 1)x
=
2k − 1
+ π
π,
− <x<
2
2
3π
π
<x<
2
2
k=1
4.10
∞
k=1
π
4
π
=−
4
BR* 168, JO (268)a
BR* 168, JO (269)
1.443
1.8
∞
cos kπx
k=1
k 2n
π 2n (2n)!
2n
= (−1)n−1 22n−1
k=0
2n
k
B2n−k ρk
x
(2π)2n
B 2n
2 (2n)!
2
n−1 1
= (−1)
2.
∞
sin kπx
k=1
k 2n+1
= (−1)n−1 22n
2n+1
π 2n+1 (2n + 1)!
k=0
+
2n + 1
k
x
(2π)2n+1
B 2n+1
2 (2n + 1)!
2
0 ≤ x ≤ 2,
ρ=
x : x ;,
−
2
2
CE 340, GE 71
B2n−k+1 ρk
n−1 1
= (−1)
+
0 < x < 1;
ρ=
x : x ;,
−
2
2
CE 340
1.445
3.
Trigonometric (Fourier) series
∞
cos kx
k=1
4.
∞
k2
k=1
5.
∞
sin kx
k=1
6.
∞
cos kx
k=1
7.
1.
k4
∞
sin kx
k=1
1.444
k3
k5
πx x2
π2
−
+
6
2
4
[0 ≤ x ≤ 2π]
FI III 547
x2
cos kx
π2
−
=
k2
12
4
[−π ≤ x ≤ π]
FI III 544
=
(−1)k−1
=
x3
π 2 x πx2
−
+
6
4
12
[0 ≤ x ≤ 2π]
=
π 2 x2
πx3
x4
π4
−
+
−
90
12
12
48
[0 ≤ x ≤ 2π]
AD (6617)
=
πx4
x5
π 4 x π 2 x3
−
+
−
90
36
48
240
[0 ≤ x ≤ 2π]
AD (6818)
∞
sin 2(k + 1)x
k=1
k(k + 1)
= sin 2x − (π − 2x) sin2 x − sin x cos x ln 4 sin2 x
[0 ≤ x ≤ π]
2.
∞
k=1
3.
∞
k=1
4.
∞
∞
k=0
BR* 168, GI III (190)
π
cos 2(k + 1)x
= cos 2x −
− x sin 2x + sin2 x ln 4 sin2 x
k(k + 1)
2
[0 ≤ x ≤ π]
!
x !!
x
sin(k + 1)x
!
= sin x − (1 + cos x) − sin x ln !2 cos !
(−1)k
k(k + 1)
2
2
(−1)k
k=1
5.
47
(−1)k
!
x !!
x
cos(k + 1)x
!
= cos x − sin x − (1 + cos x) ln !2 cos !
k(k + 1)
2
2
sin(2k + 1)x π
= x
(2k + 1)2
4
π
= (π − x)
4
BR* 168
MO 213
MO 213
+ π
π,
− ≤x≤
2
2
π
3
≤x≤ π
2
2
MO 213
6.6
∞
FI III 546
+
π,
0≤x≤
2
JO (591)
∞
k sin kx
π sinh α(π − x)
=
k 2 + α2
2
sinh απ
[0 < x < 2π]
BR* 157, JO (411)
∞
1
cos kx
π cosh α(π − x)
− 2
=
2
2
k +α
2α
sinh απ
2α
[0 ≤ x ≤ 2π]
BR* 257, JO (410)
∞
k=1
1.445
1.
π
− |x|
2
[−π ≤ x ≤ π]
k=1
7.
cos(2k − 1)x
π
=
(2k − 1)2
4
1 π
cos 2kx
= − sin x
(2k − 1)(2k + 1)
2
4
k=1
2.
k=1
48
3.
Trigonometric and Hyperbolic Functions
∞
(−1)k cos kx
k 2 + α2
k=1
4.
∞
(−1)k−1
k=1
5.
=
1
π cosh αx
− 2
2α sinh απ
2α
k sin kx
π sinh αx
=
2
2
k +α
2 sinh απ
1.446
[−π ≤ x ≤ π]
FI III 546
[−π < x < π]
FI III, 546
+
,
if x = 2mπ, then
··· = 0
∞
k sin kx
sin {α[(2m + 1)π − x]}
=π
2
2
k −α
2 sin απ
k=1
[2mπ < x < (2m + 2)π,
6.
∞
[2mπ ≤ x ≤ (2m + 2)π,
(−1)k
k=1
8.
α not an integer]
MO 213
∞
+
,
if x = (2m + 1)π, then
··· = 0 ,
k sin kx
sin[α(2mπ − x)]
=π
k 2 − α2
2 sin απ
[(2m − 1)π < x < (2m + 1)π, α not an integer]
(−1)k
k=1
9.∗
MO 213
∞
cos kx
1
π cos [α {(2m + 1)π − x}]
=
−
2
2
2
k −α
2α
2
α sin απ
k=1
7.
α not an integer]
FI III 545a
cos kx
1
π cos[α(2mπ − x)]
=
−
k 2 − α2
2α2
2
α sin απ
[(2m − 1)π ≤ x ≤ (2m + 1)π, α not an integer]
FI III 545a
∞
einα
π eiβ(α−2π) sinh(γα) + eiβα sinh [γ(2π − α)]
=
(n − β)2 + γ 2
γ
cosh(2πγ) − cos(2πβ)
n=−∞
[0 ≤ α ≤ 2π]
1.446
∞
k=1
π
(−1)k+1 cos(2k + 1)x
1
= cos2 x − cos x
(2k − 1)(2k + 1)(2k + 3)
8
3
+ π
π,
− ≤x≤
2
2
BR* 256, GI III (189)
1.447
1.
∞
pk sin kx =
k=1
2.
∞
pk cos kx =
k=0
3.
1+2
∞
k=1
p sin x
1 − 2p cos x + p2
[|p| < 1]
FI II 559
[|p| < 1]
FI II 559
[|p| < 1]
FI II 559a, MO 213
1 − p cos x
1 − 2p cos x + p2
pk cos kx =
1 − p2
1 − 2p cos x + p2
1.449
Trigonometric (Fourier) series
49
1.448
1.
∞
pk sin kx
k
k=1
2.
∞
k=1
3.
∞
k=1
4.
∞
k=1
5.
∞
k=1
6.
= arctan
p sin x
1 − p cos x
1 pk cos kx
= − ln 1 − 2p cos x + p2
k
2
2p sin x
p2k−1 sin(2k − 1)x 1
= arctan
2k − 1
2
1 − p2
p2k−1 cos(2k − 1)x 1 1 + 2p cos x + p2
= ln
2k − 1
4 1 − 2p cos x + p2
(−1)k−1 p2k−1 sin(2k − 1)x 1 1 + 2p sin x + p2
= ln
2k − 1
4 1 − 2p sin x + p2
∞
(−1)k−1 p2k−1 cos(2k − 1)x
2k − 1
k=1
=
1
2p cos x
arctan
2
1 − p2
0 < x < 2π,
p2 ≤ 1
0 < x < 2π,
p2 ≤ 1
0 < x < 2π,
p2 ≤ 1
0 < x < 2π,
p2 ≤ 1
0 < x < π,
p2 ≤ 1
0 < x < π,
p2 ≤ 1
FI II 559
FI II 559
JO (594)
JO (259)
JO (261)
JO (597)
1.449
1.
∞
pk sin kx
k!
k=1
2.
∞
k=0
= ep cos x sin (p sin x)
pk cos kx
= ep cos x cos (p sin x)
k!
Let S(x) = − x1 cos x +
3.∗
∞
n=1
4.∗
5.∗
n2
1
x
and C(x) =
1
x
p2 ≤ 1
p2 ≤ 1
JO (486)
JO (485)
sin x.
n
π
S(nx) = [C(ax) − cot(πa)S(ax)]
2
−a
2
[0 < x < 2π,
a = 0, ±1, ±2, . . .]
[0 ≤ x ≤ 2π,
a = 0, ±1, ±2, . . .]
∞
1
1
π
[S(ax) − cot(πa)C(ax)]
C(nx) = 2 −
2 − a2
n
2a
2a
n=1
∞
(−1)n−1 n
π
S(nx) = cosec(πa)S(ax)
2 − a2
n
2
n=1
[−π < x < π,
a = 0, ±1, ±2, . . .]
50
Trigonometric and Hyperbolic Functions
6.∗
∞
(−1)n−1
1
π
cosec(πa)C(ax)
C(nx) = − 2 +
2 − a2
n
2a
2a
n=1
7.∗
,
πa
2n − 1
π+
S(nx)
=
C(ax)
+
tan
S(ax)
(2n − 1)2 − a2
4
2
n=1
8.∗
1.451
a = 0, ±1, ±2, . . .]
[−π < x < π,
∞
[0 < x < π,
∞
,
πa
1
π +
S(ax)
−
tan
C(ax)
C(nx)
=
−
(2n − 1)2 − a2
4a
2
n=1
a = 0, ±1, ±2, . . .]
9.∗
∞
πa
(−1)n−1
π
sec
S(ax)
S(nx) =
2 − a2
(2n
−
1)
4a
2
n=1
[0 ≤ x ≤ π, a = 0, ±1, ±2, . . .]
+ π
,
π
− ≤ x ≤ , a = 0, ±1, ±2, . . .
2
2
10.∗
∞
πa
(−1)n−1 (2n − 1)
π
C(nx) = sec
C(ax)
2
2
(2n − 1) − a
4
2
n=1
+ π
π
− ≤x≤ ,
2
2
,
a = 0, ±1, ±2, . . .
Fourier expansions of hyperbolic functions
1.451
1.
2.
∞ 2
1 + 02 12 + 22 . . . 12 + (2k)2
sinh x = cos x
sin2k+1 x
(2k + 1)!
k=0
∞ 2
1 + 12 12 + 32 . . . 12 + (2k − 1)2
sin2k x
cosh x = cos x + cos x
(2k)!
JO (504)
JO (503)
k=1
1.452
1.
sinh (x cos θ) = sec (x sin θ)
∞
x2k+1 cos(2k + 1)θ
(2k + 1)!
k=0
2.
cosh (x cos θ) = sec (x sin θ)
∞
k=0
3.
sinh (x cos θ) = cosec (x sin θ)
x2k cos 2kθ
(2k)!
cosh (x cos θ) = cosec (x sin θ)
x2 < 1
x2 < 1
JO (391)
JO (390)
∞
x2k sin 2kθ
k=1
4.
(2k)!
x2 < 1,
x sin θ = 0
x2 < 1,
x sin θ = 0
JO (393)
∞
x2k+1 sin(2k + 1)θ
k=0
(2k + 1)!
JO (392)
1.480
Lobachevskiy’s “Angle of Parallelism”
51
1.46 Series of products of exponential and trigonometric functions
1.461
1.
∞
e−kt sin kx =
k=0
2.
1+2
∞
sin x
1
2 cosh t − cos x
e−kt cos kx =
k=1
1.4629
⎡
∞
sin kx sin ky
k
k=1
sinh t
cosh t − cos x
e−2k|t|
[t > 0]
MO 213
[t > 0]
MO 213
⎤
x+y
2
+
sinh
t
1 ⎢
⎥
2
= ln ⎣
⎦
4
2
2 x−y
+ sinh t
sin
2
sin2
MO 214
1.463
x cos ϕ
∞
xn cos nϕ
cos (x sin ϕ) =
n!
n=0
1.
e
2.
ex cos ϕ sin (x sin ϕ) =
∞
xn sin nϕ
n!
n=1
x2 < 1
x2 < 1
AD (6476.1)
AD (6476.2)
1.47 Series of hyperbolic functions
1.471
1.
∞
sinh kx
k=1
2.
∞
cosh kx
k=0
3.
k!
∞
k=0
k!
= ecosh x sinh (sinh x) .
JO (395)
= ecosh x cosh (sinh x) .
JO (394)
1
1
(2m + 1)πx
(2m + 1)π
π3
tanh
+
x
tanh
=
(2k + 1)3 x
2
2x
16
1.472
1.
∞
pk sinh kx =
p sinh x
1 − 2p cosh x + p2
pk cosh kx =
1 − p cosh x
1 − 2p cosh x + p2
k=1
2.
∞
k=0
p2 < 1
p2 < 1
JO (396)
JO (397)a
1.48 Lobachevskiy’s “Angle of Parallelism” Π(x)
1.480
1.
Deﬁnition.
Π(x) = 2 arccot ex = 2 arctan e−x
[x ≥ 0]
LO III 297, LOI 120
52
2.
1.481
Trigonometric and Hyperbolic Functions
Π(x) = π − Π(−x)
2.
Functional relations
1
sin Π(x) =
cosh x
cos Π(x) = tanh x
3.
tan Π(x) =
1.
[x < 0]
1.481
LO III 183, LOI 193
LO III 297
LO III 297
4.
1
sinh x
cot Π(x) = sinh x
5.
sin Π(x + y) =
sin Π(x) sin Π(y)
1 + cos Π(x) cos Π(y)
LO III 297
6.
cos Π(x + y) =
cos Π(x) + cos Π(y)
1 + cos Π(x) cos Π(y)
LO III 183
LO III 297
LO III 297
1.482
Connection with the Gudermannian.
π
gd(−x) = Π(x) −
2
(Deﬁnite) integral of the angle of parallelism: cf. 4.581 and 4.561.
1.49 The hyperbolic amplitude (the Gudermannian) gd x
1.490
1.
Deﬁnition.
x
gd x =
0
2.
gd x
x=
0
1.491
dt
π
= 2 arctan ex −
cosh t
2
JA
gd x π
+
2
4
JA
dt
= ln tan
cos t
Functional relations.
1.
cosh x = sec(gd x)
AD (343.1), JA
2.
sinh x = tan(gd x)
AD (343.2), JA
3.
ex = sec(gd x) + tan(gd x) = tan
4.
tanh x = sin(gd x)
5.
tanh
6.
1
gd x
2
1
arctan (tanh x) = gd 2x
2
x
= tan
2
1.492 If γ = gd x, then ix = gd iγ
1.493 Series expansion.
π gd x
+
4
2
=
1 + sin(gd x)
cos(gd x)
AD (343.5), JA
AD (343.3), JA
AD (343.4), JA
AD (343.6a)
JA
∞
1.
x
gd x (−1)k
=
tanh2k+1
2
2k + 1
2
k=0
JA
1.513
Series representation
∞
2.
x 1
=
tan2k+1
2
2k + 1
k=0
3.
4.
53
1
gd x
2
JA
x5
61x7
x3
+
−
+ ···
6
24 5040
(gd x)5
61(gd x)7
(gd x)3
+
+
+ ...
x = gd x +
6
24
5040
gd x = x −
JA
+
π,
gd x <
2
JA
1.5 The Logarithm
1.51 Series representation
∞
1.511
xk
1
1
1
ln(1 + x) = x − x2 + x3 − x4 + · · · =
(−1)k+1
2
3
4
k
k=1
[−1 < x ≤ 1]
1.512
1.
∞
(x − 1)k
1
1
(−1)k+1
ln x = (x − 1) − (x − 1)2 + (x − 1)3 − · · · =
2
3
k
k=1
%
2.
ln x = 2
x−1 1
+
x+1 3
3
x−1
x+1
+
1
5
x−1
x+1
&
5
[0 < x ≤ 2]
+ ... = 2
∞
k=1
1
2k − 1
x−1
x+1
2k−1
[0 < x]
3.
4.∗
ln x =
ln x = lim
→0
1.513
1.
x−1 1
+
x
2
x−1
x
2
+
1
3
+ ··· =
∞
1
k
k=1
x−1
x
x ≥ 12
k
AD (644.6)
x −1
∞
ln
x−1
x
3
1
1+x
=2
x2k−1
1−x
2k − 1
x2 < 1
FI II 421
k=1
∞
2.
ln
1
x+1
=2
x−1
(2k − 1)x2k−1
x2 > 1
AD (644.9)
k=1
∞
3.
1
x
=
ln
x−1
kxk
[x ≤ −1 or x > 1]
JO (88a)
[−1 ≤ x < 1]
JO (88b)
[−1 ≤ x < 1]
JO (102)
k=1
∞
4.
ln
xk
1
=
1−x
k
k=1
∞
5.
1
xk
1−x
ln
=1−
x
1−x
k(k + 1)
k=1
54
6.
The Logarithm
1.514
∞
k
1
1
1
ln
=
xk
1−x 1−x
n
n=1
x2 < 1
JO (88e)
k=1
∞
7.
xk−1
1
(1 − x)2
1
3
=
+
ln
−
2x3
1−x
2x2
4x
k(k + 1)(k + 2)
[−1 ≤ x < 1]
AD (6445.1)
k=1
1.514
∞
cos kϕ k
x ;
ln 1 − 2x cos ϕ + x2 = −2
k
ln x +
k=1
(see 1.631, 1.641, 1.642, 1.646)
1.515
1.11
2.
3.
4.
ln 1 +
ln 1 +
1 + x2 = arcsinh x
2
x ≤ 1, x cos ϕ = 1
1·1 2 1·1·3 4 1·1·3·5 6
x −
x +
x − ...
2·2
2·4·4
2·4·6·6
∞
(2k − 1)! 2k
(−1)k
= ln 2 −
2x
22k (k!)
k=1
2
x ≤1
MO 98, FI II 485
1 + x2 = ln 2 +
1
1
1·3
−
+
− ...
x 2 · 3x3
2 · 4 · 5x5
∞
(2k − 1)!
1 (−1)k 2k−1
= ln x + +
x
2
· k!(k − 1)!(2k + 1)x2k+1
k=1
2
x ≥1
JO (91)
1 + x2 = ln x +
∞
22k−1 (k − 1)!k! 2k+1
x
1 + x2 ln x + 1 + x2 = x −
(−1)k
(2k + 1)!
k=1
2
x ≤1
√
∞
2
2
ln x + 1 + x2
22k (k!) 2k+1
√
x
x ≤1
=
(−1)k
2
(2k + 1)!
1+x
AD (644.4)
JO (93)
JO (94)
k=0
1.516
1.
∞
k
k+1 k+1 (∓1)
x
1
1
2
{ln (1 ± x)} =
2
k+1
n
n=1
∞
k
n
(−1)k+1 xk+2 1 1
1
3
{ln(1 + x)} =
6
k+2
n + 1 m=1 m
n=1
x2 < 1
JO (86), JO (85)
k=1
2.
x2 < 1
AD (644.14)
k=1
∞
2k−1
2
x2k (−1)n+1
x <1
k n=1
n
k=1
∞
xk−1
1
+
+ 2 ln(1 − x) =
2x
(2k − 1)2k(2k + 1)
3.
− ln(1 + x) · ln(1 − x) =
4.
1
4x
√
1+x 1+ x
√
√ ln
1− x
x
JO (87)
k=1
[0 < x < 1]
AD (6445.2)
1.521
Series of logarithms (cf. 1.431)
1.517
1.
6
1
2x
√
1−x
1 − ln(1 + x) − √ arctan x
x
=
∞
k=1
55
(−1)k+1 xk−1
(2k − 1)2k(2k + 1)
[0 < x ≤ 1]
2.
3.
1+x
1
arctan x ln
=
2
1−x
∞
k=1
4k−2 2k−1
x
2k − 1
n=1
(−1)n−1
2n − 1
∞
2k
(−1)k+1 x2k+1 1
1
arctan x ln 1 + x2 =
2
2k + 1
n
n=1
x2 < 1
x2 ≥ 1
AD (6445.3)
BR* 163
AD (6455.3)
k=1
1.518
1.
ln sin x = ln x −
= ln x +
x4
x6
x2
−
−
− ...
6
180 2835
∞
(−1)k 22k−1 B2k x2k
k(2k)!
k=1
[0 < x < π]
2.3
3.
2
4
6
x
x
17x
x
−
−
−
− ...
2
12 45 2520
∞
∞
2k−1
2k
2
2 − 1 |B2k | 2k
1 sin2k x
x =−
=−
k(2k)!
2
k
k=1
k=1
π2
x2 <
4
ln cos x = −
7
x2
62 6
127 8
+ x4 +
x +
x + ...
3
90
18, 900
2835
∞
22k−1 − 1 22k B2k x2k
= ln x +
(−1)k+1
k(2k)!
+
π,
0<x<
2
1.52 Series of logarithms (cf. 1.431)
1.521
∞
ln 1 −
4x2
(2k − 1)2 π 2
ln 1 −
x2
k2 π2
k=1
2.
FI II 524
ln tan x = ln x +
k=1
1.
AD (643.1)a
8
∞
k=1
= ln cos x
= ln sin x − ln x
+ π
π,
− <x<
2
2
[0 < x < π]
AD (643.3)a
56
The Inverse Trigonometric and Hyperbolic Functions
1.621
1.6 The Inverse Trigonometric and Hyperbolic Functions
1.61 The domain of deﬁnition
The principal values of the inverse trigonometric functions are deﬁned by the inequalities:
1.
2.
π
π
≤ arcsin x ≤ ; 0 ≤ arccos x ≤ π
2
2
π
π
− < arctan x < ; 0 < arccot x < π
2
2
−
[−1 ≤ x ≤ 1]
FI II 553
[−∞ < x < +∞]
FI II 552
1.62–1.63 Functional relations
1.621
1.
The relationship between the inverse and the direct trigonometric functions.
+
π,
π
arcsin (sin x) = x − 2nπ
2nπ − ≤ x ≤ 2nπ +
2
2
+
π,
π
= −x + (2n + 1)π
(2n + 1)π − ≤ x ≤ (2n + 1)π +
2
2
2.
arccos (cos x) = x − 2nπ
= −x + 2(n + 1)π
3.
arctan (tan x) = x − nπ
4.
arccot (cot x) = x − nπ
[2nπ ≤ x ≤ (2n + 1)π]
[(2n + 1)π ≤ x ≤ 2(n + 1)π]
+
π,
π
nπ − < x < nπ +
2
2
[nπ < x < (n + 1)π]
1.622 The relationship between the inverse trigonometric functions, the inverse hyperbolic functions,
and the logarithm.
1.
2.
3.
4.
5.
6.
7.
8.
1
1
ln iz + 1 − z 2 = arcsinh(iz)
i
i
1
1
arccos z = ln z + z 2 − 1 = arccosh z
i
i
1
1 + iz
1
arctan z =
ln
= arctanh(iz)
2i 1 − iz
i
1
iz − 1
arccot z =
ln
= i arccoth(iz)
2i iz + 1
1
arcsinh z = ln z + z 2 + 1 = arcsin(iz)
i
arccosh z = ln z + z 2 − 1 = i arccos z
arcsin z =
1
1 1+z
ln
= arctan(iz)
2 1−z
i
1 z+1
1
arccoth z = ln
= arccot(−iz)
2 z−1
i
arctanh z =
1.624
Functional relations
57
Relations between diﬀerent inverse trigonometric functions
1.623
1.
2.
1.624
1.
2.
3.
4.
5.
6.
7.
8.
9.
π
2
π
arctan x + arccot x =
2
arcsin x + arccos x =
NV 43
NV 43
1 − x2
= − arccos 1 − x2
arcsin x = arccos
arcsin x = arctan √
[0 ≤ x ≤ 1]
NV 47 (5)
[−1 ≤ x ≤ 0]
NV 46 (2)
x
1 − x2
√
1 − x2
√ x
1 − x2
−π
= arccot
x
arccos x = arcsin 1 − x2
= π − arcsin 1 − x2
arcsin x = arccot
√
1 − x2
x√
1 − x2
= π + arctan
x
x
arccos x = arccot √
1 − x2
x
arctan x = arcsin √
1 + x2
1
arctan x = arccos √
1 + x2
1
= − arccos √
1 + x2
arccos x = arctan
1
x
1
= − arccot − π
x
arctan x = arccot
1
10.11 arccot x = arcsin √
1 + x2
1
= π − arcsin √
1 + x2
x
11.
arccot x = arccos √
1 + x2
x2 < 1
[0 < x ≤ 1]
[−1 ≤ x < 0]
NV 49 (10)
[0 ≤ x ≤ 1]
[−1 ≤ x ≤ 0]
NV 48 (6)
[0 < x ≤ 1]
[−1 ≤ x < 0]
NV 48 (8)
[−1 ≤ x < 1]
NV 46 (4)
NV 6 (3)
[x ≥ 0]
[x ≤ 0]
NV 48 (7)
[x > 0]
[x < 0]
NV 49 (9)
[x > 0]
[x < 0]
NV 49 (11)
NV 46 (4)
58
12.
1.625
1.
The Inverse Trigonometric and Hyperbolic Functions
1
x
1
= π + arctan
x
arccot x = arctan
1.625
[x > 0]
[x < 0]
NV 49 (12)
arcsin x + arcsin y = arcsin x 1 − y 2 + y 1 − x2
= π − arcsin x 1 − y 2 + y 1 − x2
= −π − arcsin x 1 − y 2 + y 1 − x2
xy ≤ 0 or x2 + y 2 ≤ 1
x > 0,
y > 0 and x2 + y 2 > 1
x < 0,
y < 0 and x2 + y 2 > 1
NV 54(1), GI I (880)
2.
3.
1 − x2 1 − y 2 − xy
= − arccos
1 − x2 1 − y 2 − xy
arcsin x + arcsin y = arccos
√
x 1 − y 2 + y 1 − x2
arcsin x + arcsin y = arctan √
1 − x2 1 − y 2 − xy
√
x 1 − y 2 + y 1 − x2
+π
= arctan √
1 − x2 1 − y 2 − xy
√
x 1 − y 2 + y 1 − x2
−π
= arctan √
1 − x2 1 − y 2 − xy
[x ≥ 0,
y ≥ 0]
[x < 0,
y < 0]
xy ≤ 0 or x2 + y 2 < 1
NV 55
x > 0,
y > 0 and x2 + y 2 > 1
x < 0,
y < 0 and x2 + y 2 > 1
NV 56
4.
arcsin x − arcsin y = arcsin x 1 − y 2 − y 1 − x2
= π − arcsin x 1 − y 2 − y 1 − x2
= −π − arcsin x 1 − y 2 − y 1 − x2
xy ≥ 0 or x2 + y 2 ≤ 1
x > 0,
y < 0 and x2 + y 2 > 1
x < 0,
y > 0 and x2 + y 2 > 1
NV 55(2)
5.
6.
7.11
arcsin x − arcsin y = arccos x 1 − x2 1 − y 2 + xy
= − arccos
1 − x2 1 − y 2 + xy
1 − x2 1 − y 2
= 2π − arccos xy − 1 − x2 1 − y 2
arccos x + arccos y = arccos xy −
arccos x − arccos y = − arccos xy + 1 − x2 1 − y 2
= arccos xy + 1 − x2 1 − y 2
[xy > y]
[x < y]
NV 56
[x + y ≥ 0]
[x + y < 0]
NV 57 (3)
[x ≥ y]
[x < y]
NV 57 (4)
1.627
8.
Functional relations
x+y
1 − xy
x+y
= π + arctan
1 − xy
x+y
= −π + arctan
1 − xy
arctan x + arctan y = arctan
59
[xy < 1]
[x > 0,
xy > 1]
[x < 0,
xy > 1]
NV 59(5), GI I (879)
9.
x−y
1 + xy
x−y
= π + arctan
1 + xy
x−y
= −π + arctan
1 + xy
arctan x − arctan y = arctan
[xy > −1]
[x > 0,
xy < −1]
[x < 0,
xy < −1]
NV 59(6)
1.626
1.
1
|x| ≤ √
2 1
√ <x≤1
2
1
−1 ≤ x < − √
2
2 arcsin x = arcsin 2x 1 − x2
= π − arcsin 2x 1 − x2
= −π − arcsin 2x 1 − x2
NV 61 (7)
2.
3.
[0 ≤ x ≤ 1]
2 arccos x = arccos 2x2 − 1
= 2π − arccos 2x2 − 1
[−1 ≤ x < 0]
2x
1 − x2
2x
+π
= arctan
1 − x2
2x
−π
= arctan
1 − x2
2 arctan x = arctan
NV 61 (8)
[|x| < 1]
[x > 1]
[x < −1]
NV 61 (9)
1.627
1.
2.
arctan x + arctan
arctan x + arctan
1 π
=
2
x
π
=−
2
[x > 0]
[x < 0]
1−x π
=
4
1+x
3
=− π
4
GI I (878)
[x > −1]
[x < −1]
NV 62, GI I (881)
60
The Inverse Trigonometric and Hyperbolic Functions
1.628
1.628
1.
arcsin
2x
= −π − 2 arctan x
1 + x2
= 2 arctan x
[x ≤ −1]
[−1 ≤ x ≤ 1]
= π − 2 arctan x
[x ≥ 1]
NV 65
1−x
= 2 arctan x
1 + x2
= −2 arctan x
2
2.
arccos
[x ≥ 0]
[x ≤ 0]
NV 66
1
2x − 1
2x − 1
− arctan tan
π = E (x)
2
π
2
1.631 Relations between the inverse hyperbolic functions.
x
1.
arcsinh x = arccosh x2 + 1 = arctanh √
x2 + 1
√
x2 − 1
2
2.
arccosh x = arcsinh x − 1 = arctanh
x
x
1
1
3.
arctanh x = arcsinh √
= arccosh √
= arccoth
2
2
x
1−x
1−x
2
4.
arcsinh x ± arcsinh y = arcsinh x 1 + y ± y 1 + x2
5.
arccosh x ± arccosh y = arccosh xy ± (x2 − 1) (y 2 − 1)
1.629
6.
arctanh x ± arctanh y = arctanh
GI (886)
JA
JA
JA
JA
JA
x±y
1 ± xy
JA
1.64 Series representations
1.641
1.
2.
1 3
1·3 5
1·3·5 7
π
− arccos x = x +
x +
x +
x + ...
2
2·3
2·4·5
2·4·6·7
∞
1 1 3 2
(2k)!
x2k+1 = x F
, ; ;x
=
2k (k!)2 (2k + 1)
2 2 2
2
k=0
2
x ≤1
arcsin x =
1 3
1·3 5
x +
x − ...;
arcsinh x = x −
2·3
2·4·5
∞
(2k)!
x2k+1
(−1)k
=
2
2k
2
(k!)
(2k
+
1)
k=0
= x F 12 , 12 ; 32 ; −x2
x2 ≤ 1
FI II 479
FI II 480
1.645
Series representations
61
1.642
1.
1 1
1·3 1
−
+ ...
2 2x2
2 · 4 4x4
∞
(2k)!x−2k
(−1)k+1
= ln 2x +
2
22k (k!) 2k
k=1
arcsinh x = ln 2x +
[x ≥ 1]
AD (6480.2)a
2.
arccosh x = ln 2x −
∞
(2k)!x−2k
[x ≥ 1]
2
k=1
22k (k!) 2k
AD (6480.3)a
1.643
1.
x3
x5
x7
+
−
+ ...
3
5
7
∞
(−1)k x2k+1
=
2k + 1
arctan x = x −
k=0
2.
arctanh x = x +
x5
x3
+
+ ··· =
3
5
∞
k=0
x2k+1
2k + 1
x2 ≤ 1
x2 < 1
FI II 479
AD (6480.4)
1.644
1.
∞
x
(2k)!
2
2k
1 + x k=0 2 (k!)2 (2k + 1)
1 1 3
x2
x
, ; ;
=√
F
2 2 2 1 + x2
1 + x2
arctan x = √
x2
1 + x2
k
2
x <∞
AD (641.3)
2.
arctan x =
1
1
π
1
1
π
− + 3 − 5 + 7 − ··· = −
2
x 3x
5x
7x
2
∞
(−1)k
k=0
1
(2k + 1)x2k+1
AD (641.4)
1.645
∞
1.
π
1
1
1·3
π (2k)!x−(2k+1)
− −
−
− ··· = −
2
3
5
2
x 2 · 3x
2 · 4 · 5x
2
(k!) 22k (2k + 1)
k=0
1 1 3 1
1
π
, ; ;
= − F
2
x
2 2 2 x2
arcsec x =
x2 > 1
AD (641.5)
2.
2
(arcsin x) =
∞
2
22k (k!) x2k+2
(2k + 1)!(k + 1)
x2 ≤ 1
AD (642.2), GI III (152)a
k=0
3.
3
(arcsin x) = x3 +
3! 2
1
3 1+ 2
5!
3
x5 +
3! 2 2
1
1
3 · 5 1 + 2 + 2 x7 + . . .
7!
3 5 x2 ≤ 1
BR* 188, AD (642.2), GI III (153)a
62
The Inverse Trigonometric and Hyperbolic Functions
1.646
1.646
∞
1.
arcsinh
1
(−1)k (2k)!
x−2k−1
= arcosech x =
2
x
22k (k!) (2k + 1)
k=0
∞
(2k)!
2.
arccosh
2
1
= arcsech x = ln −
x
x
3.
arcsinh
2 (−1)k+1 (2k)! 2k
1
x
= arcosech x = ln +
2
x
x
22k (k!) 2k
2
22k (k!) 2k
k=1
x2k
x2 ≥ 1
AD (6480.5)
[0 < x ≤ 1]
AD (6480.6)
[0 < x ≤ 1]
AD (6480.7)a
∞
k=1
4.
1.647
1.
2.
arctanh
1
= arccoth x =
x
∞
k=0
−(2k+1)
x
2k + 1
x2 > 1
AD (6480.8)
⎛
∗
n
∗
(−1)j−1 22j − 1 24n−2j+4 − 1 B2j−1
B4n−2j+3
⎝2
=
(2k − 1)4n+3
2
(2j)!(4n − 2j + 4)!
j=1
k=1
2n+2
2 ∗ 2 ⎞
n
(−1) 2
− 1 B2n+1
⎠
+
2
[(2n + 2)!]
n = 0, 1, 2, . . . ,
⎛
⎞
∞
n−1
j ∗
∗
k−1
4n+1
∗
n ∗ 2
(−1)
B
B
2B4n
(−1) B2n ⎠
(−1)
sech(2k − 1) (π/2) π
2j 4n−2j
+
+
,
= 4n+3 ⎝2
(2k − 1)4n+1
2
(2j)!(4n
−
2j)!
(4n)!
[(2n)]!2
j=1
∞
tanh(2k − 1) (π/2)
π
4n+3
k=1
n = 1, 2, . . .
(The summation term on the right is to be omitted for n = 1.) (See page xxxiii for the deﬁnition of Br∗ .)
2 Indeﬁnite Integrals of Elementary
Functions
2.0 Introduction
2.00 General remarks
We omit the constant of integration in all the formulas of this chapter. Therefore, the equality sign (=)
means that the functions on the left and right of this symbol diﬀer by a constant. For example (see
201 15), we write
dx
= arctan x = − arctan x
1 + x2
although
π
arctan x = − arctan x + .
2
we !integrate
certain
functions, we obtain the logarithm of the absolute value (for example,
!
√
" When
√ dx
= ln !x + 1 + x2 !). In such formulas, the absolute-value bars in the argument of the logarithm
1+x2
are omitted for simplicity in writing.
In certain cases, it is important to give the complete form of the primitive function. Such primitive
functions, written in the form of deﬁnite integrals, are given in Chapter 2 and in other chapters.
Closely related to these formulas are formulas in which the limits of integration and the integrand
depend on the same parameter.
A number of formulas lose their meaning for certain values of the constants (parameters) or for certain
relationships between these constants (for example, formula 2.02 8 for n = −1 or formula 2.02 15 for
a = b). These values of the constants and the relationships between them are for the most part completely
clear from the very structure of the right-hand member of the formula (the one not containing an integral
sign). Therefore, throughout the chapter, we omit remarks to this eﬀect. However, if the value of the
integral is given by means of some other formula for those values of the parameters for which the formula
in question loses meaning, we accompany this second formula with the appropriate explanation.
The letters x, y, t, . . . denote independent variables; f , g, ϕ, . . . denote functions of x, y, t, . . . ; f ,
g , ϕ , . . . , f , g , ϕ , . . . denote their ﬁrst, second, etc., derivatives; a, b, m, p, . . . denote constants, by
which we generally mean arbitrary real numbers. If a particular formula is valid only for certain values
of the constants (for example, only for positive numbers or only for integers), an appropriate remark is
made, provided the restriction that we make does not follow from the form of the formula itself. Thus,
in formulas 2.148 4 and 2.424 6, we make no remark since it is clear from the form of these formulas
themselves that n must be a natural number (that is, a positive integer).
63
64
Introduction
2.01 The basic integrals
1.
2.
3.
4.
5.
6.
11
7.
8.
xn+1
(n = −1)
x dx =
n+1
dx
= ln x
x
ex dx = ex
ax
ax dx =
ln a
sin x dx = − cos x
cos x dx = sin x
dx
= − cot x
sin2 x
dx
= tan x
cos2 x
n
11
16.
17.
18.
19.
20.
21.
22.11
23.
24.
25.
26.
dx
1 1+x
= arctanh x = ln
2
1−x
2 1−x
dx
√
= arcsin x = − arccos x
1 − x2
dx
√
= arcsinh x = ln x + x2 + 1
x2 + 1
dx
√
= arccosh x = ln x + x2 − 1
x2 − 1
sinh x dx = cosh x
cosh x dx = sinh x
dx
= − coth x
sinh2 x
dx
= tanh x
cosh2 x
tanh x dx = ln cosh x
coth x dx = ln sinh x
x
dx
= ln tanh
sinh x
2
9.
10.
sin x
dx = sec x
cos2 x
cos x
dx = − cosec x
sin2 x
tan x dx = − ln cos x
11.
12.
cot x dx = ln sin x
13.
14.
15.
x
dx
= ln tan
sin x
2
π x
dx
= ln tan
+
= ln (sec x + tan x)
cos x
4
2
dx
π
= arctan x = − arccot x
2
1+x
2
General formulas
65
2.02 General formulas
1.
af dx = a f dx
[af ± bϕ ± cψ ± . . .] dx = a f dx ± b ϕ dx ± c ψ dx ± . . .
d
f dx = f
dx
f dx = f
f ϕ dx = f ϕ − f ϕ dx
[integration by parts]
(n+1)
(n)
(n−1)
(n−2)
n (n)
n+1
f
ϕ dx = ϕf − ϕ f
+ϕ f
− . . . + (−1) ϕ f + (−1)
ϕ(n+1) f dx
f (x) dx = f [ϕ(y)]ϕ (y) dy
[x = ϕ(y)]
[change of variable]
2.
3.
4.
5.
6.
7.
11
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
(f )n f dx =
(f )n+1
n+1
For n = −1
f dx
= ln f
f
(af + b)n+1
(af + b)n f dx =
a(n + 1)
√
f dx
2 af + b
√
=
a
af + b
f
f ϕ−ϕf
dx =
ϕ2
ϕ
f
f ϕ − ϕ f
dx = ln
fϕ
ϕ
dx
dx
dx
=±
∓
f (f ± ϕ)
fϕ
ϕ (f ± ϕ)
f dx
= ln f + f 2 + a
f2 + a
a
b
f dx
dx
dx
=
−
(f + a)(f + b)
a − b (f + a) a − b (f + b)
For
a
=
b
dx
dx
f dx
−a
=
(f + a)2
f +a
(f + a)2
f dx
dx
ϕ dx
=
−
(f + ϕ)n
(f + ϕ)n−1
(f + ϕ)n
qf
f dx
1
arctan
=
2
2
2
p +q f
pq
p
[n = −1]
66
Rational Functions
2.101
qf − p
f dx
1
ln
=
q 2 f 2 − p2
2pq qf + p
f dx
dx
= −x +
1−f
1−f
1 f dx
1 f dx
f 2 dx
+
=
f 2 − a2
2 f −a 2 f +a
f
f dx
= arcsin
a
a2 − f 2
1
f
f dx
= ln
2
af + bf
b af + b
f
1
f dx
= arcsec
2
2
a
a
f f −a
(f ϕ − f ϕ ) dx
f
= arctan
2
2
f +ϕ
ϕ
1 f −ϕ
(f ϕ − f ϕ ) dx
= ln
f 2 − ϕ2
2 f +ϕ
18.
19.
20.
21.
22.
23.
24.
25.
2.1 Rational Functions
2.10 General integration rules
F (x)
, where F (x) and f (x) are polynomials with
f (x)
no common factors, we ﬁrst need to separate out the integral part E(x) [where E(x) is a polynomial],
if
there is an integral part, and then to integrate separately the integral part and the remainder; thus:
F (x) dx
ϕ(x)
= E(x) dx +
dx.
f (x)
f (x)
Integration of the remainder, which is then a proper rational function (that is, one in which the degree
of the numerator is less than the degree of the denominator) is based on the decomposition of the fraction
into elementary fractions, the so-called partial fractions.
2.102 If a, b, c, . . . , m are roots of the equation f (x) = 0 and if α, β, γ, . . . , μ are their corresponding
ϕ(x)
can be decomposed into the following
multiplicities, so that f (x) = (x−a)α (x−b)β . . . (x−m)μ , then
f (x)
partial fractions:
ϕ(x)
Aα
Bβ
Aα−1
A1
Bβ−1
B1
=
+
+ ...
+
+ ... +
+
+ ... +
α
α−1
β
β−1
f (x) (x − a)
(x − a)
x − a (x − b)
(x − b)
x−b
Mμ−1
M1
Mμ
,
+
+ ... +
+
μ
μ−1
(x − m)
(x − m)
x−m
2.101
To integrate an arbitrary rational function
where the numerators of the individual fractions are determined by the following formulas:
(k−1)
(a)
ψ1
,
(k − 1)!
ϕ(x)(x − a)α
ψ1 (x) =
,
f (x)
Aα−k+1 =
(k−1)
(b)
ψ2
,
(k − 1)!
ϕ(x)(x − b)β
ψ2 (x) =
,
f (x)
Bβ−k+1 =
(k−1)
...,
...,
ψm (m)
,
(k − 1)!
ϕ(x)(x − m)μ
ψm (x) =
f (x)
Mμ−k+1 =
2.104
General integration rules
67
TI 51a
If a, b, . . . , m are simple roots, that is, if α = β = . . . = μ=1, then
ϕ(x)
A
B
M
=
+
+ ··· +
,
f (x)
x−a x−b
x−m
where
ϕ(a)
ϕ(b)
ϕ(m)
A= ,
B= ,
... ,
M= .
f (a)
f (b)
f (m)
If some of the roots of the equation f (x) = 0 are imaginary, we group together the fractions that represent
conjugate roots of the equation. Then, after certain manipulations, we represent the corresponding pairs
of fractions in the form of real fractions of the form
M1 x + N 1
M2 x + N 2
Mp x + N p
+
p.
2 + ... +
x2 + 2Bx + C
(x2 + 2Bx + C)
(x2 + 2Bx + C)
g dx
ϕ(x)
reduces to integrals of the form
2.103 Thus, the integration of a proper rational fraction
f
(x)
(x
− a)α
Mx + N
or
p dx. Fractions of the ﬁrst form yield rational functions for α > 1 and logarithms
(A + 2Bx + Cx2 )
for α = 1. Fractions of the second form yield rational functions and logarithms or arctangents:
d(x − a)
g
g dx
=g
=−
1.
(x − a)α
(x − a)α
(α − 1)(x − a)α−1
d(x − a)
g dx
=g
= g ln |x − a|
2.
x−a
x−a
Mx + N
N B − M A + (N C − M B)x
3.
p dx =
p−1
(A + 2Bx + Cx2 )
2(p − 1) (AC − B 2 ) (A + 2Bx + Cx2 )
(2p − 3)(N C − M B)
dx
+
p−1
2(p − 1) (AC − B 2 )
(A + 2Bx + Cx2 )
dx
1
Cx + B
4.
=√
arctan √
for AC > B 2
2
2
A + 2Bx + Cx2
AC − B
Ac − B
!
!
! Cx + B − √B 2 − AC !
1
!
!
√
= √
ln !
! for AC B 2
2
2
2C
C AC − B
AC − B
!
!
! Cx + B − √B 2 − AC !
!
N
C
−
M
B
M !!
!
!
√
√
ln A + 2Bx + Cx2 ! +
=
ln !
! for AC < B 2
2
2
!
!
2C
2C B − AC
Cx + B + B − AC
The Ostrogradskiy–Hermite method
ϕ(x)
dx
f (x)
without ﬁnding the roots of the equation f (x) = 0 and without decomposing the integrand into partial
fractions:
2.104
By means of the Ostrogradskiy-Hermite method, we can ﬁnd the rational part of
68
Rational Functions
2.110
M
N dx
ϕ(x)
dx =
+
FI II 49
f (x)
D
Q
Here, M , N , D, and Q are rational functions of x. Speciﬁcally, D is the greatest common divisor of
f (x)
the function f (x) and its derivative f (x); Q =
; M is a polynomial of degree no higher than m − 1,
D
where m is the degree of the polynomial D; N is a polynomial of degree no higher than n − 1, where
n is the degree of the polynomial Q. The coeﬃcients of the polynomials M and N are determined by
equating the coeﬃcients of like powers of x in the following identity:
ϕ(x) = M Q − M (T − Q ) + N D
f (x)
where T =
and M and Q are the derivatives of the polynomials M and Q.
D
2.11–2.13 Forms containing the binomial a + bxk
2.110
1.
Reduction formulas for zk = a + bxk and an explicit expression for the general case.
amk
xn+1 zkm
+
xn zkm dx =
xn zkm−1 dx
km + n + 1 km + n + 1
p
(ak)s (m + 1)m(m − 1) . . . (m − s + 1)zkm−s
xn+1 =
m + 1 s=0 [mk + n + 1][(m − 1)k + n + 1] . . . [(m − s)k + n + 1]
(ak)p+1 m(m − 1) . . . (m − p + 1)(m − p)
xn zkm−p−1 dx
+
[mk + n + 1][(m − 1)k + n + 1] . . . [(m − p)k + n + 1]
LA 126(4)
3.
4.
5.
6.
7.∗
8.∗
km + k + n + 1
xn zkm+1 dx
ak(m + 1)
ak(m + 1)
bkm
xn+1 zkm
n m
−
x zk dx =
xn+k zkm−1 dx
n+1
n+1
xn+1−k zkm+1
n+1−k
−
xn zkm dx =
xn−k zkm+1 dx
bk(m + 1)
bk(m + 1)
xn+1−k zkm+1
a(n + 1 − k)
−
xn−k zkm dx
xn zkm dx =
b(km + n + 1) b(km + n + 1)
xn+1 zkm+1
b(km + k + n + 1)
n m
−
x zk dx =
xn+k zkm dx
a(n + 1)
a(n + 1)
b c k−i
k
i
b
k
n +n
nk (−1) k! Γ a+1
n
b
a+1
xa+1+ib
x nx + c dx =
b i=0
(k − i)! Γ b + i + 1
xn zkm dx =
2.
−xn+1 zkm+1
+
xn zkm dx =
a m−i
LA 126 (6)
LA 125 (2)
LA 126 (3)
LA 126 (5)
[a, b, k ≥ 0 are all integers]
m
xk(J+i+1)
bm (−1)i m!J! xk + b
k i=0
(m − i)!(J + i + 1)!
J=
n+1
−1
k
[a, b, k, m, n real,
k = 0,
m ≥ 0 an integer]
Forms containing the binomial a + bxk
2.114
Forms containing the binomial z1 = a + bx
2.111
1.
2.
3.8
4.
5.
6.
2.113
z1m+1
b(m + 1)
For m = −1
1
dx
= ln z1
z1
b
n
n−1
dx
na
xn
x dx
x
−
= m−1
z1m
z1m
z1 (n + 1 − m)b (n + 1 − m)b
For n = m − 1, we may use the formula
m−1
dx
1 xm−2 dx
xm−1
x
+
= − m−1
z1m
z1 (m − 1)b b
z1m−1
For m = 1
n
x dx
axn−1
an−1 x (−1)n an
xn
a2 xn−2
−
=
+
− . . . + (−1)n−1
+
ln z1
2
3
z1
nb
(n − 1)b
(n − 2)b
1 · bn
bn+1
n
n−1
n−1
kak−1 xn−k
an
x dx n−1
n+1 na
=
(−1)k−1
+
(−1)
+
(−1)
ln z1
2
z1
(n − k)bk+1
bn+1 z1
bn+1
k=1
a
x dx
x
= − 2 ln z1
z1
b
b
2
ax a2
x dx
x2
− 2 + 3 ln z1
=
z1
2b
b
b
z1m dx =
1.
2.
3.
2.114
1.
2.
3.
4.6
dx
1
=−
2
z1
bz1
x dx
x
1
a
1
=−
+ 2 ln z1 = 2 + 2 ln z1
z12
bz1
b
b z1
b
x2 dx
x
a2
2a
=
−
− 3 ln z1
2
2
3
z1
b
b z1
b
dx
1
=−
z13
2bz12
+x
a , 1
x dx
+
=
−
z13
b
2b2 z12
2
2ax 3a2 1
x dx
1
=
+
+ 3 ln z1
z13
b2
2b3 z12
b
3
3
x
a 2
a2
5 a3 1
a
x dx
+ 2 2x − 2 3x −
=
− 3 4 ln z1
3
2
4
z1
b
b
b
2 b z1
b
69
70
2.115
1.
2.
3.
4.
2.116
1.
2.
3.
4.
2.117
Rational Functions
dx
1
=−
z14
3bz13
+x
a , 1
x dx
+ 2 3
=−
4
z1
2b 6b z1
2
2
x
ax
a2 1
x dx
+
=
−
+
z14
b
b2
3b3 z13
3
3ax2
x dx
9a2 x 11a3 1
1
=
+
+
+ 4 ln z1
z14
b2
2b2
6b4 z13
b
dx
1
=−
5
z1
4bz14
+x
a , 1
x dx
+
=
−
z15
3b 12b2 z14
2
2
x
1
ax
x dx
a2
+ 2+
=−
z15
2b
3b
12b3 z14
3
3
x
3ax2
x dx
a2 x
a3 1
+
=
−
+
+
z15
b
2b2
b3
4b4 z14
1.
2.
3.
4.
dx
−1
b(2 − n − m)
=
+
xn z1m
a(n − 1)
(n − 1)axn−1 z1m−1
dx
1
=−
m
z1
(m − 1)bz1m−1
1
dx
1
+
= m−1
xz1m
z1 a(m − 1) a
dx
xz1m−1
n−1
(−1)k bk−1
dx
(−1)n bn−1 z1
=
+
ln
xn z1
(n − k)ak xn−k
an
x
k=1
2.118
1.
2.
3.
2.119
1.
dx
1 z1
= − ln ,
xz1
a
x
b
dx
1
z1
+ 2 ln
=−
2
x z1
ax a
x
b2
1
b
z1
dx
=−
+ 2 − 3 ln
3
2
x z1
2ax
a x a
x
dx
1
1
z1
=
− 2 ln
2
xz1
az1
a
x
dx
xn−1 z1m
2.115
Forms containing the binomial a + bxk
2.124
1
2b 1
dx
2b z1
+
=−
+ 3 ln
x2 z12
ax a2 z1
a
x
3b2 1
dx
1
3b
3b2 z1
= −
+ 2 + 3
− 4 ln
2
3
2
x z1
2ax
2a x
a
z1
a
x
2.
3.
2.121
1.
2.
3.
2.122
1.
2.
3.
2.123
1.11
2.
3.
2.124
1.
2.
3
dx
bx 1
1
z1
+
=
− 3 ln
3
2
2
xz1
2a a z1
a
x
2
1
9b
dx
3b x 1
3b z1
+
=−
+ 3
+ 4 ln
x2 z13
ax 2a2
a
z12
a
x
9b2
dx
1
2b
6b3 x 1
6b2 z1
= −
+ 2 + 3 + 4
− 5 ln
3
2
3
2
x z1
2ax
a x
a
a
z1
a
x
11 5bx b2 x2 1
dx
1
z1
+
=
+ 3
− 4 ln
xz14
6a 2a2
a
z13
a
x
1
22b 10b2 x 4b3 x2 1
dx
4b z1
+
=
−
+
+
+ 5 ln
x2 z14
ax 3a2
a3
a4
z13
a
x
2
3
4 2
1
55b
dx
1
5b
25b x 10b x
10b2 z1
+
=
−
+
+
+
−
ln
x3 z14
2ax2
2a2 x
3a3
a4
a5
z13
a6
x
25
13bx 7b2 x2
b3 x3 1
1
z1
dx
+
=
+
+
− 5 ln
xz15
12a
3a2
2a3
a4 z14
a
x
2
3 2
4 3
1
125b 65b x 35b x
dx
1
5b x
5b z1
−
= −
−
−
−
+ 6 ln
5
4
2
2
3
4
5
x z1
ax 12a
3a
2a
a
z1
a
x
2
3
4 2
5 3
1
125b
dx
1
3b
65b x 105b x
15b x
15b2 z1
= −
+ 2 +
+
+
+
− 7 ln
5
4
3
2
3
4
5
6
x z1
2ax
a x
4a
a
2a
a
z1
a
x
Forms containing the binomial z2 = a + bx2 .
b
dx
1
if [ab > 0]
(see also 2.141 2)
= √ arctan x
z2
a
ab
√
a + xi ab
1
√
if [ab < 0]
(see also 2.143 2 and 2.1433)
= √ ln
2i ab a − xi ab
x dx
1
=−
(see also 2.145 2, 2.145 6, and 2.18)
m
z2
2b(m − 1)z2m−1
71
72
Rational Functions
2.125
Forms containing the binomial z3 = a + bx3
a
Notation: α = 3
b
2.125
n
n−3
dx
(n − 2)a
xn−2
x dx
x
−
=
1.
m−1
m
m
z3
z3
z3 (n + 1 − 3m)b b(n + 1 − 3m)
n
n
n+1
x dx
x
n + 4 − 3m x dx
2.
=
−
m
m−1
z3
3a(m − 1)
3a(m − 1)z3
z3m−1
2.126
1.
2.
3.
4.
5.
2.127
2.
3.
4.
1.
2.
√ √
(x + α)2
x 3
1
ln
+ 3 arctan
2 x2 − αx + α2
2α − x
2
√
(x + α)
1
2x − α
√
ln 2
+
3
arctan
2 x − αx + α2
α 3
2
(x + α)
1
ln
−
2 x2 − αx + α2
x dx
1
=−
z3
3bα
√
(see also 2.141 3 and 2.143)
2x − α
√
3 arctan
α 3
(see also 2.145 3. and 2.145 7)
2
−3
x dx
1
ln 1 + x3 α
=
z3
3b
3
x dx
x a dx
= −
z3
b
b z3
4
2
x dx
a x dx
x
−
=
z3
2b
b
z3
1.
2.128
dx
α
=
z3
3a
α
=
3a
LA 133 (1)
dx
x
2
=
+
z32
3az3
3a
x2
1
x dx
=
+
2
z3
3az3
3a
x3 dx
x
1
=−
+
2
z3
3bz3
3b
(see 2.126 2)
(see 2.126 1)
x dx
z3
1
ln z3
3b
(see 2.126 1)
dx
z3
x2 dx
1
=−
z32
3bz3
=
dx
z3
(see 2.126 2)
(see 2.126 1)
dx
1
b(3m + n − 4)
dx
=
−
−
m
m−1
n
n−3
n−1
x z3
a(n − 1)
x
z3m
(n − 1)ax
z3
dx
1
n + 3m − 4
dx
=
+
xn z3m
3a(m − 1) xn z3m−1
3a(m − 1)xn−1 z3m−1
LA 133 (2)
Forms containing the binomial a + bxk
2.133
2.129
1.
2.
3.
2.131
1.
2.
3.
x3
1
dx
ln
=
xz3
3a z3
b x dx
1
dx
−
=
−
x2 z3
ax a
z3
dx
1
b dx
=−
−
x3 z3
2ax2
a z3
(see 2.126 2)
(see 2.126 1)
1
1
x3
dx
=
+
ln
xz32
3az3
3a2
z3
2
1
1
4bx
dx
4b
x dx
+
=
−
−
x2 z32
ax
3a2 z3
3a2
z3
1
5bx 1
5b
dx
dx
=−
+ 2
− 2
2
3
2
x z3
2ax
6a z3
3a
z3
(see 2.126 2)
(see 2.126 1)
Forms containing the binomial z4 = a + bx4
a
a
4
α = 4 −
Notation: α =
b
b
2.132
√
√ 2
2
+
αx
2
+
α
2
x
α
αx
dx
√
1.8
ln
= √
+ 2 arctan 2
for ab > 0
z4
α − x2
4a 2
x2 − αx 2 + α2
x
α
x + α
+
2
arctan
for ab < 0
=
ln
4a
x − α
α
b
1
x dx
2
for ab > 0
(see
= √ arctan x
2.
z4
a
2 ab
√
a + x2 i ab
1
√
for ab < 0
(see
= √ ln
4i ab a − x2 i ab
√
√ 2
x dx
x2 − αx 2 + α2
1
αx 2
√
√
3.
ln
=
+ 2 arctan 2
for ab > 0
z4
α − x2
4bα 2
x2 + αx 2 + α2
x
x + α
1
− 2 arctan ln
for ab < 0
=−
4bα
x − α
α
3
1
x dx
ln z4
=
4.
z4
4b
2.133
1.
2.
73
xn+1
4m − n − 5
xn dx
=
+
m
m−1
z4
4a(m − 1)
4a(m − 1)z4
(see also 2.141 4)
(see also 2.143 5)
also 2.145 4)
also 2.145 8)
xn dx
z4m−1
n
n−4
dx
(n − 3)a
x dx
xn−3
x
−
= m−1
z4m
z4m
z4 (n + 1 − 4m)b b(n + 1 − 4m)
LA 134 (1)
74
2.134
Rational Functions
1.
2.
3.
4.
x
3
dx
=
+
z42
4az4
4a
x dx
x2
1
=
+
z42
4az4
2a
x2 dx
x3
1
=
+
2
z4
4az4
4a
2.134
dx
z4
(see 2.132 1)
x dx
z4
(see 2.132 2)
x2 dx
z4
(see 2.132 3)
x3 dx
x4
1
=
=−
z42
4az4
4bz4
dx
1
b(4m + n − 5)
2.135
=−
−
m
m−1
n
n−1
x z4
(n − 1)a
(n − 1)ax
z4
For n = 1
dx
dx
dx
1
b
=
−
m
m−1
−3
xz4
a xz4
a x z4m
2.136 dx
1
x4
ln x ln z4
1.
−
=
ln
=
xz4
a
4a
4a z4
2
b x dx
dx
1
−
2.
=−
2
x z4
ax a
z4
dx
xn−4 z4m
(see 2.132 3)
2.14 Forms containing the binomial 1 ± xn
2.141 1.
2.11
4.
dx
= arctan x = − arctan
1 + x2
1
x
(see also 2.124 1)
√
1+x
x 3
1
1
dx
(see also 2.126 1)
= ln √
+ √ arctan
1 + x3
3
2−x
3
1 − x + x2
√
√
dx
1 + x 2 + x2
x 2
1
1
√
√
√
ln
arctan
=
+
1 + x4
1 − x2
4 2 1 − x 2 + x2
2 2
3.
dx
= ln(1 + x)
1+x
(see also 2.132 1)
2.142
n
−1
2
dx
2
=
−
Pk cos
1 + xn
n
k=0
n
−1
2
2k + 1
2
π +
Qk sin
n
n
k=0
2k + 1
π
n
for n a positive even number
TI (43)a
n−3
2
=
2
1
ln(1 + x) −
Pk cos
n
n
k=0
n−3
2
2k + 1
2
π +
Qk sin
n
n
k=0
2k + 1
π
n
for n a positive odd number
TI (45)
Forms containing the binomial 1 ± xn
2.145
75
where
2k + 1
1
ln x2 − 2x cos
π +1
2
n
x − cos 2k+1
x sin 2k+1
π
π
n2k+1 = arctan
2k+1n Qk = arctan
1 − x cos n π
sin n π
Pk =
2.143
1.
2.
3.
4.
5.
dx
= − ln(1 − x)
1−x
dx
1 1+x
= arctanh x
[−1 < x < 1]
(see also 2.141 1)
= ln
2
1−x
2 1−x
1 x−1
dx
= ln
= − arccoth x
[x > 1, x < −1]
2
x −1
2 x+1
√
√
dx
1
1 + x + x2
x 3
1
+ √ arctan
(see also 2.126 1)
= ln
1 − x3
3
1−x
2+x
3
dx
1
1 1+x 1
+ arctan x = (arctanh x + arctan x)
= ln
4
1−x
4 1−x 2
2
(see also 2.132 1)
2.144
1.
n
−1
n
−1
2
2
2
2
dx
1 1+x
2k
2k
ln
−
π
+
=
P
cos
Qk sin π
k
1 − xn
n 1−x n
n
n
n
k=1
k=1
for n a positive even number
where Pk =
2.
1
2k + 1
ln x2 + 2x cos
π+1 ,
2
n
Qk = arctan
n−3
n−3
k=0
k=0
x + cos 2k+1
n π
sin 2k+1
n π
2
2
2
2
dx
1
2k + 1
2k + 1
ln(1
−
x)
+
π
+
π
=
−
P
cos
Qk sin
k
n
1−x
n
n
n
n
n
for n a positive odd number
where Pk =
2.145
1.
2.
3.
TI (47)
1
2k
ln x2 − 2x cos π + 1 ,
2
n
Qk = arctan
x − cos
2k
n π
sin 2k
n π
x dx
= x − ln(1 + x)
1+x
x dx
1 = ln 1 + x2
2
1+x
2
(1 + x)2
2x − 1
1
1
x dx
ln
=
−
+ √ arctan √
1 + x3
6 1 − x + x2
3
3
(see also 2.126 2)
TI (49)
76
Rational Functions
2.146
4.
5.
6.
7.
8.
2.146
1.
x dx
1
= arctan x2
1 + x4
2
x dx
= − ln(1 − x) − x
1−x
x dx
1 = − ln 1 − x2
2
1−x
2
(1 − x)2
x dx
2x + 1
1
1
ln
=
−
− √ arctan √
1 − x3
6 1 + x + x2
3
3
x dx
1 1 + x2
= ln
1 − x4
4 1 − x2
(see also 2.126 2)
(see also 2.132 2)
For m and n natural numbers.
m−1
n
x
2k − 1
dx
1 mπ(2k − 1)
2
ln
π
+
x
1
−
2x
cos
=
−
cos
1 + x2n
2n
2n
2n
k=1
n
2k−1
x − cos 2n π
mπ(2k − 1)
1
arctan
sin
+
n
2n
sin 2k−1
2n π
k=1
[m < 2n]
2.
n
k=1
3.11
TI (44)a
1
2k − 1
dx
ln(1 + x)
x
mπ(2k − 1)
−
ln 1 − 2x cos
π + x2
= (−1)m+1
cos
2n+1
1+x
2n + 1
2n + 1
2n + 1
2n + 1
k=1
n
2k−1
x − cos 2n+1
π
mπ(2k − 1)
2 arctan
sin
+
2k−1
2n + 1
2n + 1
sin 2n+1 π
m−1
[m ≤ 2n]
TI (46)a
n−1
kπ
1 1 kmπ
xm−1 dx
m+1
ln 1 − 2x cos
+ x2
(−1)
=
ln(1
+
x)
−
ln(1
−
x)
−
cos
2n
1−x
2n
2n
n
n
k=1
n−1
x − cos kπ
1
kmπ
n
+
arctan
sin
n
n
sin kπ
n
k=1
4.
[m < 2n]
TI (48)
m−1
dx
x
1
ln(1 − x)
=−
1 − x2n+1
2n + 1
n
2k − 1
1 mπ(2k − 1)
+(−1)m+1
ln 1 + 2x cos
π + x2
cos
2n + 1
2n + 1
2n + 1
k=1
n
2k−1
x + cos 2n+1
π
2
mπ(2k − 1)
arctan
sin
+(−1)m+1
2k−1
2n + 1
2n + 1
sin 2n+1 π
k=1
[m ≤ 2n]
2.147 1 xm dx
1 xm dx
xm dx
=
+
1.
1 − x2n
2 1 − xn
2 1 + xn
m−2
xm−1
x
dx
xm dx
1
m−1
·
2.
+
n =−
2n − m − 1 (1 + x2 )n−1
2n − m − 1 (1 + x2 )n
(1 + x2 )
TI (50)
LA 139 (28)
Forms containing the binomial 1 ± xn
2.149
3.
4.
5.
2.148 1.
2.
3.
4.
xm−1
xm
−
dx
=
1 + x2
m−1
77
xm−2
dx
1 + x2
m−2
xm−1
x
dx
1
m−1
xm dx
=
−
n
n−1
2
2
2
2n − m − 1 (1 − x )
2n − m − 1 (1 − x )n
(1 − x )
m−1
x
1
m−1
xm−2 dx
=
−
2n − 2 (1 − x2 )n−1
2n − 2 (1 − x2 )n−1
m
m−1
x dx
x
+
=−
2
1−x
m−1
LA 139 (33)
m−2
x
dx
1 − x2
dx
1
1
2n + m − 3
−
n =−
m − 1 xm−1 (1 + x2 )n−1
m−1
xm (1 + x2 )
dx
n
xm−2 (1 + x2 )
LA 139 (29)
For m = 1
1
dx
1
dx
LA 139 (31)
n =
n−1 +
n−1
2
2
2n
−
2
x (1 + x )
(1 + x )
x (1 + x2 )
For
m = 1 and n = 1
x
dx
= ln √
2
x (1 + x )
1 + x2
1
dx
dx
=
−
−
m
2
m−1
m−2
x (1 + x )
(m − 1)x
x
(1 + x2 )
dx
x
dx
1
2n − 3
=
+
FI II 40
n
2n − 2 (1 + x2 )n−1
2n − 2 (1 + x2 )n−1
(1 + x2 )
n−1
dx
x (2n − 1)(2n − 3)(2n − 5) · · · (2n − 2k + 1)
(2n − 3)!!
arctan x
=
+ n−1
n
n−k
2
k
2
2n
−
1
2
(n − 1)!
(1 + x )
k=1 2 (n − 1)(n − 2) . . . (n − k) (1 + x )
TI (91)
2.149 1.
2.
3.
dx
1
2n + m − 3
n =−
n−1 +
m−1
2
m−1
xm (1 − x2 )
(m − 1)x
(1 − x )
dx
n
xm−2 (1 − x2 )
LA 139 (34)
For m = 1
dx
1
dx
LA 139 (36)
n =
n−1 +
n−1
2
x (1 − x2 )
2(n − 1) (1 − x )
x (1 − x2 )
For
m = 1 and n = 1
x
dx
= ln √
2
x (1 − x )
1 − x2
x
dx
1
2n − 3
dx
=
+
LA 139 (35)
n
n−1
2
2
2n − 2 (1 − x )
2n − 2 (1 − x2 )n−1
(1 − x )
n−1
1+x
dx
x (2n − 1)(2n − 3)(2n − 5) . . . (2n − 2k + 1)
(2n − 3)!!
ln
+ n
n =
n−k
k
2
2n − 1
2
·
(n
−
1)!
1
−x
(1 − x2 )
k=1 2 (n − 1)(n − 2) . . . (n − k) (1 − x )
TI (91)
78
Rational Functions
2.151
2.15 Forms containing pairs of binomials: a + bx and α + βx
Notation:
t = α + βx; Δ = aβ − αb
z = a + bx; n+1
mΔ
tm
z
n m
−
2.151
z t dx =
z n tm−1 dx
(m + n + 1)b (m + n + 1)b
2.152
z
bx
Δ
1.
dx =
+ 2 ln t
t
β
β
βx Δ
t
dx =
− 2 ln z
2.
z
b
b
m
m−1
tm
dx
t dx
1
mΔ
t
2.153
=
−
zn
(m − n + 1)b z n−1
(m − n + 1)b
zn
tm+1
1
(m − n + 2)β tm dx
=
−
n−1
(n − 1)Δ z n−1
(n − 1)Δ
m−1z
tm
1
mβ
t
=−
+
dx
(n − 1)b z n−1
(n − 1)b z n−1
1
t
dx
=
ln
2.154
zt
Δ
z
1
1
(m + n − 2)b
dx
dx
=
−
−
2.155
z n tm
(m − 1)Δ tm−1 z n−1
(m − 1)Δ tm−1 z n
1
1
(m + n − 2)β
dx
=
+
(n − 1)Δ tm−1 z n−1
(n − 1)Δ
tm z n−1
1 a
α
x dx
=
ln z − ln t
2.156
zt
Δ b
β
2.16 Forms containing the trinomial a + bxk + cx2k
2.160
1.
2.
3.
2.161
Reduction formulas for Rk = a + bxk + cx2k .
xm Rkn+1
(m + k + nk)b
(m + 2k + 2kn)c
m−1 n
m+k−1 n
x
−
Rk dx =
Rk dx −
x
xm+2k−1 Rkn dx
ma
ma
ma
bkn
xm Rkn
2ckn
m−1 n
m+k−1 n−1
−
Rk dx =
Rk dx −
x
x
xm+2k−1 Rkn−1 dx
m
m
m
xm−2k Rkn+1
(m − 2k)a
(m − k + kn)b
xm−1 Rkn dx =
−
xm−2k−1 Rkn dx −
xm−k−1 Rkn dx
(m + 2kn)c
(m + 2kn)c
(m
+
2kn)c
2kna
xm Rkn
bkn
+
=
xm−1 Rkn−1 dx +
xm+k−1 Rkn−1 dx
m + 2kn m + 2kn
m + 2kn
Forms containing the trinomial R2 = a + bx2 + cx4 .
b 1 2
b 1 2
−
b − 4ac, g = +
b − 4ac,
2 2
2 2
a
h = b2 − 4ac, q = 4 , l = 2a(n − 1) b2 − 4ac ,
c
Notation: f =
b
cos α = − √
2 ac
Forms containing a + bx + cx2 and powers of x
2.172
1.
dx
R2
2
dx
dx
−
h >0
LA 146 (5)
2
2
cx + f
cx + g
2
x2 − q 2
1
α
α x2 + 2qx cos α2 + q 2
h < 0 LA 146 (8)a
ln
arctan
=
+
2
cos
sin
α
4cq 3 sin α
2 x2 − 2qx cos 2 + q 2
2
2qx sin α2
2
x dx
cx2 + f
1
h >0
LA 146 (6)
ln 2
=
R2
2h cx + g
2
2
2
x − q cos α
1
h <0
LA 146 (9)a
arctan
=
2
2
2cq sin α
q sin α
2
2
f
x dx
g
dx
dx
−
h >0
=
LA 146 (7)
2
2
R2
h cx + g
h cx + f
bcx3 + b2 − 2ac x b2 − 6ac dx bc x2 dx
dx
=
+
+
R22
lR2
l
R2
l
R2
2
2
3
bcx + b − 2ac x (4n − 7)bc x dx 2(n − 1)h2 + 2ac − b2
dx
dx
=
+
+
n
2
n−1
R2
lRn−1
l
l
R2
R2n−1
=
2.
3.
4.
5.
79
6.9
c
h
1
(m + 2n − 3)b
dx
=−
−
xm R2n
(m − 1)a
(m − 1)axm−1 R2n−1
[n > 1]
(m + 4n − 5)bc
dx
−
xm−2 R2n
(m − 1)a
LA 146
dx
xm−4 R2n
LA 147 (12)a
2.17 Forms containing the quadratic trinomial a + bx + cx2 and powers of x
Notation: R = a + bx + cx2 ; Δ = 4ac − b2
2.171 am
xm Rn+1
b(m + n + 1)
m+1 n
m−1 n
−
R dx =
R dx −
1.
x
x
xm Rn dx
c(m + 2n + 2) c(m + 2n + 2)
c(m + 2n + 2)
2.
3.
4.
TI (97)
R dx
R
b(n − m + 1) R dx c(2n − m + 2) R dx
=−
+
+
LA 142(3), TI (96)a
m+1
m
x
amx
am
xm
am
xm−1
dx
b + 2cx (4n − 2)c dx
=
+
TI (94)a
Rn+1
nΔRn
nΔ
Rn
n−1
n
dx
(2cx + b) 2k(2n + 1)(2n − 1)(2n − 3) . . . (2n − 2k + 1)ck
dx
n (2n − 1)!!c
=
+
2
n+1
k+1
n−k
n
R
2n + 1
n(n − 1) · · · (n − k)Δ
R
n!Δ
R
n
n+1
n
n
k=0
2.17211
TI (96)a
√
1
−Δ − (b + 2cx)
−2
b + 2cx
dx
√
=√
ln
=√
arctanh √
R
−Δ (b + 2cx) + −Δ
−Δ
−Δ
−2
=
b + 2cx
b + 2cx
2
= √ arctan √
Δ
Δ
for [Δ < 0]
for [Δ = 0, b and c non-zero]
for [Δ > 0]
80
Rational Functions
2.173
2.173 dx
b + 2cx 2c dx
+
1.
=
R2
ΔR
Δ
R
dx
1
b + 2cx
3c
6c2 dx
2.
=
+
+
R3
Δ
2R2
ΔR
Δ2
R
2.174
1.
2.
2.175
1.
2.
3.
4.
5.
6.
m−1
m−2
xm dx
dx
dx
xm−1
(n − m)b
(m − 1)a
x
x
=
−
−
+
n
n−1
n
n
R
(2n − m − 1)cR
(2n − m − 1)c
R
(2n − m − 1)c
R
For m = 2n − 1, this formula is inapplicable. Instead, we may use
2n−1
x
dx
1 x2n−3 dx a x2n−3 dx b x2n−2 dx
=
−
−
Rn
c
Rn−1
c
Rn
c
Rn
1
b
x dx
dx
=
ln R −
R
2c
2c R
x dx
b
2a + bx
dx
−
=−
R2
ΔR
Δ R
x dx
2a + bx 3b(b + 2cx) 3bc dx
− 2
=−
−
R3
2ΔR2
2Δ2 R
Δ
R
2
x dx
x
b
b2 − 2ac dx
= − 2 ln R +
R
c
2c
2c2
R
2
2
ab + b − 2ac x 2a dx
x dx
+
=
R2
cΔR
Δ
R
2
2
ab + b − 2ac x
2ac + b2 (b + 2cx)
x dx
+
=
+
R3
2cΔR2
2cΔ2 R
7.
8.
9.
2.176
1.
(see 2.172)
2.177
(see 2.172)
(see 2.172)
(see 2.172)
(see 2.172)
(see 2.172)
2ac + b2
Δ2
b b2 − 3ac
x3 dx
x2
bx b2 − ac
dx
=
− 2 +
ln R −
3
3
R
2c
c
2c
2c
R
dx
R
(see 2.172)
(see 2.172)
2
a 2ac − b + b 3ac − b x b 6ac − b2
x dx
1
dx
−
= 2 ln R +
2
2
2
R
2c
c ΔR
2c Δ
R
3
3
x dx
=−
R3
dx
xm Rn
(see 2.172)
=
2
2
2
x
abx 2a
+
+
c
cΔ
cΔ
−1
(m − 1)axm−1 Rn−1
1
x2
b
dx
=
ln
−
xR
2a
R
2a
dx
R
1
3ab
−
2R2
2cΔ
(see 2.172)
dx
(see 2.173 1)
R2
dx
dx
b(m + n − 2)
c(m + 2n − 3)
−
−
m−1
n
m−2
a(m − 1)
x
R
a(m − 1)
x
Rn
(see 2.172)
2.177
Quadratic trinomials and binomials
2.
3.
1
1
x2
dx
+
=
ln
2
2
xR
2a
R
2aR
2
dx
1
b
1
1
x
−
=
+ 2 + 3 ln
xR3
4aR2
2a R 2a
R
2a
4.
5.
b
− 2
2a
2ac
1+
Δ
dx
R
(see 2.172)
b
b
dx
dx
dx
− 2
− 3
R3
2a
R2
2a
R
(see 2.172, 2.173)
dx
b
1
b − 2ac dx
x
= − 2 ln
−
+
(see 2.172)
x2 R
2a
R
ax
2a2
R
2
b − 3ac (b + 2cx)
dx
a + bx
1 b4
b
x2
6b2 c 6c2
−
+
−
=
−
ln
−
+
x2 R2
a3
R
a2 xR
a2 ΔR
Δ a3
a2
a
6.
7.
b(b + 2cx)
1−
Δ
81
2
2
dx
1
3b
=−
−
x2 R3
axR2
a
5c
dx
−
xR3
a
dx
R
(see 2.172)
dx
R3
(see 2.173 and 2.177 3)
b 3ac − b2
ac − b2 x2
b
1
dx
dx
=−
+ 2 −
ln
+
3
3
2
3
x R
2a
R
a x 2ax
2a
R
8.
dx
x3 R2
9.
=
dx
=
x3 R3
−
1
3b
+ 2
2
2ax
2a x
−1
2b
+ 2
2ax2
a x
3b
2c
−
2
a
a
1
+
R
1
+
R2
2
2
6b
3c
−
a2
a
(see 2.172)
9bc dx
dx
+
xR2
2a2 R2
(see 2.173 1 and 2.177 2)
10bc dx
dx
+ 2
xR3
a
R3
(see 2.173 2 and 2.177 3)
2.18 Forms containing the quadratic trinomial a + bx + cx2 and the binomial α + βx
Notation: R = a + bx + cx2 ;
B = bβ − 2cα;
1.
2.
z = α + βx;
A = aβ 2 − αbβ + cα2 ;
Δ = 4ac − b2
(m − 1)A
R dx −
z
z m−2 Rn dx
(m + 2n + 1)c
n
n−1
Rn
dx
R dx
R
1
2nA
=
−
−
zm
(m − 2n − 1)β z m−1
(m − 2n − 1)β 2
zm
nB
Rn−1 dx
−
;
LA 184 (4)a
(m − 2n − 1)β 2
z m−1
n
n
n+1
R
−β
(m − n − 2)B R dx (m − 2n − 3)c R dx
=
−
−
LA 148 (5)
m−1
(m − 1)A z
(m − 1)A z m−1
(m − 1)A
z m−2
Rn
Rn−1 dx
Rn−1 dx
1
nB
2nc
=−
+
+
LA 418 (6)
m−1
2
m−1
2
(m − 1)β z
(m − 1)β
z
(m − 1)β
z m−2
(m + n)B
βz m−1 Rn+1
−
z R dx =
(m + 2n + 1)c (m + 2n + 1)c
m
n
m−1
n
82
Algebraic Functions
3.
z m−1
β
(m − n)B
z m dx
=
−
n
n−1
R
(m − 2n + 1)c R
(m − 2n + 1)c
b + 2cx z
2(m − 2n + 3)c
−
(n − 1)Δ Rn−1
(n − 1)Δ
m
=
2.201
z m−1 dx
(m − 1)A
−
n
R
(m − 2n + 1)c
m
Bm
z dx
−
Rn−1
(n − 1)Δ
z m−2 dx
Rn
LA 147 (1)
z
m−1
dx
Rn−1
LA 148 (3)
4.3
1
β
(m + n − 2)B
dx
=−
−
m
n
m−1
n−1
z R
(m − 1)A z
R
(m − 1)A
=
1
β
B
−
2(n − 1)A z m−1 Rn−1
2A
dx
(m + 2n − 3)c
−
m−1
n
z
R
(m − 1)A
(m + 2n − 3)β
dx
+
z m−1 Rn
2(n − 1)A
2
dx
z m−2 Rn
LA 148 (7)
dx
z m Rn−1
LA 148 (8)
For m = 1 and n = 1
β
z2
B
dx
dx
=
ln
−
zR
2A
R
2A R
For A = 0
1
β
(m + 2n − 2)c
dx
dx
=−
−
z m Rn
(m + n − 1)B z m Rn−1
(m + n − 1)B z m−1 Rn
LA 148 (9)
2.2 Algebraic Functions
2.20 Introduction
r
s
αx + β
αx + β
,
, . . . dx, where r, s, . . . are rational numbers,
γx + δ
γx + δ
can be reduced to integrals of rational functions by means of the substitution
αx + β
= tm ,
FI II 57
γx + δ
where m is the common denominator of the fractions r, s, . . . .
p
2.202 Integrals of the form xm (a + bxn ) dx,∗ where m, n, and p are rational numbers, can be
2.201
The integrals
R x,
expressed in terms of elementary functions only in the following cases:
(a)
When p is an integer; then, this integral takes the form of a sum of the integrals shown in 2.201;
(b)
is an integer: by means of the substitution xn = z, this integral can be transformed
m+1
1
to the form
(a + bz)p z n −1 dz, which we considered in 2.201;
n
(c)
When
When
m+1
n
m+1
n
+ p is an integer; by means of the same substitution xn = z, this integral can be
p
m+1
a + bz
1
z n +p−1 dz, considered in 2.201;
reduced to an integral of the form
n
z
For reduction formulas for integrals of binomial diﬀerentials, see 2.110.
∗ Translator:
The authors term such integrals “integrals of binomial diﬀerentials.”
2.214
Forms containing the binomial a + bxk and
2.21 Forms containing the binomial a + bxk and
√
√
x
x
Notation: z1 = a + bx.
dx
bx
2
√ = √ arctan
[ab > 0]
2.211
a √
z1 x
ab
a − bx + 2i xab
1
[ab < 0]
= √ ln
z1
i ab
m√
m
m+1 √ (−1)k ak xm−k
x
x
dx
m+1 a
√
dx = 2 x
+ (−1)
2.212
z1
(2m − 2k + 1)bk+1
bm+1 z1 x
k=0
(see 2.211)
2.213
√
√
dx
x dx
2 x a
√
1.
−
(see 2.211)
=
z1
b
b z1 x
√
√
a
x x dx
x
a2
dx
√
− 2 2 x+ 2
(see 2.211)
2.
=
z1
3b b
b
z1 x
2√
√
x2
xa
x x dx
a2
a3
dx
√
− 2 + 3 2 x− 3
(see 2.211)
3.
=
z1
5b
3b
b
b
z1 x
√
dx
x
1
dx
√
√
4.
=
(see 2.211)
+
az1
2a z1 x
z12 x
√
√
x dx
x
1
dx
√
5.
(see 2.211)
=−
+
2
z1
bz1
2b z1 x
√
√
√
x x dx
x dx
2x x 3a
6.
=
−
(see 2.213 5)
z12
bz1
b
z12
√
√
2√
x2
5ax 2 x 5a2
x dx
x x dx
− 2
=
+ 2
(see 2.213 5)
7.
2
z1
3b
3b
z1
b
z12
√
1
dx
3
3
dx
√
√
8.
=
(see 2.211)
+
x
+
2az12
4a2 z1
8a2 z1 x
z13 x
√
√
x dx
1
1
1
dx
√
9.
(see 2.211)
=
−
+
x
+
3
2
z1
2bz1
4abz1
8ab z1 x
√
√
√
x x dx
x dx
2x x 3a
10.
=−
+
(see 2.213 9)
z13
bz12
b
z13
√
√
2√
x2
5ax 2 x 15a2
x dx
x x dx
+
=
−
(see 2.213 9)
11.
3
2
2
2
z1
b
b
z1
b
z13
a
a
4
2
, α = 4 − .
Notation: z2 = a + bx , α =
b
b
%
√
√ & +
,
2
a
x + α 2x + α
1
α 2x
dx
√ =
√ ln
>0
+ arctan 2
2.214
√
z2
α −x
b
z2 x bα3 2
√
√
+
,
a
x
1
α − x
<0
=
ln √ − 2 arctan 3
2bα
α
b
α + x
83
84
Algebraic Functions
2.215
%
√
√ &
√
x dx
x + α 2x + α2
1
α 2x
2.215
= √ − ln
+ arctan 2
√
z2
z2
α −x
bα 2
√
√ 1
x
α − x
=
ln √ + 2 arctan 2bα
α
α + x
2.216
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
√
√
2 x a
x x dx
dx
√
−
=
z2
b
b z2 x
√
√
√
x dx
x2 x dx
2x x a
−
=
z2
3b
b
z2
√
x
dx
3
dx
√ =
√
+
2az2
4a z2 x
z22 x
√
√
√
x dx
x dx
x x
1
=
+
z22
2az2
4a
z2
√
√
dx
x
x x dx
1
√
=−
+
z22
2bz2
4b z2 x
√
√
√
x dx
x2 x dx
x x
3
=
−
+
z22
2bz2
4b
z2
√
1
dx
7
21
dx
√ =
√
+
x+
2
3
2
2
4az2
16a z2
32a
z2 x
z2 x
√
√
√
1
x dx
x dx
5
5
=
+
x
+
x
z23
4az22
16a2 z2
32a2
z2
2
√
√
bx − 3a x
x x dx
dx
3
√
=
+
3
z2
16abz22
32ab z2 x
√
√
2√
x dx
2x x 3a
x x dx
=−
+
z23
5bz22
5b
z23
2.22–2.23 Forms containing
Notation: z = a + bx.
n
√
n l lm+f
2.220
x z
dx =
k=0
k=0
2.222
1.
dx
2√
√ =
z
b
z
b
+a
b
,
>0
,
<0
(see 2.214)
(see 2.215)
(see 2.214)
(see 2.215)
(see 2.214)
(see 2.215)
(see 2.214)
(see 2.215)
(see 2.214)
(see 2.216 8)
n
(a + bx)k
√
l
(−1)k nk z n−k ak
l z l(m+1)+f
ln − lk + l(m + 1) + f
bn+1
The square root
n
√
(−1)k n z n−k ak 2 z 2m+1
√
n
k
2.221
x z 2m−1 dx =
2n − 2k + 2m + 1
bn+1
+a
Forms containing the binomial a + bxk and
2.225
2.
3.
x dx
√ =
z
x2 dx
√ =
z
1
z−a
3
√
2 z
b2
1 2 2
z − az + a2
5
3
2.223 dx
2
√ =− √
1.
b z
z3
x dx
2
√ = (z + a) 2 √
2.
3
b
z
z
2
x dx
z2
√
− 2az − a2
3.
=
3
z3
2.224
1.
2.
√
x
√
2 z
b3
b3
2
√
z
√
z m dx
z m dx
zm z
2m − 2n + 3 b
√
√
=
−
+
(n − 1)axn−1
2(n − 1) a xn−1 z
xn z
⎧
m
⎨
√
1
z dx
m
√
=
−z
z
n
⎩
(n − 1)axn−1
x z
⎫
(2m − 2n + 3)(2m − 2n + 5) . . . (2m − 2n + 2k + 1) bk ⎬
+
2k (n − 1)(n − 2) . . . (n − k − 1)xn−k−1
ak+1 ⎭
k=1
(2m − 2n + 3)(2m − 2n + 5) . . . (2m − 3)(2m − 1) bn−1 z m dx
√
+
2n−1 (n − 1)!x
an−1
x z
n−2
3.
4.
5.6
2.225
1.
2.
3.
For n = 1
m−1
m
2z m
z
z
√ dx =
√ +a
√ dx
x z
(2m − 1) z
x z
m
m
z
2am−k z k
dx
√ dx =
√ + am
√
x z
(2k − 1) z
x z
k=1
!√
√ !
! z − a!
1
dx
!
√ = √ ln ! √
√ !
x z
a
z + √a !
2
z
=√
arctan √
−a
−a
√
√
z dx
dx
√
=2 z+a
x
x z
√
√
b
dx
z dx
z
√
+
=−
2
x
x
2 x z
√
√
√
b2
dx
z dx
z3
b z
√
−
=−
+
x3
2ax2
4ax
8a x z
[a > 0]
[a < 0]
(see 2.224 4)
(see 2.224 4)
(see 2.224 4)
85
86
2.226
1.
2.
3.
Algebraic Functions
√ 3
√
z
z dx
dx
√
=
+ a 2 z + a2
x
3
x z
√
√ 3
√ 3
z dx
z5
z dx
3b
+
=−
2
x
ax
2a
x
√ 3
√ 3
√
1
z dx
z dx
b
3b2
5
=−
+ 2
z + 2
x3
2ax2
4a x
8a
x
2.226
(see 2.224 4)
(see 2.226 1)
(see 2.226 1)
2.227
m−1
dx
2
1
dx
√
√ =
√ + m
m
m−k
k
a
xz
z
(2k + 1)a
z z
x z
(see 2.224 4)
k=0
2.228
1.
2.
2.229
1.
2.
3.
√
b
dx
z
dx
√
√
−
=
−
2
ax
2a x z
x z
dx
√ =
x3 z
1
3b
−
+ 2
2ax2
4a x
(see 2.224 4)
√
3b2
z+ 2
8a
dx
√
x z
(see 2.224 4)
dx
1
2
dx
√ = √ +
√
(see 2.224 4)
a x z
a z
x z3
1
dx
3b
3b
1
dx
√
√ = −
− 2 √ − 2
(see 2.224 4)
2
3
ax
a
2a
z
x
z
x z
1
15b2
15b2
1
5b
dx
dx
√
√ = −
√
+
+
+
2ax2
4a2 x
4a3
8a3 x z
z
x3 z 3
(see 2.224 4)
Cube root
2.231
1.
2.
3.
4.
√
3
(−1)k nk z n−k ak
3 z 3(m+1)+1
3n − 3k + 3(m + 1) + 1
bn+1
k=0
n
(−1)k n z n−k ak
3
xn dx
k
√
√
=
3
3n − 3k − 3(m − 1) − 2 bn+1 3 z 3(m−1)+2
z 3m+2
k=0
√
n
√
3
(−1)k n z n−k ak
3 z 3(m+1)+2
3
n
k
z 3m+2 x dx =
3n − 3k + 3(m + 1) + 2
bn+1
k=0
n
(−1)k n z n−k ak
xn dx
3
k
√
√
=
3
3
3m+1
n+1
3n
−
3k
−
3(m
−
1)
−
1
z
b
z 3(m−1)+1
n
√
3
n
3m+1
z
x dx =
k=0
Forms containing the binomial a + bxk and
2.236
√
x
1
z n+ 3
3n − 3m + 4 b
z n dx
z n dx
√
√
=
−
+
3
m−1
(m − 1)ax
3(m − 1) a xm−1 3 z 2
xm x2
For m = 1
n−1
n
dx
3z n
z
z dx
√
√
√
=
+
a
3
3
3
2
2
x z
(3n − 2) z
x z2
√
√
3
z dx
dx
33z
1
√
6.
=
+
3
n
(3n − 1)az
a
xz n
xz n z 2
√ √
√
√
3
z− 3a √
33z
3
1
dx
√
√
√
ln
− 3 arctan √
2.232
= √
3
3
3
3
x
z+23a
x z2
a2 2
5.
2.233
1.
2.
3.
√
3
√
z dx
dx
√
=33z+a
3
x
x z2
√
√
3
b√
z dx
z3z
+ 3z+
=−
2
x
ax
a
√
3
z dx
1
b
= −
+ 2
3
2
x
2ax
3a x
√
3
4.
5.
2.234
(see 2.232)
b
3
dx
√
(see 2.232)
3
x z2
√
b2 √
b2
dx
√
z3z− 2 3z−
3a
9a x 3 z 2
z
2b
dx
dx
√
√
−
=−
3
3
2
ax
3a
z
x z2
1
5b √
5b2
dx
dx
3
√
√
=
−
+
z
+
3
2ax2
6a2 x
9a2 x 3 z 2
x3 z 2
x2
(see 2.232)
(see 2.232)
(see 2.232)
√
3
z n dx
zn z2
3n − 3m + 5 b
z n dx
√
√
1.
=
−
+
3
(m − 1)axm−1
3(m − 1) a xm−1 3 z
xm z 2
For m = 1:
n
n−1
z dx
dx
3z n
z
√
√
√
2.
=
+
a
x3z
(3n − 1) 3 z
x3z
√
√
3
3
3 z2
z 2 dx
1
dx
√
=
+
3.
3
n
n
(3n − 2)az
a
xz n
xz z
√ √
√
√
3
z− 3a √
33z
1
dx
3
√ = √
√
√
ln
+ 3 arctan √
2.235
3
3
3
x3z
x
z+23a
a2 2
2.236
1.
2.
√
3
3√
z 2 dx
dx
3
√
=
z2 + a
x
2
x3z
√
√
3
3
b√
z 2 dx
z5
2b
dx
3
2+
√
+
=
−
z
2
x
ax
a
3 x3z
(see 2.235)
(see 2.235)
87
88
Algebraic Functions
√
3
3.
4.
5.
z 2 dx
1
b
b2 √
b2
dx
3
5/3
2−
√
=
−
+
−
z
z
x3
2ax2
6a2 x
6a2
9a x 3 z
dx
√
x2 3 z
dx
√
3
z
x3
Notation: z = a + bx,
1.
2.
1.
11
2.
3.
4.
5.
6.
7.
8.
(see 2.235)
(see 2.235)
√
a + bx and the binomial α + βx
t = α + βx,
Δ = aβ − bα.
√
z m tn dx
2
(2m − 1)Δ
√
tn+1 z m−1 z +
=
(2n + 2m + 1)β
(2n + 2m + 1)β
z
n
√
tn z m dx
√
= 2 z 2m+1
k
z
n
k=0
2.242
(see 2.235)
√
3
b
z2
dx
√
−
=−
ax
3a x 3 z
1
2b √
2b2
dx
3
√
= −
+ 2
z+ 2
2
2ax
3a x
9a
x3z
2.24 Forms containing
2.241
2.241
k
αn−k β k (−1)p
bk+1 p=0
k
p
z m−1 tn dx
√
z
z k−p ap
2k − 2p + 2m + 1
√
√
z
2 z
2α z
t dx
√ =
+β
−a
b
3
b2
z
√
√
√
z2
2 z
2
z
2 z
2α2 z
t2 dx
2
2
√ =
+ 2αβ
−a
−
za
+
a
+
β
b
3
b2
5
3
b3
z
√
√
√
z2
2 z
2 z
2
t3 dx 2α3 z
z
2
2
2
√ =
+ 3α β
−α
− za + a
+ 3αβ
b
3
b2
5
3
b3
z
√
3
2
z
2 z
3z a
−
+ za2 − a3
+β α
7
5
b4
√
√
z
a 2 z3
2α z 3
tz dx
√ =
+β
−
3b
5 3
b2
z
√
√
√
z2
z
a 2 z3
2za a2 2 z 3
2α2 z 3
t2 z dx
2
√
+ 2αβ
−
−
+
=
+β
3b
5 3
b2
7
5
3
b3
z
√
√
√
z2
a 2 z3
2za a2 2 z 3
t3 z dx 2α3 z 3
z
2
2
√ =
+ 3α β
−
−
+
+ 3αβ
3b
5 3
b2
7
5
3
b3
z
√
3
2
2
3
z
3z a 3za
a
2 z3
−
+
−
+β 3
9
7
5
3
b4
√
√
z
a 2 z5
2α z 5
tz 2 dx
√
+β
−
=
5b
7 5
b2
z
√
√
√
z
z2
a 2 z5
2za a2 2 z 5
2α2 z 5
t2 z 2 dx
2
√
+ 2αβ
−
−
+
=
+β
5b
7 5
b2
9
7
5
b3
z
LA 176 (1)
2.244
Forms with
√
a + bx and α + βx
√
√
a 2 z5
t3 z 2 dx 2α3 z 5
z
2
√
+ 3α β
−
=
+ 3αβ 2
5b
7 5
b2
z
√
z3
3z 2 a 3za2
a3 2 z 5
3
−
+
−
+β
11
9
7
5
b4
√
√
3
z
a 2 z7
2α z 7
tz dx
√
+β
−
=
7b
9 7
b2
z
√
√
2 3
z2
z
a 2 z7
2α2 z 7
t z dx
2
√
+ 2αβ
−
=
+
β
7b
9 7
b2
11
z
√
√
3 3
3
7
7
a 2 z
t z dx 2α z
z
√
+ 3α2 β
−
=
+ 3αβ 2
7b
9 7
b2
z
√
z3
3z 2 a 3za2
a3 2 z 7
3
−
+
−
+β
13
11
9
7
b4
9.
10.
11.
12.
2.243
1.
89
z2
2za a2
−
+
9
7
5
√
2 z5
b3
√
2 z7
b3
√
z2
2za a2 2 z 7
−
+
11
9
7
b3
2za a2
−
+
9
7
tn dx
tn+1 √
2
(2n − 2m + 3)β
tn dx
√
√
=
z
−
(2m − 1)Δ z m
(2m − 1)Δ
zm z
z m−1 z
tn √
tn−1 dx
2nβ
2
√
z+
=−
m
(2m − 1)b z
(2m − 1)b z m−1 z
LA 176 (2)
2.
2.244
1.
2.
3.
4.
5.
6.
7.
n
n
2
tn dx
√
√
=
2m−1
k
zm z
z
k=0
k
an−k β k (−1)p
bk+1 p=0
k
p
z k−p ap
2k − 2p − 2m + 1
t dx
2a
2β(z + a)
√ =− √ +
√
z z
b z
b2 z
2
t2 dx
2α2
4αβ(z + a) 2β
√ =− √ +
√
+
z z
b z
b2 z
z2
3
− 2za − a2
√
b3 z
2
3
2 z
2
2β 3 z5 − z 2 a + 3za2 + a3
3 − 2za − a
t3 dx
2α3
6α2 β(z + a) 6αβ
√ =− √ +
√
√
√
+
+
z z
b z
b2 z
b3 z
b4 z
2β z − a3
2a
t dx
√
√
√
=
−
−
z2 z
3b z 3
b2 z 3
2
2β 2 z 2 + 2az − a3
4αβ z − a3
2a2
t2 dx
√ =− √ −
√
√
+
z2 z
3b z 3
b2 z 3
b3 z 3
2
3
6αβ 2 z 2 + 2za − a3
2β 3 z3 − 3z 2 a − 3za2 +
6α2 β z − a3
2α3
t3 dx
√ =− √ −
√
√
√
+
+
z2 z
3b z 3
b2 z 3
b3 z 3
b4 z 3
2β z3 − a5
2α
t dx
√ =− √ −
√
z3 z
5b z 5
b2 z 5
a3
3
90
Algebraic Functions
2.245
a2
2β 2 z 2 − 2za
4αβ z3 − a5
3 + 5
2α2
t2 dx
√ =− √ −
√
√
−
z3 z
5b z 5
b2 z 5
b3 z 5
a2
3
6αβ 2 z 2 − 2za
6α2 β z3 − a5
3 + 5
2α3
t dx
√ =− √ −
√
√
−
z3 z
5b z 5
b2 z 5
b3 z 5
3
2β 3 z 3 + 3z 2 a − za2 + a5
√
+
b4 z 5
8.
9.
2.245
1.
2.
m−1
z m dx
z m−1 √
dx
2
(2m − 1)Δ
z
√ =−
√
z−
(2n − 2m − 1)β tn−1
(2n− 2m − 1)β
tn z
tn z
z m−1
z m−1 √
1
(2m − 1)b
√ dx
=−
z
+
(n − 1)β tn−1
2(n − 1)β tn−1
zm
m √
z
1
(2n − 2m − 3)b
z dx
√
=−
z−
n−1
n−1
(n − 1)Δ t
2(n − 1)Δ
t
z
⎡
m
√
1
1
z dz
√ = −z m z ⎣
n−1
n
(n − 1)Δ t
t z
+
n−1
k=2
−
3.
4.
(2n − 2m − 3)(2n − 2m − 5) . . . (2n − 2m − 2k + 1)b
2k−1 (n − 1)(n − 2) . . . (n − k)Δk
k=0
2.246
2.247
√
√
1
dx
β z − βΔ
√ √
√
ln √
[βΔ > 0]
t z βΔ β z + βΔ
√
2
β z
=√
arctan √
[βΔ < 0]
−βΔ
−βΔ
√
2 z
[Δ = 0]
=−
bt
m
dx
βm
2
β k−1 z k
dx
√
√ = m−1 √ +
+ m
k
m
Δ (2m − 2k + 1) Δ
z
tz
z
z
t z
k=1
(see 2.246)
2.248
1.
k−1
(2n − 2m − 3)(2n − 2m − 5) . . . (−2m + 3)(−2m + 1)bn−1
2n−1 · (n − 1)!Δn
For n = 1
m
zm
2
Δ z m−1 dx
z dx
√ =
√
√ +
(2m − 1)β z
β
t z
t z
m
m−1
Δk
z m−k
Δm
z dx
dx
√ =2
√
√
+
k+1
m
(2m − 2k − 1)β
β
t z
z
t z
LA 176 (3)
2
β
dx
√ = √ +
Δ
tz z
Δ z
dx
√
t z
(see 2.246)
⎤
1
⎦
tn−k
z m dx
√
t z
2.248
Forms with
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
2
2β
β2
dx
√
√
√
=
+
+
Δ2
tz 2 z
3Δz z
Δ2 z
√
a + bx and α + βx
dx
√
t z
2
2β
2β 2
β3
dx
√ + 3√ + 3
√ =
√ +
3
2
2
Δ
tz z
5Δz z
3Δ z z
Δ z
√
b
z
dx
√ =−
−
Δt
2Δ
z
t2
91
(see 2.246)
dx
√
t z
(see 2.246)
dx
√
t z
dx
1
3b
3bβ
√ = − √ − 2√ −
2Δ2
t2 z z
Δt z
Δ z
(see 2.246)
dx
√
t z
1
5b
5bβ
5bβ 2
dx
√
√
√
√
=
−
−
−
−
2Δ3
t2 z 2 z
Δtz 2 z
3Δ2 z z
Δ3 z
(see 2.246)
dx
√
t z
(see 2.246)
dx
1
7b
7bβ
7bβ
7bβ 3
dx
√ − 4√ −
√
√ =−
√ −
√ −
2Δ4 t z
t2 z 3 z
Δtz 2 z
5Δ2 z 2 z
3Δ3 z z
Δ z
2
√
√
dx
3b2
z
3b z
dx
√
√
+
=
−
+
2Δt2
4Δ2 t
8Δ2 t z
t3 z
dx
1
5b
15b2
15b2 β
√
√
√
√
=
−
+
+
+
8Δ3
t3 z z
2Δt2 z
4Δ2 t z
4Δ3 z
(see 2.246)
(see 2.246)
dx
√
t z
(see 2.246)
√
2
dx
1
7b z
35b
35b2 β
35b2 β 2
dx
√ +
√ +
√ +
√
√ =−
√ +
4
3
2
2
2
2
4
8Δ
t z z
2Δt z z
4Δ tz z
12Δ z z
4Δ z
t z
(see 2.246)
dx
1
9b
63b2
21b2 β
63b2 β 2
63b2 β 3
√
√
√
√
√
√
=
−
+
+
+
+
+
8Δ5
t3 z 3 z
2Δt2 z 2 z
4Δ2 tz 2 z
20Δ3 z 2 z
4Δ4 z z
4Δ5 z
(see 2.246)
√
Δ
2 z
z dx
dx
√ =
√
+
(see 2.246)
β
β t z
t z
√
√
2Δ z
z 2 dx
2z z
Δ2
dx
√ =
√
+
(see 2.246)
+ 2
2
3β
β
β
t z
t z
√
√
√
2Δz z
2z 2 z
z 3 dx
2Δ2 z
Δ3
dx
√ =
√
+
(see 2.246)
+
+
5β
3β 2
β3
β3 t z
t z
√
√
z dx
b z
b
z z
dx
√
√ =−
+
+
(see 2.246)
2
Δt
βΔ
2β t z
t z
√
√
√
bz z
3b z
z2 z
z 2 dx
3bΔ
dx
√
√
+
+
=
−
(see 2.246)
+
Δt
βΔ
β2
2β 2 t z
t2 z
dx
√
t z
92
Algebraic Functions
17.
18.3
19.
20.
2.249
√
√
√
√
bz 2 z
5bz z
z3 z
5bΔ z
5Δ2 b
z 3 dx
dx
√
√ =−
+
+
+
+
2
3
3
2
Δt
βΔ
3β
β
2β
t z
t z
z dx
√
t3 z
z 2 dx
√
t3 z
(see 2.246)
√
√
2
2 b z
z z
bz z
b
dx
√
−
=−
(see 2.246)
+
+
2Δt2
4Δ2 t
4βΔ2
8βΔ t z
√
√
√
√
b2 z z
3b2
z2 z
bz 2 z
3b2 z
dx
√
+
+
=−
+
+
2
2
2
2
2
2Δt
4Δ t
4βΔ
4β Δ
8β
t z
√
(see 2.246)
√
√
√
√
√
3
2 2
2
z dx
3b z z
15b2 z
z z
3bz z
5b z z
15b2 Δ
dx
√
√
+
+
=
−
+
+
+
2
2
2
2
3
3
3
2Δt
Δ t
4βΔ
4β Δ
4β
8β
t z
t z
3
3
(see 2.246)
2.249
1.
2.
√
z
2
(2n + 2m − 3)β
dx
dx
√ =
√
+
m
n z m−1 z
(2m − 1)Δ tn−1
z
(2m
−
1)Δ
z m tn z
t
√
z
1
(2n + 2m − 3)b
dx
√
=−
−
m
n−1
n−1
(n − 1)Δ z t
2(n − 1)Δ
t
zm z
⎡
√
1
z ⎣ −1
dx
√ = m
n−1
m
n
z
(n
−
1)Δ
t
z t z
LA 177 (4)
⎤
k−1
(2n
+
2m
−
3)(2n
+
2m
−
5)
.
.
.
(2n
+
2m
−
2k
+
1)b
1
+
(−1)k
· n−k ⎦
2k−1 (n − 1)(n − 2) . . . (n − k)Δk
t
k=2
n−1 (2n + 2m − 3)(2n + 2m − 5) . . . (−2m + 3)(−2m + 1)b
dx
√
+(−1)n−1
n−1
n−1
m
2
(n − 1)!Δ
tz
z
n−1
For n = 1
1
2
β
dx
dx
√
√
√
=
+
m
m−1
m−1
(2m − 1)Δ z
Δ tz
z t z
z
z
2.25 Forms containing
√
a + bx + cx2
Integration techniques
2.251
It is possible to rationalize the integrand in integrals of the form
R x,
a + bx + cx2 dx by
using one or more of the following three substitutions, known as the “Euler substitutions”:
√
√
a + bx + cx2 = xt ± a for a > 0;
1.
√
√
a + bx + cx2 = t ± x c for c > 0;
2.
c (x − x1 ) (x − x2 ) = t (x − x1 ) when x1 and x2 are real roots of the equation a + bx + cx2 = 0.
3.
2.252
Forms containing
√
a + bx + cx2
93
2.252
Besides
the Euler substitutions, there is also the following method of calculating integrals of
the form R x, a + bx + cx2 dx. By removing the irrational expressions in the denominator and
performing simple algebraic operations, we can reduce the integrand to the sum of some rational function
P (x)
√ 1
, where P 1 (x) and P 2 (x) are both polynomials.
of x and an expression of the form
P 2 (x) a + bx + cx2
P 1 (x)
By separating the integral portion of the rational function
from the remainder and decomposing
P 2 (x)
the latter into partial fractions, we can reduce the integral of these partial fractions to the sum of integrals,
each of which is in one of the following three forms:
P (x) dx
√
, where P (x) is a polynomial of some degree r;
1.
a + bx + cx2
dx
√
2.
;
k
(x + p) a + bx + cx2
(M x + N ) dx
a
b
,
a1 = , b 1 =
3.
.
m
2
2
c
c
(a + βx + x )
c (a1 + b1 x + x )
In more detail:
P (x) dx
dx
√
1.
= Q(x) a + bx + cx2 + λ √
, where Q(x) is a polynomial of
2
a + bx + cx
a + bx + cx2
degree (r − 1). Its coeﬃcients, and also the number λ, can be calculated by the method of
undetermined coeﬃcients from the identity
2.
3.
1
P (x) = Q (x) a + bx + cx2 + Q(x)(b + 2cx) + λ
LI II 77
2
P (x) dx
Integrals of the form √
(where r ≤ 3) can also be calculated by use of formulas
a + bx + cx2
2.26.
P (x) dx
√
, where the degree n of the polynomial P (x) is
Integrals of the form
k
(x + p) a + bx + cx2
1
lower than k can, by means of the substitution t =
, be reduced to an integral of the form
x
+
p
P (t) dt
. (See also 2.281).
a + βt + γt2
(M x + N ) dx
can be calculated by the following
Integrals of the form
m
(α + βx + x2 )
c (a1 + b1 x + x2 )
procedure:
• If b1 = β, by using the substitution
x=
<
2
t − 1 (a1 − α) − (αb1 − a1 β) (β − b1 )
a1 − α
+
βb1
t+1
β − b1
94
Algebraic Functions
2.260
P (t) dt
, where P (t)
m
+ p)
c (t2 + q)
P (t) dt
is a polynomial of degree no higher than 2m − 1. The integral
can be
2 + p)m
(t
t2 + q
t dt
dt
reduced to the sum of integrals of the forms
and
.
k
k
2
2
2
(t + p)
t +q
(t + p)
t2 + q
P (t) dt
• If b1 = β, we can reduce it to integrals of the form
by means of the
m
2
c (t2 + q)
(t + p)
b1
substitution t = x + .
2
t dt
can be evaluated by means of the substitution t2 + q =
The integral
k
2
(t + p)
c (t2 + q)
u2 .
t
dt
=
can be evaluated by means of the substitution The integral
k
2
2
2
t +q
(t + p)
c (t + q)
υ (see also 2.283).
FI II 78-82
we can reduce this integral to an integral of the form
2.26 Forms containing
(t2
√
a + bx + cx2 and integral powers of x
Notation: R = a + bx + cx2 , Δ = 4ac − b2
For simpliﬁed formulas for the case b = 0, see 2.27.
2.260 √
√
√
(2m + 2n + 1)b
xm−1 R2n+3
m
2n+1
1.
x
−
R
dx =
xm−1 R2n+1 dx
(m + 2n + 2)c 2(m + 2n + 2)c
√
(m − 1)a
−
xm−2 R2n+1 dx
(m + 2n + 2)c
2.
3.
2cx + b √ 2n+1
2n + 1 Δ √ 2n−1
R
R
R2n+1 dx =
+
dx
4(n + 1)c
8(n + 1) c
√ √
n−1
(2n + 1)(2n − 1) . . . (2n − 2k + 1)
R
(2cx
+
b)
Rn +
R2n+1 dx =
4(n + 1)c
8k+1 n(n − 1) . . . (n − k)
k=0
n+1 Δ
(2n + 1)!!
dx
√
+ n+1
8
(n + 1)! c
R
TI (192)a
√
TI (188)
Δ
c
k+1
R
n−k−1
TI (190)
2.26111
For n = −1 √
1
dx
2 cR + 2cx + b
√ = √ ln
√
c
R
Δ
2cx + b
1
√
= √ arcsinh
c
Δ
1
= √ ln(2cx + b)
c
2cx + b
−1
= √ arcsin √
−c
−Δ
[c > 0]
TI (127)
[c > 0,
Δ > 0]
DW
[c > 0,
Δ = 0]
DW
[c < 0,
Δ < 0]
TI (128)
2.263
2.262
1.
2.
3.
Forms containing
√
Δ
dx
(2cx + b) R
√
+
R dx =
4c
8c
R
√
√
3
(2cx + b)b √
R
bΔ
x R dx =
−
R−
3c
8c2
16c2
√
x
5b2
5b √ 3
x2 R dx =
−
R
+
−
4c 24c2
16c2
√
4.
5.
6.
7.
8.
1.
95
(see 2.261)
dx
√
R
a
4c
(see 2.261)
√
(2cx + b) R
+
4c
5b2
a
−
16c2
4c
Δ
8c
dx
√
R
(see 2.261)
√
x3 R dx =
√
2.263
√
a + bx + cx2 and xn
2
2
x
7bx
7b
2a
−
+
−
5c
40c2
48c3 15c2
7b3
3ab Δ
dx
√
−
− 2
3
32c
8c
8c
R
R3 dx =
R
3Δ
+
8c 64c2
R3
√
3Δ2
(2cx + b) R +
128c2
√
√
R5
− (2cx + b)
x R3 dx =
5c
√
x2 R3 dx =
√
7b3
3ab
− 2
32c3
8c
−
√
(2cx + b) R
4c
(see 2.261)
dx
√
R
(see 2.261)
b √ 3
3Δb √
3Δ2 b
dx
√
R
+
R
−
16c2
128c3
256c3
R
x
7b2
7b √ 5
a
−
R
+
−
2
2
6c 60c
24c
6c
Δ2
7b2
dx
√
−a
+
4c
256c3
R
(see 2.261)
√
R3
3Δ √
b
+
R
2x +
c
8
64c
(see 2.261)
√
x3 R3 dx =
x2
3bx
3b2
2a √ 5
−
+
−
R
2
3
7c
28c
40c
35c2 √
3b3
3Δ √
R3
ab
b
+
−
− 2
R
2x +
16c3
4c
c
8
64c
3Δ2 b
3b2
dx
√
−a
−
4c
512c4
R
(see 2.261)
xm dx
xm−1
(2m − 2n − 1)b
√
√
=
−
2n+1
2n−1
2(m − 2n)c
R
(m − 2n)c R
(m − 1)a
xm−1 dx
√
−
2n+1
(m
− 2n)c
R
xm−2 dx
√
R2n+1
TI (193)a
2.
For m = 2n
x2n dx
x2n−1
b
1
x2n−1
x2n−2
√
√
√
√
=−
−
dx +
dx
2c
c
R2n+1
(2n − 1)c R2n−1
R2n+1
R2n−1
TI (194)a
96
Algebraic Functions
dx
√
R2n+1
3.
4.
dx
√
R2n+1
2.264
dx
2(2cx + b)
8(n − 1)c
√
√
=
+
2n−1
(2n
−
1)Δ
(2n − 1)Δ R
R2n−1
n−1
ck k
8k (n − 1)(n − 2) . . . (n − k)
2(2cx + b)
√
1+
=
R
(2n − 3)(2n − 5) . . . (2n − 2k − 1) Δk
(2n − 1)Δ R2n−1
k=1
[n ≥ 1] .
2.264
1.
3.
4.
5.
6.
7.
8.
TI (191)
(see 2.261)
√
x dx
b
R
dx
√ =
√
(see 2.261)
−
c
2c
R
R
2
x
3b2
x dx
3b √
a
dx
√
√
=
(see 2.261)
− 2
R+
−
2
2c 4c
8c
2c
R
R
3
x2
5b3
5bx
x dx
5b2
2a √
3ab
dx
√
√
=
−
+ 3− 2
R−
− 2
2
3
3c
12c
8c
3c
16c
4c
R
R
2.
dx
√
R
TI (189)
(see 2.261)
dx
2(2cx + b)
√
√
=
3
Δ R
R
x dx
2(2a + bx)
√
√
=−
3
Δ R
R
2
Δ − b2 x − 2ab 1
x dx
dx
√
√
√
+
(see 2.261)
=−
3
c
cΔ R
R
R
3
cΔx2 + b 10ac − 3b2 x + a 8ac − 3b2
3b
x dx
dx
√
√
− 2 √
=
2c
c2 Δ R
R
R3
√ 2n+1
R
2.265
xm
(see 2.261)
√
√ 2n+1
R2n+3
R
(2n − 2m + 5)b
dx = −
+
dx
(m − 1)axm−1 √ 2(m − 1)a
xm−1
R2n+1
(2n − m + 4)c
dx
+
(m − 1)a
xm−2
TI (195)
For√m = 1
√
√ 2n−1
R2n+1
R2n+1
b √ 2n−1
R
dx =
+
dx
R
dx + a
x
2n + 1
2
x
For<a = 0
<
<
2n+1
2n+3
2n+1
(bx + cx2 )
(bx + cx2 )
2 (bx + cx2 )
2(m − 2n − 3)c
dx
=
+
xm
(2n − 2m + 3)bxm
(2n − 2m + 3)b
xm−1
For m = 0 see 2.260 2 and 2.260 3.
For n = −1 and m = 1:
TI (198)
LA 169 (3)
2.267
Forms containing
8
2.266
√
dx
2a + bx + 2 aR
1
√ = − √ ln
x
a
x R
1
2a + bx
=√
arcsin √
−a
x b2 − 4ac
1
2a + bx
=√
arctan √ √
−a
2 −a R
2a + bx
1
= − √ arcsinh √
a
x Δ
2a + bx
1
= − √ arctanh √ √
a
2 a R
x
1
= √ ln
a√ 2a + bx
2 bx + cx2
=−
bx
2a + bx
1
√
√
arccosh
=
a
x −Δ
√
a + bx + cx2 and xn
[a > 0]
[a < 0,
TI (137)
Δ < 0]
TI (138)
[a < 0]
[a > 0,
97
LA 178 (6)a
Δ > 0]
DW
[a > 0]
[a > 0,
Δ = 0]
[a = 0,
b = 0]
LA 170 (16)
[a > 0, Δ < 0]
2.267 √
b
dx
R dx √
dx
√ +
√
(see 2.261 and 2.266)
= R+a
1.
x
x R 2
R
√
√
dx
b
R dx
R
dx
√
√
2.
+
c
(see 2.261 and 2.266)
+
=
−
2
x
x
2 x R
R
For a = 0
√
√
dx
bx + cx2
2 bx + cx2
√
+
c
dx
=
−
(see 2.261)
2
x
x
bx + cx2
√
√
1
b2
c
R dx
b
dx
√
−
3.
=−
+
R−
x3
2x2
4ax
8a 2
x R
(see 2.266)
4.
5.
For a = 0
<
3
√
2
2 (bx + cx2 )
bx + cx
dx = −
x3
3bx3
√
√
b 12ac − b2
2bcx + b2 + 8ac √
R3
R3
dx
dx
2
√ +
√
dx =
+
R+a
x
3
8c
16c
x R
R
(see 2.261 and 2.266)
√
√ 3
√
3 4ac + b2
R5
R
cx + b √ 3 3
dx
dx
3
√ +
√
+
dx = −
R + (2cx + 3b) R + ab
2
x
ax
a
4
2
8
x R
R
(see 2.261 and 2.266)
For a = 0
<
3
(bx + cx2 )
x2
<
=
3
(bx + cx2 )
2x
+
3b 3b2
bx + cx2 +
4
8
dx
√
bx + cx2
(see 2.261)
98
6.
Algebraic Functions
2.268
√ 3
R
dx = −
x3
√
1
b
bcx + 2ac + b2 √ 3 3 bcx + 2ac + b2 √
5
+ 2
R +
R +
R
2
2ax2
4ax
4a
4a
3
dx
dx
3
√ + bc √
+ 4ac + b2
8
x R 2
R
(see 2.261 and 2.266)
For a = 0
<
3
(bx + cx2 )
x3
c−
dx =
2b
x
dx
3bc
√
bx + cx2 +
2
bx + cx2
(see 2.261)
2.268
xm
dx
1
√
√
=−
2n+1
m−1
2n−1
R
(m − 1)ax
R
(2n + 2m − 3)b
(2n + m − 2)c
dx
dx
√
√
−
−
2(m − 1)a
(m − 1)a
xm−1 R2n+1
xm−2 R2n+1
TI (196)
For m = 1
For a = 0 2.269
1.
3.
√
dx
R2n+1
+
1
a
dx
√
x R2n−1
dx
2
<
<
=−
2n+1
2n−1
xm (bx + cx2 )
(2n + 2m − 1)bxm (bx + cx2 )
dx
(4n + 2m − 2)c
<
−
(2n + 2m − 1)b
2n+1
xm−1 (bx + cx2 )
(cf. 2.265)
dx
√
x R
√
dx
R
√ =−
−
2
ax
x R
For a = 0
2
dx
√
=
2
2
3
x bx + cx
1
dx
√ = −
2ax2
x3 R
2.
dx
1
b
√
√
=
−
2n+1
2n−1
2a
x R
(2n − 1)a R
(see 2.266)
b
2a
dx
√
x R
(see 2.266)
2c 1
+
bx + cx2
bx2
b2 x
√
3b2
3b
c
+ 2
R+
−
4a x
8a2
2a
−
dx
√
x R
(see 2.266)
4.
For a = 0
2
4c
8c2
dx
1
√
=
− 3+ 2 2− 3
5
bx
3b x
3b x
x3 bx + cx2
2
2 bcx − 2ac + b
1
dx
dx
√
√
√
+
=−
3
a x R
aΔ R
x R
For a = 0
bx + cx2
(see 2.266)
TI (199)
2.271
Forms containing
5.11
2
dx
<
=
3
3
x (bx + cx2 )
−
3b
A
dx
√
= −√ − 2
2
3
R 2a
x R
4c 8c2 x
1
+ 2 + 3
bx b
b
dx
√
x R
where A =
6.
√
2.27 Forms containing
1
√
bx + cx2
b 10ac − 3b2
c 8ac − 3b2 x
1
−
−
−
ax
a2 Δ
a2 Δ
−
64c3
8c
16c2
128c4 x
1
+
+
−
+
bx3
5b2 x2
5b3 x
5b4
5b5
2.
3.
4.
(see 2.266)
√
1
bx + cx2
√
a + cx2 and integral powers of x
√
Notation: u = a + cx2 .
√
1
I1 = √ ln x c + u
c
1
c
= √ arcsin x −
a
−c
√
u− a
1
√
I2 = √ ln
2 a √
u+ a
a−u
1
= √ ln √
2 a
a +
u
1
1
c
a
1
=√
−
arcsec x − = √
arccos
a
x
c
−a
−a
1.
99
For a = 0
1
dx
8c2
2
2c
16c3 x
1
<
√
=
− 2+ 2 − 3 −
4
5
bx
b
x
b
b
3
bx + cx2
x2 (bx + cx2 )
dx
√
3
x R3
bc 15b2 − 52ac x
15b2 − 12ac
15b4 − 62acb2 + 24a2 c2
1
1
5b
dx
√ +
√
−
= − 2+ 2 −
3
3
3
ax
2a x
2a Δ
2a Δ
8a
2 R
x R
(see 2.266)
For a = 0
2
dx
<
=
7
3
x3 (bx + cx2 )
2.271
a + cx2 and xn
[c > 0]
[c < 0 and a > 0]
[a > 0 and c > 0]
[a > 0 and c > 0]
[a < 0 and c > 0]
1 5
5
5
5
xu + axu3 + a2 xu + a3 I1
6
24
16
16
3
1
3
u3 dx = xu3 + axu + a2 I1
4
8
8
1
1
u dx = xu + aI1
2
2
dx
= I1
u
u5 dx =
DW
DW
DW
DW
100
Algebraic Functions
5.
6.
dx
u2n+1
7.
2.272
dx
1x
=
u3
au
=
k=0
x2 u3 dx =
x2 u dx =
2.
3.
4.
5.
6.
7.
4.
5.
6.
1 axu3
1 a2 xu
1 a3
1 xu5
−
−
−
I1
6 c
24 c
16 c
16 c
DW
DW
1
x2
x
dx = − + I1
u3
cu c
DW
x2
1 x3
dx
=
u5
3 au3
DW
n−2
x2 dx
1 (−1)k
=
u2n+1
an−1
2k + 3
k=0
n−2
k
ck x2k+3
u2k+3
3
x dx
1
a
=−
+
u2n+1
(2n − 3)c2 u2n−3
(2n − 1)c2 u2n−1
x4 u dx =
3.
DW
DW
ck x2k+1
u2k+1
1 xu 1 a
x2
dx =
−
I1
u
2 c
2c
x4 u3 dx =
2.
n−1
k
1 axu 1 a2
1 xu3
−
−
I1
4 c
8 c
8 c
1.
7.
n−1
1 (−1)k
an
2k + 1
1
x dx
=−
2n+1
u
(2n − 1)cu2n−1
2.273
DW
1.
2.272
axu5
1 x3 u5
a2 xu3
3a3 xu
3a4
−
+
+
+
I1
8 c
16c2
64c2
128c2
128c2
axu3
1 x3 u3
a2 xu
a3
−
+
+
I1
2
2
6 c
8c
16c
16c2
DW
DW
DW
1 x3 u 3 axu 3 a2
x4
dx =
−
+
I1
u
4 c
8 c2
8 c2
DW
3 a
x4
1 xu
ax
dx =
+ 2 −
I1
3
2
u
2c
c u 2 c2
DW
1 x3
x4
x
1
dx = − 2 −
+ 2 I1
5
u
c u 3 cu3
c
DW
x4
1 x5
dx
=
u7
5 au5
DW
n−3
x4 dx
1 (−1)k
=
u2n+1
an−2
2k + 5
k=0
n−3
k
ck x2k+5
u2k+5
2.275
Forms containing
8.
2.274
x6 u3 dx =
x6 u dx =
2.
3.
4.
5.
6.
7.
8.
9.
2.275
1.
2.
3.
4.
5.
6.
7.
8.
a + cx2 and xn
1
2a
a2
x5 dx
=
−
+
−
u2n+1
(2n − 5)c3 u2n−5
(2n − 3)cu2n−3
(2n − 1)c3 u2n−1
1.
√
101
DW
ax3 u5
1 x5 u5
a2 xu5
a3 xu3
3a4 xu
3 a5
−
+
−
−
−
I1
2
3
3
3
10 c
16c
32c
128c
256c
256 c3
5 ax3 u3
1 x5 u3
5a2 xu3
5a3 xu
5 a4
−
+
−
−
I1
8 c
48 c2
64c3
128c3
128 c3
1 x5 u
5 ax3 u
x6
5 a2 xu
5 a3
dx =
−
+
−
I1
u
6 c
24 c2
16 c3
16 c3
DW
5 ax3
15 a2 x 15 a2
x6
1 x5
−
−
+
dx =
I1
3
2
u
4 cu 8 c u
8 c3 u
8 c3
DW
x6
1 x5
10 ax3
5 a2 x
5 a
dx
=
+
+
−
I1
u5
2 cu3
3 c2 u3
2 c3 u3
2 c3
DW
x6
23 x5
7 ax3
a2 x
1
dx
=
−
−
−
+ 3 I1
7
5
2
5
3
5
u
15 cu
3c u
c u
c
DW
x6
1 x7
dx =
9
u
7 au7
DW
n−4
x6 dx
1 (−1)k
=
u2n+1
an−3
2k + 7
k=0
n−4
k
ck x2k+7
u2k+7
7
x dx
1
3a
3a2
a3
=
−
+
−
+
u2n+1
(2n − 7)c4 u2n−7 (2n − 5)c4 u2n−5 (2n − 3)c4 u2n−3 (2n − 1)c4 u2n−1
DW
u5
1
u5
dx =
+ au3 + a2 u + a3 I2
x
5
3
DW
u3
u3
dx =
+ au + a2 I2
x
3
u
dx = u + aI2
x
dx
= I2
xu
n−1
1
dx
1
=
I
+
2
2n+1
n
xu
a
(2k + 1)an−k u2k+1
k=0
5
5
15
u
u5
15
dx = − + cxu3 + acxu + a2 I1
2
x
x
4
8
8
3
3
u
3
3
u
dx = − + cxu + aI1
x2
x
2
2
u
u
dx = − + cI1
x2
x
DW
DW
DW
DW
DW
DW
102
Algebraic Functions
9.
2.276
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
2.277
1.
2.
3.
4.
dx
1
= − n+1
x2 u2n+1
a
u (−1)k+1
+
x
2k − 1
n
k=1
n k x
c
k
u
2.276
2k−1
5
u5
u5
5
5
dx = − 2 + cu3 + acu + a2 cI2
3
x
2x
6
2
2
DW
u3
3
u3
3
dx = − 2 + cu + acI2
3
x
2x
2
2
u
c
u
dx = − 2 + I2
x3
2x
2
dx
u
c
=−
− I2
x3 u
2ax2
2a
3c
3c
dx
1
−
−
=−
I2
x3 u3
2ax2 u 2a2 u 2a2
5 c
dx
1
5 c
5 c
−
=−
−
−
I2
x3 u5
2ax2 u3
6 a2 u 3
2 a3 u 2 a3
DW
DW
DW
DW
DW
u5
au3
2acu c2 xu 5
+
+ acI1
dx
=
−
−
x4
3x3
x
2
2
DW
u3
u3
cu
+ cI1
dx = − 3 −
4
x
3x
x
DW
u
u3
dx
=
−
x4
3ax3
dx
1
= n+2
x4 u2n+1
a
DW
−
cu (−1)k
u3
+
+
(n
+
1)
3x3
x
2k − 3
n+1
k=2
n+1
k
ck
x
u
2k−3
u3
u3
3 cu3
3 c2 u 3 2
+ c I2
dx = − 4 −
+
5
2
x
4x
8 ax
8 a
8
DW
u
u
1 cu
1 c2
I2
dx
=
−
−
−
x5
4x4
8 ax2
8 a
DW
u
dx
3 cu
3 c2
=
−
+
+
I2
x5 u
4ax4
8 a2 x2
8 a2
DW
5 c
15 c2
15 c2
dx
1
+
+
+
=−
I2
5
3
4
2
2
3
x u
4ax u 8 a x u
8 a u
8 a3
DW
2.278
1.
2.
u3
u5
dx
=
−
x6
5ax5
DW
u
u3
2 cu3
dx = −
+
6
5
x
5ax
15 a2 x3
DW
2.284
Forms containing
3.
4.
2 cu3
c2 u
u5
+
−
5x5
3 x3
x
1
1 n+2
u5
= n+3 − 5 +
a
5x
3
1
1
dx
= 3
6
x u
a
dx
x6 u2n+1
√
a + bx + cx2 and polynomials
103
−
2.28 Forms containing
DW
cu3
−
x3
n+2
2
c2 u (−1)k
+
x
2k − 5
n+2
k=3
See also 2.252
tn−1 dt
dx
3
√
=− 2.281
(x + p)n R
c + (b − 2pc)t + (a − bp + cp2 ) t2
t=
1.
3
3.
4.
5.
[x + p > 0]
1
1
dx
dx
dx
√ =
√ +
√
q
−
p
p
−
q
(x + p)(x + q) R
(x + p) R
(x + q) R
√
√
√
1
1
R dx
R dx
R dx
=
+
(x + p)(x + q)
q−p
x+p
p−q
x+q
√
√
√
R dx
(x + p) R dx
=
R dx + (p − q)
x+q
x+q
(rx + s) dx
s − pr
s − qr
dx
dx
√ =
√ +
√
q
−
p
p
−
q
(x + p)(x + q) R
(x + p) R
(x + q) R
2.283
1
>0
x+p
√
dx
x dx
dx
R dx
√
= c √ + (b − cp) √ + a − bp + cp2
x+p
R
R
(x + p) R
2.
(Ax + B) dx
A
√ =
n
c
(p + R) R
c
k
x
u
2k−5
√
a + bx + cx2 and ﬁrst- and second-degree polynomials
Notation: R = a + bx + cx2
2.282
n+2
k
2Bc − Ab
du
n +
2c
(p + u2 )
n−1
dυ
1 − cυ 2
n ,
2
b
− cpυ 2
p+a−
4c
√
b + 2cx
√ .
where u = R and υ =
2c R
2Bc − Ab
A
Ax + B
√ dx = I1 + I2 ,
2.284
c
(p + R) R
c2 p [b2 − 4(a + p)c]
where
.
R
1
[p > 0]
I1 = √ arctan
p
p
√
√
1
−p − R
√
= √
[p < 0]
ln √
2 −p
−p + R
104
Algebraic Functions
b + 2cx
p
√
b2 − 4(a + p)c
R
b + 2cx
p
√
= − arctan
b2 − 4(a + p)c
R
√
√
4(a + p)c − b2 R + p(b + 2cx)
1
√
ln =
√
2i
4(a + p)c − b2 R − p(b + 2cx)
√
√
b2 − 4(a + p)c R − −p(b + 2cx)
1
√
ln =
√
2i
b2 − 4(a + p)c R + −p(b + 2cx)
I2 = arctan
2.290
2
p b − 4(a + p)c > 0,
p<0
2
p b − 4(a + p)c > 0,
p>0
2
p b − 4(a + p)c < 0,
p>0
2
p b − 4(a + p)c < 0,
p<0
2.29 Integrals that can be reduced to elliptic or pseudo-elliptic integrals
2.290
Integrals of the form
R x,
P (x) dx, where P (x) is a third- or fourth-degree polynomial,
can, by means of algebraic transformations, be reduced to a sum of integrals expressed in terms of elementary functions and elliptic integrals (see 8.11). Since the substitutions that transform the given integral
into an elliptic integral in the normal Legendre form are diﬀerent for diﬀerent intervals of integration, the
corresponding formulas are given in the chapter on deﬁnite integrals (see 3.13, 3.17).
"
2.291 Certain integrals of the form R x, P (x) dx, where Pn (x) is a polynomial of not more than
"
fourth degree, can be reduced to integrals of the form R x, k Pn (x) dx with k ≥2. Below are examples
of this procedure.
dz
dx
1
√
=− √
x2 =
1.
1 + z2
1 − x6
3 + 3z 2 + z 4
dz
dx
1
√
√
2.
=
2
4
6
2
2
a + bx + cx + dx
az + bz + cz 3 + dz 4
2
x =z
1
±1/3
3 z 2 A± 3 dz
dx =
3.
a + 2bx + cx2 + gx3
2
B
⎤
⎡
3
2 + (z 3 − a) c
b
−b
+
⎣a + 2bx + cx2 = z 3 , A = g
+ z 3 , B = b2 + (z 3 − a) c⎦
c
dx
√
4.
6
a + bx + cx2 + dx3 + cx4 + bx5 + ax
+
,
dx
dz
1
1
−√
x = z + z2 − 1
= −√
2
2
(z + 1)p
(z − 1)p
+
,
d
1
dz
1
+√
x = z − z2 − 1
= −√
2
2
(z + 1)p
(z − 1)p
where p = 2a 4z 3 − 3z + 2b 2z 2 − 1 + 2cz + d.
2.292
Integrals reducible to elliptic integrals
105
dy
√
[x = y]
√ 2
3
4
y a + by + cy + by + ay
,
+
1
dz
dz
1
=− √
+ √
y = z + z2 − 1
2 2
(z + 1)p 2 2
(z − 1)p
+
,
dz
dz
1
1
= √
− √
y = z − z2 − 1
2 2
(z + 1)p 2 2
(z − 1)p
where p = 2a 2z 2 − 1 + 2bz + c.
a√
dx
18 a
dt
8
√
√√
6.
=
t
;
x
=
2 c
c
t ab t2 + at4
a + bx4 + cx8
1
+
,
dz
a
dz
1
− t = z + z2 − 1
=− √ 8
2 2 c
(z + 1)p
(z − 1)p
+
,
1 8 a
dz
dz
=− √
+ t = z − z2 − 1
2 2 c
(z + 1)p
(z − 1)p
2
where p = 2a 2z − 1 + b1 ; b1 = b ac .
x dx
z 2 dz
√
√
7.
a + bx2 + cx4 = z 4 , A = b2 − 4ac, B = 4c
=
2
4
a + bx2 + cx4
A + Bz 4
4 2
R2 z 4 z 2 dz
b2 − a (c − z 4 ) + b
dx
2
√
8.
z dz = R1 z z dz + ,
=
4
a + 2bx2 + cx4
(c − z 4 ) b2 − a (c − z 4 )
b2 − a (c − z 4 )
where R1 z 4 and R2 z 4 are rational functions of z 4 and a + 2bx2 + cx4 = x4 z 4 .
"
2.292 In certain cases, integrals of the form R x, P (x) dx, where P (x) is a third- or fourth-degree
polynomial, can be expressed in terms of elementary functions. Such integrals are called pseudo-elliptic
integrals.
Thus, if the relations
5.
dx
1
√
=
2
4
6
8
2
a + bx + cx + bx + ax
f1 (x) = f1
1
k2 x
,
f2 (x) = f2
1 − k2 x
k 2 (1 − x)
hold, then
f1 (x) dx
= R1 (z) dz
1.
x(1 − x) (1 − k 2 x)
f2 (x) dx
2.
= R2 (z) dz
x(1 − x) (1 − k 2 x)
f3 (x) dx
3.
= R3 (z) dz
x(1 − x) (1 − k 2 x)
where R1 (z), R2 (z), and R3 (z) are rational functions of z.
f3 (x) = f3
,
1−x
1 − k2 x
,
+
,
z = x(1 − x) (1 − k 2 x)
%
&
x (1 − k 2 x)
√
z=
1−x
%
&
x(1 − x)
z= √
1 − k2 x
106
The Exponential Function
2.311
2.3 The Exponential Function
2.31 Forms containing eax
eax
a
2.312 ax in the integrands should be replaced with ex ln a = ax
2.313 dx
1
[mx − ln (a + bemx )]
1.
=
a + bemx
am
dx
ex
2.
= ln
= x − ln (1 + ex )
x
1+e
1 + ex
dx
a
1
mx
2.314
[ab > 0]
√
arctan
e
=
aemx + be−mx
b
m ab
√
!
!
! b + emx −ab !
1
!
[ab < 0]
√
√
=
ln !!
2m −ab
b − emx −ab !
√
√
dx
a + bemx − a
1
√
2.315
= √ ln √
[a > 0]
√
mx
mx
m a
a + be
a + be
√ + a
2
a + bemx
√
= √
arctan
[a < 0]
m −a
−a
eax dx =
2.311
PE (410)
PE (409)
PE (411)
2.32 The exponential combined with rational functions of x
2.321 n
xn eax
−
xn−1 eax dx
1.
xn eax dx =
a
a
n
(−1)k k! n
n ax
ax
n−k
k
11
x
x e dx = e
2.
ak+1
k=0
2.322
xeax dx = eax
1.
2.
x2 eax dx = eax
x2
2x
2
− 2 + 3
a
a
a
x3 eax dx = eax
x3
3x2
6x
6
− 2 + 3 − 4
a
a
a
a
x4 eax dx = eax
x4
4x3
12x2
24x 24
− 2 + 3 − 4 + 5
a
a
a
a
a
3.
4.10
2.323
x
1
−
a a2
P m (x)e
ax
m
P (k) (x)
eax dx =
(−1)k
,
a
ak
k=0
where P m (x) is a polynomial in x of degree m and P (k) (x) is the k th derivative of P m (x) with respect
to x.
2.325
2.324
Exponentials and rational functions of x
107
ax 1
e dx
eax dx
eax
=
+
a
−
xm
m−1
xm−1
xm−1
ak−1
eax
an−1
ax
Ei(ax)
dx
=
−e
+
xn
(n − 1)(n − 2) . . . (n − k)xn−k
(n − 1)!
1.
2.
n−1
k=1
2.325
eax
dx = Ei(ax)
x
ax
e
eax
+ a Ei(ax)
dx = −
2
x
x
ax
e
a2
eax
aeax
+
Ei(ax)
dx
=
−
−
x3
2x2
2x
2
ax
a3
eax
aeax
a2 eax
e
+
Ei(ax)
dx
=
−
−
−
x4
3x3
6x2
6x
6
±axn
±axn ±axn
1
e
e
e
dx =
dx
− m−1 ± na
xm
m−1
x
xm−n
axn
(−1)z+1 az Γ (−z, −axn )
e
dx =
m
x
n
(−1)z+1 az ∞ e−t
=
dt
z+1
n
−axn t
1.
2.
3.
4.∗
5.∗
6.∗
z=
[m = 1]
m−1
,
n
8.∗
9.∗
10.
∗
11.
∗
12.∗
[n = 0]
n
Ei (axn )
eax
dx =
x
n
#z−1 az−k−1
axn
az Ei (axn )
e
k=0 k! xn(k+1)
axn
+
dx
=
−e
xm
nz!
nz!
a = 0,
7.∗
for Γ(α, x) see 8.350.2
n
n
n
n
n
eax
aeax
a2 Ei (axn )
eax
dx
=
−
−
+
xm
2nx2n
2nxn
2n
n
2
n = 0]
m−1
= 1, 2, . . . ,
n
a = 0,
z=
m = 2, 3, . . .
m−1
=1
n
m−1
=2
z=
n
n
eax
eax
a2 eax
a3 Ei (axn )
eax
dx
=
−
−
−
+
xm
3nx3n
6nx2n
6nxn
6n
√ √
eax
eax
+ aπ erﬁ ax
dx = −
2
x
x
z=
a = 0,
n
eax
a Ei (axn )
eax
dx = − n +
m
x
nx
n
n
[a = 0,
a = 0,
z=
m−1
=3
n
2
where erﬁ(z) =
erf(iz)
i
108
13.
The Exponential Function
∗
2
1
e(ax +2bx+c) dx =
2
2.326
2.33
1.
8
2.∗
3.∗
4.∗
5.∗
∗
e
7.
8.∗
9.
∗
10.∗
11.∗
ac − b2
a
π
exp
a
xe dx
e
= 2
(1 + ax)2
a (1 + ax)
−(ax2 +2bx+c)
2
1
dx =
2
[a = 0]
b2 − ac
a
π
exp
a
1
2
n
eax
dx =
na
m axn
eax
dx =
n
n
n
n
xm eax dx =
eax
n
n
m axn
x e
(γ − 1)!
γ−1
(−1)k+1−γ
k=0
eax
dx =
n
n
[a = 0]
where erﬁ(z) =
erf(iz)
i
[a = 0]
n
xnk
k!aγ−k
xn
1
− 2
a
a
x2n
2xn
2
− 2 + 3
a
a
a
x3n
3x2n
6xn
6
− 2 + 3 − 4
a
a
a
a
Γ (γ, βxn )
γ
nβ
∞
1
tγ−1 e−t dx
=− γ
nβ βxn
xm e−βx dx = −
erf(iz)
i
xm−n e±ax dx
n
m axn
x e
eax
n
%
where erﬁ(z) =
√
b
ax + √
a
erﬁ
m+1−n
xm+1−n
∓
na
na
n
x e
√
b
ax + √
a
[a = 0]
n
xm eax dx =
erf
xm e±ax dx = ±
√
b
ax + √
a
ax
√ π
erﬁ ax
a
ac − b2
1 π
ax2 +bx+c
exp
dx =
e
2 a
a
erﬁ
[a = 0]
ax
eax dx =
6.
∗
2.326
&
[a = 0,
n = 0]
a = 0,
γ=
a = 0,
γ=
a = 0,
γ=
a = 0,
γ=
a = 0,
γ=
m+1
= 1, 2, . . .
n
m+1
=1
n
m+1
=2
n
m+1
=3
n
m+1
=4
n
for Γ(α, x) see 8.350.2
m+1
, β = 0,
γ=
n
&
%γ−1
xnk
(γ
−
1)!
exp (−βxn )
xm exp (−βxn ) dx = −
n
k!β γ−k
k=0
m+1
= 1, 2, . . .
γ=
n
n=0
2.33
12.
Exponentials and rational functions of x
∗
13.∗
14.
∗
15.∗
16.
∗
17.∗
exp (−βxn )
x exp (−βx ) dx = −
nβ
m
n
exp (−βxn )
n
xn
1
+ 2
β
β
xm exp (−βxn ) dx = −
exp (−βxn )
n
x2n
2xn
2
+ 2 + 3
β
β
β
m+1
=3
γ=
n
xm exp (−βxn ) dx = −
exp (−βxn )
n
x3n
3x2n
6xn
6
+ 2 + 3 + 4
β
β
β
β
m+1
=4
γ=
n
e
m+1
=1
γ=
n
m+1
=2
γ=
n
xm exp (−βxn ) dx = −
−βxn
1
dx =
2
π
erf
βx
β
[β = 0]
β z Γ (−z, βxn )
exp (−βxn )
dx
=
−
xm
n
β z ∞ e−t
=−
dt
n βxn tz+a
z=
18.
∗
19.
∗
20.∗
21.∗
22.∗
23.∗
109
m−1
n
Ei (−βxn )
exp (−βxn )
dx =
x
n
n z−k−1
z
exp (−βxn )
z exp (−βx )
k! β
z β
Ei (−βxn )
dx
=
(−1)
(−1)
+
(−1)
n(k+1)
xm
nz!
nz!
x
k=0
m−1
= 1, 2, . . . , m = 2, 3, . . .
z=
n
n
n
n
exp (−βx ) β Ei (−βx )
exp (−βx )
m−1
=1
dx = −
−
z=
xm
nxn
n
n
exp (−βxn ) β exp (−βxn ) β 2 Ei (−βxn )
exp (−βxn )
dx = −
+
+
m
x
2nx2n
2nxn
2n
m−1
=2
z=
n
exp (−βxn ) β exp (−βxn ) β 2 exp (−βxn ) β 3 Ei (−βxn )
exp (−βxn )
dx
=
−
+
−
−
xm
3nx3n
6nx2n
6nxn
6n
m−1
=3
z=
n
exp −βx2
exp −βx2
− βπ erf
dx = −
βx
2
x
x
z−1
110
Hyperbolic Functions
2.411
2.4 Hyperbolic Functions
2.41–2.43 Powers of sinh x, cosh x, tanh x, and coth x
sinhp+1 x coshq−1 x q − 1
+
sinh x cosh x dx =
sinhp x coshq−2 x dx
p+q
p+q
sinhp−1 x coshq+1 x p − 1
=
−
sinhp−2 x coshq x dx
p+q
p+q
sinhp−1 x coshq+1 x p − 1
=
−
sinhp−2 x coshq+2 x dx
q+1
q+1
sinhp+1 x coshq−1 x q − 1
=
−
sinhp+2 x coshq−2 x dx
p+1
p+1
sinhp+1 x coshq+1 x p + q + 2
=
−
sinhp+2 x coshq x dx
p+1
p+1
sinhp+1 x coshq+1 x p + q + 2
=−
+
sinhp x coshq+2 x dx
q+1
q+1
p
2.411
2.412
1.
q
⎡
p+1
x
sinh
⎣ cosh2n−1 x
sinhp x cosh2n x dx =
2n + p
k=0
sinh2m+1 x dx =
3.
m
1 22m
(−1)k
k=0
m
n
= (−1)
k=0
4.
⎤
(2n − 1)(2n − 3) . . . (2n − 2k + 1)
cosh2n−2k−1 x⎦
(2n + p − 2)(2n + p − 4) . . . (2n + p − 2k)
k=1
(2n − 1)!!
+
sinhp x dx
(2n + p)(2n + p − 2) . . . (p + 2)
This formula is applicable for arbitrary real p, except for the following negative even integers:
−2, −4, . . . , −2n. If p is a natural number and n = 0, we have
m−1
2m
x
2m sinh(2m − 2k)x
1
+
(−1)k
TI (543)
sinh2m x dx = (−1)m
m 22m
k
22m−1
2m − 2k
+
2.
n−1
sinhp x cosh2n+1 x dx
(−1)k
2m + 1
k
m
k
cosh(2m − 2k + 1)x
;
2m − 2k + 1
cosh2k+1 x
2k + 1
TI (544)
GU (351) (5)
n
sinhp+1 x
2k n(n − 1) . . . (n − k + 1) cosh2n−2k x
2n
=
cosh x +
2n + p + 1
(2n + p − 1)(2n + p − 3) . . . (2n + p − 2k + 1)
k=1
This formula is applicable for arbitrary real p, except for the following negative odd integers: −1,
−3, . . . , −(2n + 1).
2.414
Powers of hyperbolic functions
2.413
1.
111
⎡
p+1
x
cosh
⎣ sinh2n−1 x
coshp x sinh2n x dx =
2n + p
⎤
2n−2k−1
(2n
−
1)(2n
−
3)
.
.
.
(2n
−
2k
+
1)
sinh
x
⎦
+
(−1)k
(2n + p − 2)(2n + p − 4) . . . (2n + p − 2k)
k=1
(2n − 1)!!
n
+(−1)
coshp x dx
(2n + p)(2n + p − 2) . . . (p + 2)
This formula is applicable for arbitrary real p, except for the following negative even integers:
−2, −4, . . . , −2n. If p is a natural number and n = 0, we have
m−1
2m sinh(2m − 2k)x
2m
x
1
2m
+
TI (541)
cosh x dx =
m 22m
k
22m−1
2m − 2k
n−1
2.
k=0
3.
4.
m
1 2m + 1 sinh(2m − 2k + 1)x
k
22m
2m − 2k + 1
k=0
m
2k+1
m sinh
x
=
k
2k + 1
k=0
⎡
p+1
x
cosh
⎣ sinh2n x
coshp x sinh2n+1 x dx =
2n + p + 1
cosh2m+1 x dx =
+
n
k=1
TI (542)
GU (351) (8)
⎤
2n−2k
k
2
n(n
−
1)
.
.
.
(n
−
k
+
1)
sinh
x
⎦
(−1)k
(2n + p − 1)(2n + p − 3) . . . (2n + p − 2k + 1)
This formula is applicable for arbitrary real p, except for the following negative odd integers: −1,
−3, . . . , −(2n + 1).
2.414
1.
2.
3.
4.
5.
6.
1
cosh ax
a
x
1
sinh2 ax dx =
sinh 2ax −
4a
2
1
1
3
cosh 3x = cosh3 x − cosh x
sinh3 x dx = − cosh x +
4
12
3
1
1
3
3
1
3
sinh4 x dx = x − sinh 2x +
sinh 4x = x − sinh x cosh x + sinh3 x cosh x
8
4
32
8
8
4
5
1
5
sinh5 x dx = cosh x −
cosh 3x +
cosh 5x
8
48
80
4
1
4
= cosh x + sinh4 x cosh x −
cosh3 x
5
5
15
15
3
1
5
sinh6 x dx = − x +
sinh 2x −
sinh 4x +
sinh 6x
16
64
64
192
5
1
5
5
= − x + sinh5 x cosh x −
sinh3 x cosh x +
sinh x cosh x
16
6
24
16
sinh ax dx =
112
Hyperbolic Functions
2.415
7.
8.
9.
10.
11.
12.
sinh7 x dx = − 35
64 cosh x +
7
64
cosh 3x −
7
320
= − 24
35 cosh x +
8
35
cosh3 x −
6
35
1
448
cosh 7x
cosh x sinh4 x +
1
7
cosh x sinh6 x
1
sinh ax
a
1
x
cosh2 ax dx = +
sinh 2ax
2 4a
1
cosh3 x dx = 34 sinh x + 12
sinh 3x = sinh x + 13 sinh3 x
1
cosh4 x dx = 38 x + 14 sinh 2x + 32
sinh 4x = 38 x + 38 sinh x cosh x +
5
1
cosh5 x dx = 58 sinh x + 48
sinh 3x + 80
sinh 5x
cosh ax dx =
=
13.
4
5
sinh x +
+
15
64
sinh 2x +
=
5
16 x
+
5
16
sinh x cosh x +
4
15
sinh 4x +
sinh x +
7
64
sinh 3x +
7
320
=
24
35
sinh x +
8
35
sinh3 x +
6
35
1
192
sinh 5x +
1
448
cosh(a + b)x cosh(a − b)x
+
2(a + b)
2(a − b)
sinh ax cosh ax dx =
1
cosh 2ax
4a
sinh2 x cosh x dx =
1
3
sinh3 x
sinh3 x cosh x dx =
1
4
sinh4 x
sinh4 x cosh x dx =
1
5
sinh5 x
sinh x cosh2 x dx =
1
3
cosh3 x
sinh x cosh5 x
sinh 7x
sinh x cosh4 x +
sinh ax cosh bx dx =
1
6
1
7
sinh x cosh6 x
5.
6.
7.
8.
9.
1
x
+
sinh 4x
8 32
sinh3 x cosh2 x dx = 15 sinh2 x − 23 cosh3 x
1
1
1
x
sinh4 x cosh2 x dx =
−
sinh 2x −
sinh 4x +
sinh 6x
16 64
64
192
sinh2 x cosh2 x dx = −
sinh x cosh3 x
sinh 6x
4.
1
4
sinh3 x
sinh x cosh3 x +
5
24
35
64
3.
3
64
cosh7 x dx =
2.
cosh4 x sinh x +
5
16 x
1.
1
5
cosh6 x dx =
14.
2.415
cosh 5x +
2.416
Powers of hyperbolic functions
113
sinh x cosh3 x dx =
10.
cosh4 x
1
4
sinh2 x cosh3 x dx =
11.
1
5
cosh2 x + 23 sinh3 x
3
sinh3 x cosh3 x dx = − 64
cosh 2x +
12.
1
192
4
cosh 6x =
1
48
cosh3 2x −
1
16
cosh 2x
cosh6 x cosh4 x
sinh6 x sinh x
+
=
−
6
4
6
4
sinh4 x cosh3 x dx = 17 sinh3 x cosh4 x − 35 cosh2 x − 25 = 17 cosh2 x + 25 sinh5 x
sinh x cosh4 x dx = 15 cosh5 x
1
1
1
x
sinh2 x cosh4 x dx = − −
sinh 2x +
sinh 4x +
sinh 6x
16 64
64
192
sinh3 x cosh4 x dx = 17 cosh3 x sinh4 x + 35 sinh2 x − 25 = 17 sinh2 x − 25 cosh5 x
1
1
3x
sinh4 x cosh4 x dx =
−
sinh 4x +
sinh 8x
128 128
1024
=
13.
14.
15.
16.
17.
2.416
1.10
⎡
p
p+1
x⎣
sinh x
sinh
sech2n−1 x
dx =
2n − 1
cosh2n x
⎤
(2n − p − 2)(2n − p − 4) . . . (2n − p − 2k)
sec h2n−2k−1 x⎦
+
(2n − 3)(2n − 5) . . . (2n − 2k − 1)
k=1
(2n − p − 2)(2n − p − 4) . . . (−p + 2)(−p)
+
sinhp x dx
(2n − 1)!!
"
This formula is applicable for arbitrary real p. For sinhp x dx, where p is a natural number, see
2.412 2 and 2.412 3. For n = 0 and p a negative integer, we have for this integral:
⎡
cosh x ⎣
dx
=
− cosech2m−1 x
sinh2m x 2m − 1
⎤
m−1
k
(m
−
1)(m
−
2)
.
.
.
(m
−
k)
2
+
cosec h2m−2k−1 x⎦
(−1)k−1 ·
(2m − 3)(2m − 5) . . . (2m − 2k − 1)
n−1
2.
k=1
3.
⎡
cosh x ⎣
− cosech2m x
=
2m
sinh2m+1 x
dx
+
⎤
(2m
−
1)(2m
−
3)
.
.
.
(2m
−
2k
+
1)
cosec h2m−2k x⎦
(−1)k−1 ·
2k (m − 1)(m − 2) . . . (m − k)
m−1
k=1
+(−1)m
(2m − 1)!!
x
ln tanh
(2m)!!
2
114
Hyperbolic Functions
2.417
1.
2.417
⎡
sinhp x
sinhp+1 x ⎣
sech2n x
dx =
2n
cosh2n+1 x
⎤
(2n − p − 1)(2n − p − 3) . . . (2n − p − 2k + 1)
sec h2n−2k x⎦
+
2k (n − 1)(n − 2) . . . (n − k)
k=1
(2n − p − 1)(2n − p − 3) . . . (3 − p)(1 − p) sinhp x
dx
+
2n n!
cosh x
This formula is applicable for arbitrary real p. For n = 0 and p integral, we have
m
sinh2m+1 x
(−1)m+k
dx =
sinh2k x + (−1)m ln cosh x
cosh x
2k
k=1
m
(−1)m+k m
cosh2k x + (−1)m ln cosh x [m ≥ 1]
=
2k
k
n−1
2.
3.
k=1
2m
sinh x
dx =
cosh x
2.418
1.
dx
(−1)k cosech2m−2k+2 x
+ (−1)m ln tanh x
=
2m − 2k + 2
x cosh x k=1
2m+1
(−1)k cosech2m−2k+2 x
dx
+ (−1)m arctan sinh x
=
2m
2m − 2k + 1
sinh x cosh x k=1
m
⎡
coshp x
coshp+1 x ⎣
cosech2n−1 x
dx = −
2n − 1
sinh2n x
⎤
(−1)k (2n − p − 2)(2n − p − 4) . . . (2n − p − 2k)
+
cosec h2n−2k−1 x⎦
(2n − 3)(2n − 5) . . . (2n − 2k − 1)
k=1
(−1)n (2n − p − 2)(2n − p − 4) . . . (−p + 2)(−p)
+
coshp x dx
(2n − 1)!!
"
This formula is applicable for arbitrary real p. For the integral coshp x dx, where p is a natural
number, see 2.413 2 and 2.413 3. If p is a negative integer, we have for this integral:
m−1
2k (m − 1)(m − 2) . . . (m − k)
dx
sinh x
2m−1
2m−2k−1
sech
sech
x+
x
=
2m − 1
(2m − 3)(2m − 5) . . . (2m − 2k − 1)
cosh2m x
k=1
m−1
(2m − 1)(2m − 3) . . . (2m − 2k + 1)
dx
sinh x
sech2m x +
sech2m−2k x
=
2m
2k (m − 1)(m − 2) . . . (m − k)
cosh2m+1 x
k=1
(2m − 1)!!
arctan sinh x
+
(2m)!!
n−1
2.
3.
sinh2k−1 x + (−1)m arctan (sinh x)
m
sinh
5.
k=1
2k − 1
[m ≥ 1]
4.
m
(−1)m+k
2.423
2.419
Powers of hyperbolic functions
1.
115
⎡
coshp x
coshp+1 x ⎣
cosech2n x
dx = −
2n
sinh2n+1 x
⎤
(−1)k (2n − p − 1)(2n − p − 3) . . . (2n − p − 2k + 1)
cosec h2n−2k x⎦
+
2k (n − 1)(n − 2) . . . (n − k)
k=1
(−1)n (2n − p − 1)(2n − p − 3) . . . (3 − p)(1 − p) coshp x
dx
+
2n n!
sinh x
This formula is applicable for arbitrary real p. For n = 0 and p an integer
m
x
cosh2k−1 x
cosh2m x
dx =
+ ln tanh
sinh x
2k − 1
2
k=1
m
cosh2m+1 x
cosh2k x
dx =
+ ln sinh x
sinh x
2k
k=1
m
m sinh2k x
=
+ ln sinh x
k
2k
n−1
2.
3.
4.
5.
2.421
k=1
sech2m−2k+1 x
x
dx
+ ln tanh
=
2m
2
sinh x cosh x k=1 2m − 2k + 1
m
sech2m−2k+2 x
dx
=
+ ln tanh x
sinh x cosh2m+1 x k=1 2m − 2k + 2
m
In formulas 2.421 1 and 2.421 2, s = 1 for m odd and m < 2n + 1; in all other cases, s = 0.
GI (351)(11, 13)
1.10
2.
n
n
sinh2n+1 x
dx
=
(−1)n+k
m
k
cosh x
k=0
k= m−1
2
n
cosh2n+1 x
dx
=
sinhm x
k=0
k= m−1
2
2.422
1.
sinh
2.
2m
n
k
m−1
cosh2k−m+1 x
+ s(−1)n+ 2
2k − m + 1
sinh2k−m+1 x
+s
2k − m + 1
m+n−1
(−1)k+1
dx
=
2n
2m − 2k − 1
x cosh x
k=0
m+n
(−1)k+1
dx
=
2m − 2k
sinh2m+1 x cosh2n+1 x
k=0
n
m−1
2
m+n−1
k
m+n
k
n
m−1
2
ln cosh x
ln sinh x
tanh2k−2m+1 x
tanh2k−2m x + (−1)m
m+n
m
ln tanh x
k=m
GI (351)(15)
2.423 1.
x
1 cosh x − 1
dx
= ln tanh = ln
sinh x
2
2 cosh x + 1
116
Hyperbolic Functions
2.
3.
4.
5.
6.
7.
8.
9.
dx
= − coth x
sinh2 x
cosh x
1
x
dx
=−
− ln tanh
3
2
2
2
sinh x
2 sinh x
1
dx
cosh x
2
=−
+ coth x = − coth3 x + coth x
4
3
3
3
sinh x
3 sinh x
x
dx
cosh x
3 cosh x
3
=−
+
+ ln tanh
5
4
2
8
8
2
sinh x
4 sinh x
sinh x
dx
4
cosh x
4
coth3 x − coth x
=−
+
6
5
15
5
sinh x
5 sinh x
1
2
= − coth5 x + coth3 x − coth x
5
3
dx
1
x
cosh x
5
15
5
ln tanh
=−
−
+
−
8
16
2
sinh7 x
6 sinh2 x sinh4 x 4 sinh2 x
3
1
dx
= coth x − coth3 x + coth5 x − coth7 x
8
5
7
sinh x
dx
= arctan (sinh x)
cosh x
= arcsin (tanh x)
= 2 arctan (ex )
= gd x
10.
11.
12.
13.
14.
15.
16.
17.
dx
= tanh x
cosh2 x
sinh x
1
dx
=
+ arctan (sinh x)
cosh3 x
2 cosh2 x 2
dx
sinh x
2
=
+ tanh x
cosh4 x 3 cosh3 x 3
1
= − tanh3 x + tanh x
3
dx
sinh x
3 sinh x
3
=
+
+ arctan (sinh x)
5
4
2
cosh x
4 cosh x 8 cosh x 8
dx
4
sinh x
4
tanh3 x + tanh x
=
−
6
5
5
cosh x 5 cosh x 15
1
2
= tanh5 x − tanh3 x + tanh x
5
3
1
dx
sinh x
5
15
5
arctan (sinh x)
=
+
+
+
8
16
cosh7 x
6 cosh2 x cosh4 x 4 cosh2 x
dx
3
1
= − tanh7 x + tanh5 x − tanh3 x + tanh x
8
7
5
cosh x
sinh x
dx = ln cosh x
cosh x
2.423
2.423
Powers of hyperbolic functions
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
31.
32.
33.
34.
sinh2 x
dx = sinh x − arctan (sinh x)
cosh x
sinh3 x
1
dx = sinh2 x − ln cosh x
cosh x
2
1
= cosh2 x − ln cosh x
2
1
sinh4 x
dx = sinh3 x − sinh x + arctan (sinh x)
cosh x
3
sinh x
1
dx = −
cosh x
cosh2 x
sinh2 x
dx = x − tanh x
cosh2 x
sinh3 x
1
dx = cosh x +
cosh x
cosh2 x
1
sinh4 x
3
dx = − x + sinh 2x + tanh x
2
2
4
cosh x
sinh x
1
dx = −
cosh3 x
2 cosh2 x
1
= tanh2 x
2
2
sinh x
sinh x
1
dx = −
+ arctan (sinh x)
3
2
cosh x
2 cosh x 2
3
sinh x
1
dx = − tanh2 x + ln cosh x
3
2
cosh x
1
=
+ ln cosh x
2 cosh2 x
sinh x
3
sinh4 x
dx =
+ sinh x − arctan (sinh x)
2 cosh x
2
cosh3 x
sinh x
1
dx = −
cosh4 x
3 cosh3 x
1
sinh2 x
dx = tanh3 x
4
3
cosh x
3
sinh x
1
1
+
dx = −
4
cosh
x
cosh x
3 cosh3 x
4
sinh x
1
dx = − tanh3 x − tanh x + x
3
cosh4 x
cosh x
dx = ln sinh x
sinh x
cosh2 x
x
dx = cosh x + ln tanh
sinh x
2
117
118
Hyperbolic Functions
35.
36.
37.
38.
39.
40.
41.
42.
43.
44.
45.
46.
47.
48.
49.
50.
51.
1
cosh3 x
dx = cosh2 x + ln sinh x
sinh x
2
cosh4 x
1
x
dx = cosh3 x + cosh x + ln tanh
sinh x
3
2
cosh x
1
dx = −
2
sinh
x
sinh x
2
cosh x
dx = x − coth x
sinh2 x
1
cosh3 x
dx = sinh x −
2
sinh
x
sinh x
1
cosh4 x
3
dx = x + sinh 2x − coth x
2
4
sinh2 x
cosh x
1
dx = −
sinh3 x
2 sinh2 x
1
= − coth2 x
2
2
cosh x
x
cosh x
dx = −
+ ln tanh
2
sinh3 x
2 sinh2 x
3
cosh x
1
dx = −
+ ln sinh x
3
sinh x
2 sinh2 x
1
= − coth2 x + ln sinh x
2
4
x
cosh x
cosh x
3
dx = −
+ cosh x + ln tanh
3
2
2
2
sinh x
2 sinh x
1
cosh x
dx = −
sinh4 x
3 sinh3 x
cosh2 x
1
dx = − coth3 x
3
sinh4 x
3
cosh x
1
1
dx = −
−
4
sinh x 3 sinh3 x
sinh x
cosh4 x
1
dx = − coth3 x − coth x + x
3
sinh4 x
dx
= ln tanh x
sinh x cosh x
x
dx
1
+ ln tanh
=
2
cosh
x
2
sinh x cosh x
dx
1
=
+ ln tanh x
3
sinh x cosh x 2 cosh2 x
1
= − tanh2 x + ln tanh x
2
2.423
2.424
Powers of hyperbolic functions
52.
53.
54.
55.
56.
57.
58.
59.
60.
61.
62.
63.
64.
2.424
1.
2.
1
dx
1
x
+
=
+ ln tanh
4
3
cosh
x
2
sinh x cosh x
3 cosh x
dx
1
− arctan sinh x
=−
sinh x
sinh2 x cosh x
dx
= −2 coth 2x
sinh2 x cosh2 x
sinh x
1
3
dx
=−
−
− arctan sinh x
sinh2 x cosh3 x
2 cosh2 x sinh x 2
dx
1
8
=
− coth 2x
sinh2 x cosh4 x
3 sinh x cosh3 x 3
dx
1
=−
− ln tanh x
sinh3 x cosh x
2 sinh2 x
1
= − coth2 x + ln coth x
2
dx
cosh x
x
1
3
−
=−
− ln tanh
3
2
2
cosh
x
2
2
sinh x cosh x
2 sinh x
dx
2 cosh 2x
=−
− 2 ln tanh x
3
3
sinh x cosh x
sinh2 2x
1
1
= tanh2 x − coth2 x − 2 ln tanh x
2
2
dx
1
x
2
cosh x
5
−
=−
−
− ln tanh
3
4
2
2
cosh x 3 cosh x 2 sinh x 2
2
sinh x cosh x
1
dx
1
−
=
+ arctan sinh x
sinh x 3 sinh3 x
sinh4 x cosh x
1
8
dx
=−
+ coth 2x
4
2
3
sinh x cosh x
3 cosh x sinh x 3
1
dx
2
sinh x
5
−
=
+
+ arctan sinh x
4
3
3
2
sinh x 3 sinh x 2 cosh x 2
sinh x cosh x
dx
8
= 8 coth 2x − coth3 2x
4
4
3
sinh x cosh x
tanhp−1 x
+ tanhp−2 x dx
[p = 1]
p−1
n
1
(−1)k−1 n
+ ln cosh x
tanh2n+1 x dx =
2k
k cosh2k x
k=1
n
tanh2n−2k+2 x
=−
+ ln cosh x
2n − 2k + 2
tanhp x dx = −
k=1
3.
4.
119
n
tanh2n−2k+1 x
+x
2n − 2k + 1
k=1
cothp−1 x
+ cothp−2 x dx
cothp x dx = −
p−1
tanh2n x dx = −
GU (351)(12)
[p = 1]
120
Hyperbolic Functions
2.425
n
1
1 n
+ ln sinh x
2n k sinh2k x
k=1
n
coth2n−2k+2 x
=−
+ ln sinh x
2n − 2k + 2
coth2n+1 x dx = −
5.
k=1
coth2n x dx = −
6.
n
k=1
coth2n−2k+1 x
+x
2n − 2k + 1
GU (351)(14)
For formulas containing powers of tanh x and coth x equal to n = 1, 2, 3, 4, see 2.423 17, 2.423 22,
2.423 27, 2.423 32, 2.423 33, 2.423 38, 2.423 43, 2.423 48.
Powers of hyperbolic functions and hyperbolic functions of linear functions of the argument
2.425
1.
sinh(ax + b) sinh(cx + d) dx =
2.
sinh(ax + b) cosh(cx + d) dx =
3.
4.
5.
6.
2.426
1.
cosh(ax + b) cosh(cx + d) dx =
1
sinh[(a + c)x + b + d]
2(a + c)
1
−
sinh[(a − c)x + b − d]
2(a − c)
2
a = c2
GU (352)(2a)
1
cosh[(a + c)x + b + d]
2(a + c)
1
+
cosh[(a − c)x + b − d]
2(a − c)
2
a = c2
GU (352)(2c)
1
sinh[(a + c)x + b + d]
2(a + c)
1
+
sinh[(a − c)x + b − d]
2(a − c)
2
a = c2
GU (352)(2b)
When a = c:
1
x
sinh(2ax + b + d)
sinh(ax + b) sinh(ax + d) dx = − cosh(b − d) +
2
4a
1
x
cosh(2ax + b + d)
sinh(ax + b) cosh(ax + d) dx = sinh(b − d) +
2
4a
1
x
sinh(2ax + b + d)
cosh(ax + b) cosh(ax + d) dx = cosh(b − d) +
2
4a
sinh ax sinh bx sinh cx dx =
GU (352)(3a)
GU (352)(3c)
GU (352)(3b)
cosh(a + b + c)x cosh(−a + b + c)x
−
4(a + b + c)
4(−a + b + c)
cosh(a − b + c)x cosh(a + b − c)x
−
−
4(a − b + c)
4(a + b − c)
GU (352)(4a)
2.428
Powers of hyperbolic functions
2.
sinh ax sinh bx cosh cx dx =
sinh ax cosh bx cosh cx dx =
cosh(a + b + c)x cosh(−a + b + c)x
−
4(a + b + c)
4(−a + b + c)
cosh(a − b + c)x cosh(a + b − c)x
+
+
4(a − b + c)
4(a + b − c)
GU (352)(4c)
4.
sinh(a + b + c)x sinh(−a + b + c)x
−
4(a + b + c)
4(−a + b + c)
sinh(a − b + c)x sinh(a + b − c)x
−
+
4(a − b + c)
4(a + b − c)
GU (352)(4b)
3.
121
cosh ax cosh bx cosh cx dx =
sinh(a + b + c)x sinh(−a + b + c)x
+
4(a + b + c)
4(−a + b + c)
sinh(a − b + c)x sinh(a + b − c)x
+
+
4(a − b + c)
4(a + b − c)
GU (352)(4d)
2.427
1
sinh x sinh ax dx =
p+a
p
1.
sinhp x sinh(2n + 1)x dx =
2.
sinhp x sinh 2nx dx =
3.
p−1
x cosh(a − 1)x dx
sinh px cosh ax − p sinh
Γ(p + 1)
Γ ⎡p+3
2 +n
n−1
Γ p+1 + n − 2k
2
sinhp−2k x cosh(2n − 2k + 1)x
×⎣
22k+1 Γ (p − 2k + 1)
k=0
⎤
p−1
Γ 2 + n − 2k
− 2k+2
sinhp−2k−1 x sinh(2n − 2k)x⎦
2
Γ(p − 2k)
Γ p+3
2 −n
+ 2n
sinhp−2n x sinh x dx
2 Γ(p + 1 − 2n)
[p is not a negative integer]
Γ(p + 1)
p
Γ 2 +⎡n + 1
n−1
Γ p2 + n − 2k
⎣
sinhp−2k x cosh(2n − 2k)x
×
22k+1 Γ(p − 2k + 1)
k=0
⎤
Γ p2 + n − 2k − 1
−
sinhp−2k−1 x sinh(2n − 2k − 1)x⎦
22k+2 Γ(p − 2k)
[p is not a negative integer]
2.428
1.
1
sinh x cosh ax dx =
p+a
p
p−1
x sinh(a − 1)x dx
sinh x sinh ax − p sinh
p
GU (352)(5)a
122
Hyperbolic Functions
sinhp x cosh(2n + 1)x dx =
2.
Γ(p + 1)
Γ ⎧p+3
2 +n
⎡
⎨ n−1
Γ p+1 + n − 2k
2
⎣
sinhp−2k x sinh(2n − 2k + 1)x
×
⎩
22k+1 Γ(p − 2k + 1)
k=0
⎤
p−1
Γ 2 + n − 2k
− 2k+2
sinhp−2k−1 x cosh(2n − 2k)x⎦
2
Γ(p − 2k)
⎫
⎬
Γ p+3
−
n
2
sinhp−2n x cosh x dx
+ 2n
⎭
2 Γ(p + 1 − 2n)
[p is not a negative integer]
sinhp x cosh 2nx dx =
3.
2.429
Γ(p + 1)
p
Γ ⎧2 + n⎡+ 1
⎨n−1
Γ p2 + n − 2k
⎣
sinhp−2k x sinh(2n − 2k)x
×
⎩
22k+1 Γ(p − 2k + 1)
k=0
⎤
Γ p2 + n − 2k − 1
−
sinhp−2k−1 x cosh(2n − 2k − 1)x⎦
22k+2 Γ(p − 2k)
⎫
p
⎬
Γ −n+1
+ 2n 2
sinhp−2n x dx
⎭
2 Γ(p + 1 − 2n)
[p is not a negative integer]
2.429
coshp x sinh ax dx =
1.
2.
1
p+a
GU (352)(6)a
coshp x cosh ax + p coshp−1 x sinh(a − 1)x dx
⎡
p+1
n−1
Γ(p + 1) ⎣ Γ 2 + n − k
cosh x sinh(2n + 1)x dx = p+3
coshp−k x cosh(2n − k + 1)x
2k+1 Γ(p − k + 1)
Γ 2 +n
k=0
⎤
p+3 Γ 2
+ n
coshp−n x sinh(n + 1)x dx⎦
2 Γ(p − n + 1)
p
[p is not a negative integer]
p
n−1
Γ(p + 1) ⎣ Γ 2 + n − k
p
coshp−k x cosh(2n − k)x
cosh x sinh 2nx dx =
2k+1 Γ(p − k + 1)
Γ p2 + n + 1
k=0
⎤
p
Γ 2 +1
+ n
coshp−n x sinh nx dx⎦
2 Γ(p − n + 1)
⎡
3.
[p is not a negative integer]
GU (352)(7)a
2.433
2.431
Powers of hyperbolic functions
coshp x cosh ax dx =
1.
coshp x cosh(2n + 1)x dx =
2.
3.
coshp x sinh ax + p coshp−1 x cosh(a − 1)x dx
1
p+a
⎡
p+1
n−1
Γ(p + 1) ⎣ Γ 2 + n − k
p+3
coshp−k x sinh(2n − k + 1)x
2k+1 Γ(p − k + 1)
Γ 2 +n
k=0
⎤
p+3 Γ 2
+ n
coshp−n x cosh(n + 1)x dx⎦
2 Γ(p − n + 1)
[p is not a negative integer]
n−1
Γ(p + 1) ⎣ Γ 2 + n − k
p
coshp−k x sinh(2n − k)x
cosh x cosh 2nx dx = p
2k+1 Γ(p − k + 1)
Γ 2 +n+1
k=0
⎤
p
Γ 2 +1
coshp−n x cosh nx dx⎦
+ n
2 Γ(p − n + 1)
⎡
p
[p is not a negative integer]
2.432
1.
2.
3.
4.
2.433
123
GU (352)(8)a
1
sinhn x sinh nx
n
1
sinh(n + 1)x coshn−1 x dx = coshn x cosh nx
n
1
cosh(n + 1)x sinhn−1 x dx = sinhn x cosh nx
n
1
cosh(n + 1)x coshn−1 x dx = coshn x sinh nx
n
sinh(n + 1)x sinhn−1 x dx =
1.
sinh(2n − 2k)x
sinh(2n + 1)x
dx = 2
+x
sinh x
2n − 2k
n−1
k=0
2.
3.
sinh 2nx
dx = 2
sinh x
n−1
k=0
sinh(2n − 2k − 1)x
2n − 2k − 1
GU (352)(5d)
cosh(2n − 2k)x
cosh(2n + 1)x
dx = 2
+ ln sinh x
sinh x
2n − 2k
n−1
k=0
4.
5.
cosh 2nx
dx = 2
sinh x
n−1
k=0
x
cosh(2n − 2k − 1)x
+ ln tanh
2n − 2k − 1
2
cosh(2n − 2k)x
sinh(2n + 1)x
dx = 2
+ (−1)n ln cosh x
(−1)k
cosh x
2n − 2k
n−1
k=0
GU (352)(6d)
124
Hyperbolic Functions
6.
sinh 2nx
cosh(2n − 2k − 1)x
dx = 2
(−1)k
cosh x
2n − 2k − 1
2.433
n−1
GU (352)(7d)
k=0
7.
sinh(2n − 2k)x
cosh(2n + 1)x
dx = 2
+ (−1)n x
(−1)k
cosh x
2n − 2k
n−1
k=0
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
cosh 2nx
dx = 2
cosh x
n−1
(−1)k
k=0
sinh(2n − 2k − 1)x
+ (−1)n arcsin (tanh x)
2n − 2k − 1
2
sinh 2x
dx = −
sinhn x
(n − 2) sinhn−2 x
For n = 2:
sinh 2x
dx = 2 ln sinh x
sinh2 x
sinh 2x dx
2
=
coshn x
(2 − n) coshn−2 x
For n = 2:
sinh 2x
dx = 2 ln cosh x
cosh2 x
cosh 2x
x
dx = 2 cosh x + ln tanh
sinh x
2
cosh 2x
dx = − coth x + 2x
sinh2 x
x
cosh x
3
cosh 2x
dx = −
+ ln tanh
2
sinh3 x
2 sinh2 x 2
cosh 2x
dx = 2 sinh x − arcsin (tanh x)
cosh x
cosh 2x
dx = − tanh x + 2x
cosh2 x
sinh x
3
cosh 2x
dx = −
+ arcsin (tanh x)
cosh3 x
2 cosh2 x 2
sinh 3x
dx = x + sinh 2x
sinh x
sinh 3x
x
dx = 3 ln tanh + 4 cosh x
2
2
sinh x
sinh 3x
dx = −3 coth x + 4x
sinh3 x
4
1
sinh 3x
dx =
−
n−3
coshn x
(3 − n) cosh
x (1 − n) coshn−1 x
For n = 1 and n = 3:
sinh 3x
dx = 2 sinh2 x − ln cosh x
cosh x
GU (352)(8d)
2.442
Rational functions of hyperbolic functions
24.
25.
26.
27.
28.
29.
30.
sinh 3x
1
dx =
+ 4 ln cosh x
3
cosh x
2 cosh2 x
1
4
cosh 3x
+
dx =
n−3
sinhn x
(3 − n) sinh
x (1 − n) sinhn−1 x
For n = 1 and n = 3:
cosh 3x
dx = 2 sinh2 x + ln sinh x
sinh x
cosh 3x
1
dx = −
+ 4 ln sinh x
3
sinh x
2 sinh2 x
cosh 3x
dx = sinh 2x − x
cosh x
cosh 3x
dx = 4 sinh x − 3 arcsin (tanh x)
cosh2 x
cosh 3x
dx = 4x − 3 tanh x
cosh3 x
2.44–2.45 Rational functions of hyperbolic functions
2.441
1.
2.
3.
2.442
1.
2.
cosh x
A + B sinh x
aB − bA
·
n dx =
(n − 1) (a2 + b2 ) (a + b sinh x)n−1
(a + b sinh x)
1
(n − 1)(aA + bB) + (n − 2)(aB − bA) sinh x
+
dx
n−1
(n − 1) (a2 + b2 )
(a + b sinh x)
For n = 1:
B
aB − bA
A + B sinh x
dx
dx = x −
a + b sinh x
b
b
a + b sinh x
√
a tanh x2 − b + a2 + b2
dx
1
√
=√
ln
a + b sinh x
a2 + b2 a tanh x2 − b − a2 + b2
a tanh x2 − b
2
=√
arctanh √
a2 + b 2
a2 + b 2
(see 2.441 3)
A + B cosh x
B
dx
dx
=
−
+
A
n
n
n−1
(a + b sinh x)
(a
+
b
sinh x)
(n − 1)b (a + b sinh x)
For n = 1:
B
dx
A + B cosh x
dx = ln (a + b sinh x) + A
a + b sinh x
b
a + b sinh x
(see 2.441 3)
125
126
2.443
Hyperbolic Functions
1.
2.
3.
2.444
1.
2.11
2.445
1.
2.
2.443
A + B cosh x
sinh x
aB − bA
·
n dx =
n−1
2
2
(n
−
1)
(a
−
b
)
(a + b cosh x)
(a + b cosh x)
1
(n − 1)(aA − bB) + (n − 2)(aB − bA) cosh x
+
dx
n−1
(n − 1) (a2 − b2 )
(a + b cosh x)
For n = 1:
B
aB − bA
A + B cosh x
dx
dx = x −
a + b cosh x
b
b
a + b cosh x
1
dx
b + a cosh x
=√
arcsin
a + b cosh x
a + b cosh x
b 2 − a2
b + a cosh x
1
arcsin
= −√
a + b cosh x
b 2 − a2
√
x
a2 − b2 tanh
a
+
b
+
1
2
=√
ln
√
a2 − b2 a + b − a2 − b2 tanh x
2
(see 2.443 3)
2
b > a2 ,
x<0
2
b > a2 ,
x>0
2
a > b2
dx
x+a
x−a
= cosech a ln cosh
− ln cosh
cosh a + cosh x
2
2
a
x
= 2 cosech a arctanh tanh tanh
2
2
x
a
dx
= 2 cosec a arctan tanh tan
cos a + cosh x
2
2
B
B sinh x
dx = −
n
(a + b cosh x)
(n − 1)b(a + b cosh x)n−1
For n = 1:
B
B sinh x
dx = ln (a + b cosh x)
a + b cosh x
b
[n = 1]
(see 2.443 3)
In evaluating deﬁnite integrals by use of formulas 2.441–2.443 and 2.445, one may not take the integral
over points at which the integrand becomes inﬁnite, that is, over the points
a
x = arcsinh −
b
in formulas 2.441 or 2.442 or over the points
a
x = arccosh −
b
in formulas 2.443 or 2.445. Formulas 2.443 are not applicable for a2 = b2 . Instead, we may use the
following formulas in these cases:
2.449
2.446
Rational functions of hyperbolic functions
1.
A + B cosh x
n dx
(ε + cosh x)
=
B sinh x
n
B
n + εA +
n−1
(1 − n) (ε + cosh x)
×
2.
127
(2n − 2k − 3)!!
(n − 1)!
sinh x
(2n − 1)!!
(n − k − 1)!
n−1
k=0
εh
(ε + cosh x)
n−k
[ε = ±1,
For n = 1:
cosh x − ε
A + B cosh x
dx = Bx + (εA − B)
ε + cosh x
sinh x
n > 1]
[ε = ±1]
2.447
1.
2.
3.
2.448
b
sinh x dx
a ln cosh x + arctanh
bx
a
a cosh x + b sinh x =
a2 − b 2
a
bx − a ln sinh x + arctanh
b
=
b 2 − a2
For a = b = 1:
x 1
sinh x dx
= + e−2x
cosh x + sinh x
2 4
For a = −b = 1:
x 1
sinh x dx
= − + e2x
cosh x − sinh x
2
4
1.
2.
3.
2.449
1.6
[a > |b|]
[b > |a|]
MZ 215
MZ 215
ax − b ln cosh x + arctanh ab
cosh x dx
=
a cosh x + b sinh x
a2 −b2
−ax + b ln sinh x + arctanh ab
=
b 2 − a2
[a > |b|]
[b > |a|]
For a = b = 1:
x 1
cosh x dx
= − e−2x
cosh x + sinh x
2 4
For a = −b = 1:
x 1
cosh x dx
= + e2x
cosh x − sinh x
2 4
1
dx
n = n
2
(a cosh x + b sinh x)
(a − b2 )
1
=
n
2
(b − a2 )
MZ 214, 215
dx
sinhn x + arctanh
b
a
dx
cosh
n
x + arctanh
a
b
[a > |b|]
[b > |a|]
128
2.
3.
4.
2.451
Hyperbolic Functions
For n = 1:
!
!
b
1
dx
=√
arctan !!sinh x + arctanh
2
2
a cosh x + b sinh x
a
a −b
!
!
!
x + arctanh ab !!
1
!
=√
ln tanh
!
2
b 2 − a2 !
3.
!
!
!
!
[a > |b|]
[b > |a|]
For a = b = 1:
ax
= −e−x = sinh x − cosh x
cosh x + sinh x
For a = −b = 1:
dx
= ex = sinh x + cosh x
cosh x − sinh x
1.
2.
2.451
MZ 214
A + B cosh x + C sinh x
n dx
(a + b cosh x + c sinh x)
Bc − Cb + (Ac − Ca) cosh x + (Ab − Ba) sinh x
1
=
n−1 + (n − 1) (a2 − b2 + c2 )
(1 − n) (a2 − b2 + c2 ) (a + b cosh x + c sinh x)
(n − 1)(Aa − Bb + Cc) − (n − 2)(Ab − Ba) cosh x − (n − 2)(Ac − Ca) sinh x
×
dx
n−1
(a + b cosh x + c sinh x)
2
a + c2 = b 2
A n(Bb − Cc)
Bc − Cb − Ca cosh x − Ba sinh x
(n − 1)!
=
+
(c cosh x + b sinh x)
n +
2
a
(n − 1)a
(2n − 1)!!
(n − 1)a (a + b cosh x + c sinh x)
n−1
(2n − 2k − 3)!!
1
×
(n − k − 1)!ak (a + b cosh x + c sinh x)n−k
k=0
2
a + c2 = b 2
Cb − Bc
A + B cosh x + C sinh x
dx = 2
ln (a + b cosh x + c sinh x)
a + b cosh x + c sinh x
b − c2
Bb − Cc
dx
Bb − Cc
x
+
A
−
a
+ 2
2
2
2
b −c
b − c a + b cosh x + c sinh x
(see 2.451 4)
b 2 = c2
A (B ∓ C) b
C ∓B
A + B cosh x + C sinh x
dx =
(cosh x ∓ sinh x) +
−
x
a + b cosh x ± b sinh x
a
2a2
2a
C ±B
A (C ∓ B) b
± −
+
ln (a + b cosh x ± b sinh x)
2b
a
2a2
[ab = 0]
2.452
Rational functions of hyperbolic functions
4.
dx
a + b cosh x + c sinh x
(b − a) tanh x2 + c
2
arctan √
=√
b 2 − a2 − c 2
b 2 − a2 − c 2 √
x
(a
−
b)
tanh
a2 − b 2 + c 2
1
2 −c+
√
=√
ln
x
a2 − b2 + c2 (a − b) tanh 2 − c − a2 − b2 + c2
x
1
= ln a + c tanh
c
2
2
=
(a − b) tanh x2 + c
129
b2 > a2 + c2 and a = b
b2 < a2 + c2 and a = b
[a = b and c = 0]
2
b = a2 + c 2
GU (351)(18)
2.452
1.
A + B cosh x + C sinh x
dx
(a1 + b1 cosh x + c1 sinh x) (a2 + b2 cosh
x + c2 sinh x)
a1 + b1 cosh x + c1 sinh x
dx
dx
= A0 ln
+ A1
+ A2
a2 + b2 cosh x + c2 sinh x
a1 + b1 cosh x + c1 sinh x
a2 + b2 cosh x + c2 sinh x
where
GU (351)(19)
!
!
!a1 b1 c1 !
!
!
!A B C!
!
!
!a2 b2 c2 !
!2 !
!
!
!b1 c1 !2 !c1
b1 !!
!
!
+!
− !!
b2 !
b 2 c2 !
c2
A0 = !
!2
!a1
a1 !!
!
!a2
a2 !
!
!
! a1
!
!!
! ! b 1 ! ! c1 ! !
!! C B ! ! C A ! ! B A !!
!!
! !
! !
!!
!!c2 b2 ! !c2 a2 ! !b2 a2 !!
!
!
! a2
b2
c2 !
A2 = !
!
!
! ,
!
!
!a1 b1 !2 !b1 c1 !2 !c1 a1 !2
!
! +!
! −!
!
!a2 b2 !
!b2 c2 !
!c2 a2 !
2.
!
!
!
! a1
!!
! ! b1 ! ! c1 !!
!!b1 c1 ! !c1 a1 ! !a1 b1 !!
!!
! !
! !
!!
!! B C ! ! C A ! ! A B !!
!
!
! a2
b2
c2 !
A1 = !
!
!
! ,
!
!
!a1 b1 !2 !b1 c1 !2 !c1 a1 !2
!
! +!
! −!
!
! a2 b 2 !
! b 2 c2 !
! c 2 a2 !
%!
! a1
!
! a2
!2 !
!b
b1 !!
+ !! 1
!
b2
b2
!2 !
!c
c1 !!
= !! 1
!
c2
c2
!2 &
a1 !!
.
a2 !
A cosh2 x + 2B sinh x cosh x + C sinh2 x
dx
a cosh2 x + 2b sinh x cosh x + c sinh2 x ⎧
⎨
1
[4Bb − (A + C)(a + c)]x
= 2
4b − (a + c)2 ⎩
+ [(A + C)b − B(a + c)] ln a cosh2 x + 2b sinh x cosh x + c sinh2 x
⎫
⎬
+ 2(A − C)b2 − 2Bb(a − c) + (Ca − Ac)(a + c) f (x)
⎭
130
Hyperbolic Functions
2.453
where
√
1
c tanh x + b − b2 − ac
√
f (x) = √
ln
2 b2 − ac c tanh x + b + b2 − ac
c tanh x + b
1
arctan √
=√
ac − b2
ac − b2
1
=−
c tanh x + b
2.453
1.
2.
GU (351)(24)
b2 > ac
b2 < ac
b2 = ac
!
dx
1
x !!
(A + B sinh x) dx
!
=
A ln !tanh ! + (aB − bA)
sinh x (a + b sinh x)
a
2
a + b sinh x
A
(A + B sinh x) dx
= 2
sinh x (a + b cosh x)
a − b2
(see 2.441 3)
!
!
!
!
! a + b cosh x !
dx
x!
!
!
!
+B
a ln !tanh ! + b ln !
2
sinh x !
a + b cosh x
(see 2.443 3)
2
3.
4.
2.454
1.
2.
2
For a = b = 1:
A
(A + B sinh x) dx
=
sinh x (1 + cosh x)
2
(A + B sinh x) dx
A
=
sinh x (1 − cosh x)
2
!
x
x !! 1
x
!
ln !tanh ! − tanh2
+ B tanh
2
2
2
2
!
!
x! 1
x
x
!
− ln !coth ! + coth2
+ B coth
2
2
2
2
!
!
! a + b sinh x !
1
(A + B sinh x) dx
!
!
= 2
(Aa + Bb) arctan (sinh x) + (Ab − Ba) ln !
cosh x (a + b sinh x)
a + b2
cosh x !
!
!
!
! sinh x !
dx
(A + B cosh x) dx
1
x !!
!
!
! − Ab
=
A ln !tanh ! + B ln !
!
sinh x (a + b sinh x)
a
2
a + b sinh x
a + b sinh x
(see 2.441 3)
2.455
1.
!
!
!
! a + b cosh x !
x !!
(A + B cosh x) dx
1
!
!
!
= 2
(Aa + Bb) ln !tanh ! + (Ab − Ba) ln !
sinh x (a + b cosh x)
a − b2
2
sinh x !
For a2 = b2 = 1:
x !! A − B
x
A + B !!
(A + B cosh x) dx
=
ln !tanh ! −
tanh2
2.
sinh x (1 + cosh x)
2
2
4
2
A+B
x
x A−B
(A + B cosh x) dx
=
coth2 −
ln coth
3.
sinh x (1 − cosh x)
4
2
2
2
!
!
! a + b sinh x !
dx
(A + B cosh x) dx
A
! +B
!
2.456
= 2
a
arctan
(sinh
x)
+
b
ln
! cosh x !
cosh x (a + b sinh x) a + b2
a + b sinh x
(see 2.441 3)
2.459
2.457
Rational functions of hyperbolic functions
1.
131
1
(A + B cosh x) dx
dx
=
A arctan sinh x − (Ab − Ba)
cosh x (a + b cosh x)
a
a + b cosh x
(see 2.443 3)
2.458
1.
dx
a + b sinh2 x
b
=
− 1 tanh x
arctan
a
a(b − a)
1
b
=
arctanh
1 − tanh x
a
a(a − b)
1
b
=
arccoth
1 − tanh x
a
a(a − b)
1
b
>1
a
b
b
a
0 < < 1 or < 0 and sinh2 x < −
a
a
b
b
a
< 0 and sinh2 x > −
a
b
MZ 195
2.
dx
a + b cosh2 x
.
1
b
=
arctan
− 1+
coth x
a
−a(a + b)
1
b
=
arctanh
1 + coth x
a
a(a + b)
b
1
arccoth
1 + coth x
=
a
a(a + b)
b
< −1
a
b
a
−1 < < 0 and cosh2 x > −
a
b
b
b
a
2
> 0 or − 1 < < 0 and cosh x < −
a
a
b
MZ 202
2
3.
4.
5.
6.
2.459
1.
2
For a = b = 1:
dx
= tanh x
1 + sinh2 x
√
dx
1
2 tanh x
= √ arctanh
2
1 − sinh x
2
√
1
2 tanh x
= √ arccoth
2
√
dx
1
2 coth x
= √ arccoth
2
1 + cosh x
2
dx
= coth x
1 − cosh2 x
sinh2 x < 1
sinh2 x > 1
b sinh x cosh x
1
dx
=
+
(b
−
2a)
2
2a(b − a) a + b sinh2 x
a + b sinh2 x
a + b sinh2 x
dx
(see 2.458 1)
MZ 196
132
Hyperbolic Functions
2.
1
dx
b sinh x cosh x
+ (2a + b)
−
2 =
2a(a + b)
a + b cosh2 x
a + b cosh2 x
a + b cosh2 x
dx
(see 2.458 2)
⎡
3.
2
1 ⎣
3
3− 2 + 4
3 =
3
8pa
p
p
a + b sinh x
dx
2
+
+
arctan (p tanh x) + 3 −
⎤
2
1
1 + 2 − 2 tanh2 x
p
p
⎡
1 ⎣
2
3
=
3+ 2 + 4
8qa3
q
q
MZ 203
3
2
− 4
p2
p
2p tanh x
2 ⎦
1 + p2 tanh2 x
b
2
p = −1>0
a
arctanh (q tanh x) + 3 +
⎤
1
2
1 − 2 + 2 tanh2 x
q
q
p tanh x
1 + p2 tanh2 x
3
2
− 4
q2
q
q tanh x
1 − q 2 tanh2 x
2q tanh x
2 ⎦
1 − q 2 tanh2 x
b
2
q =1− >0
a
MZ 196
⎡
4.
2.461
2
1 ⎣
3
3− 2 + 4
3 =
3
8pa
p
p
a + b cosh x
dx
2
+
1+
1−
⎤
2
1
− 2 coth2 x
p2
p
⎡
1 ⎣
2
3
=
3+ 2 + 4
3
8qa
q
q
+
arctan (p coth x) + 3 −
p coth x
1 + p2 coth2 x
2p coth x
2 ⎦
1 + p2 coth2 x
2
3
− 4
2
q
q
⎤
2q coth x
2 ⎦
1 − q 2 coth2 x
ϕ(x)∗ + 3 +
1
2
+ 2 coth2 x
q2
q
3
2
− 4
2
p
p
b
p2 = −1 − > 0
a
q coth x
1 − q 2 coth2 x
b
q2 = 1 + > 0
a
2.46 Algebraic functions of hyperbolic functions
2.461 √
√
√
tanh x dx = arctanh tanh x − arctan tanh x
1.
∗ In 2.459.4, if b < 0 and cosh2 x > − a , then ϕ(x) = arctanh (q coth x). If
a
b
ϕ(x) = arccoth (q coth x).
MZ 221
b
a
< 0, but cosh2 x < − ab , or if
b
a
> 0, then
2.462
2.
2.462
Algebraic functions of hyperbolic functions
√
1.
coth x dx = arccoth
133
√
√
coth x − arctan coth x
cosh x
sinh x dx
= arcsinh √
= ln cosh x + a2 + sinh2 x
a2 − 1
a2 + sinh2 x
cosh x
= ln cosh x + a2 + sinh2 x
= arccosh √
1 − a2
= ln cosh x
MZ 222
2
a >1
2
a <1
2
a =1
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
cosh x
sinh x dx
= arcsin √
sinh2 x < a2
2
2
2
a +1
a − sinh x
sinh x dx
cosh x
= arccosh √
= ln cosh x + sinh2 x − a2
a2 + 1
sinh2 x − a2
sinh2 x > a2
sinh x
cosh x dx
= ln sinh x + a2 + sinh2 x
= arcsinh
a
a2 + sinh2 x
sinh x
cosh x dx
sinh2 x < a2
= arcsin
2
a
a2 − sinh x
cosh x dx
sinh x
= ln sinh x + sinh2 x − a2
= arccosh
a
sinh2 x − a2
sinh2 x > a2
cosh x
sinh x dx
= arcsinh
= ln cosh x + a2 + cosh2 x
a
a2 + cosh2 x
cosh x
sinh x dx
cosh2 x < a2
= arcsin
a
a2 − cosh2 x
sinh x dx
cosh x
= ln cosh x + cosh2 x − a2
= arccosh
a
cosh2 x − a2
cosh2 x > a2
sinh x
cosh x dx
= arcsinh √
= ln sinh x + a2 + cosh2 x
2
a +1
a2 + cosh2 x
sinh x
cosh x dx
= arcsin √
cosh2 x < a2
2
2
a −1
a2 − cosh x
2
cosh x dx
sinh x
= arccosh √
a >1
2
2
2
a −1
cosh x − a
2
= ln sinh x
a =1
MZ 199
MZ 215, 216
MZ 206
134
Hyperbolic Functions
√
coth x dx
b
√
= 2 a arccoth 1 + sinh x
a
a + b sinh x
√
b
= 2 a arctanh 1 + sinh x
a
.
√
b
= 2 −a arctanh − 1 + sinh x
a
√
tanh x dx
b
√
= 2 a arccoth 1 + cosh x
a
a + b cosh x
√
b
= 2 a arctanh 1 + cosh x
a
.
√
b
= 2 −a arctanh − 1 + cosh x
a
2.463
13.
14.
[b sinh x > 0,
a > 0]
[b sinh x < 0,
a > 0]
a<0
[b cosh x > 0,
a > 0]
[b cosh x < 0,
a > 0]
[a < 0]
MZ 220, 221
2.463 √
sinh x a + b cosh x
dx
1.
p + q cosh x
.
=2
.
=2
.
=2
2.
aq − bp
arccoth
q
aq − bp
arctanh
q
bp − aq
arctanh
q
√
cosh x a + b sinh x
dx
p + q sinh x
.
=2
.
=2
.
=2
.
.
q (a + b cosh x)
aq − bp
.
q (a + b cosh x)
aq − bp
q (a + b cosh x)
bp − aq
b cosh x > 0,
aq − bp
>0
q
b cosh x < 0,
aq − bp
<0
q
aq − bp
>0
q
MZ 220
aq − bp
arccoth
q
aq − bp
arctanh
q
bp − aq
arctanh
q
.
.
.
q (a + b sinh x)
aq − bp
q (a + b sinh x)
aq − bp
q (a + b sinh x)
bp − aq
b sinh x > 0,
aq − bp
>0
q
b sinh x < 0,
aq − bp
<0
q
aq − bp
>0
q
MZ 221
2.464 dx
dx
= = F (arcsin (tanh x) , k)
1.
2
2
2
k + k cosh x
1 + k 2 sinh2 x
[x > 0]
1
dx
dx
2.
= = F arcsin
,k
2
2
2
cosh x
cosh x − k 2
sinh x + k [x > 0]
BY (295.00)(295.10)
BY (295.40)(295.30)
2.464
Algebraic functions of hyperbolic functions
dx
=F
1 − k 2 cosh2 x
3.
arcsin
tanh x
k
1
0 < x < arccosh k
,k
Notation: In 2.464 4–2.464 8, we set α = arccos
4.
5.
√
6.
7.
8.
1
dx
√
F (α, r)
=
2a
sinh 2ax
1
1
[F (α, r) − 2 E (α, r)] +
sinh 2ax dx =
2a
a
1
1 − sinh 2ax
,r= √
1 + sinh 2ax
2
11.
12.
13.
14.
15.
<
sinh 2ax 1 + sinh2 2ax
BY (296.53)
1 + sinh 2ax
BY (296.51)
2
1
(1 − sinh 2ax) dx
√
=
[2 E (α, r) − F (α, r)]
2
2a
(1 + sinh 2ax) sinh 2ax
√
sinh 2ax dx
1
2 = 4a [F (α, r) − E (α, r)]
(1 + sinh 2ax)
BY (296.55)
BY (296.54)
1
cosh 2ax − 1
,r= √
cosh 2ax
2
1
dx
√
= √ F (α, r)
a 2
cosh 2ax
√
sinh 2ax
1
cosh 2ax dx = √ [F (α, r) − 2 E (α, r)] + √
a 2
a cosh 2ax
1
dx
√
= √ [2 E (α, r) − F (α, r)]
3
a
2
cosh 2ax
tanh 2ax
dx
1
√
= √ F (α, r) + √
5
3
2a
3a cosh 2ax
cosh 2ax
√
√
2
1
sinh2 2ax dx
√
F (α, r) +
sinh 2ax cosh 2ax
=−
3a
3a
cosh 2ax
√
2
2
tanh 2ax
tanh 2ax dx
√
F (α, r) − √
=
3a
cosh 2ax
3a cosh 2ax
√
1
cosh 2ax dx
= √ Π α, p2 , r
p2 + (1 − p2 ) cosh 2ax
a 2
Notation: In 2.464 16–2.464 20, we set:
√
a2 + b2 − a − b sinh x
,
α = arccos √
a2 + b2 + a + b sinh x
.
√
+
a + a2 + b 2
√
r=
a > 0, b > 0,
2 a2 + b 2
16.
[ax > 0]
1
cosh2 2ax dx
=
E (α, r)
2√
2a
(1 + sinh 2ax) sinh 2ax
10.
BY (295.20)
BY (296.50)
Notation: In 2.464 9–2.464 15, we set α = arcsin
9.
135
dx
1
√
F (α, r)
= √
4
2
a + b sinh x
a + b2
[x = 0]:
BY (296.00)
BY (296.03)
BY (296.04)
BY (296.04)
BY (296.07)
BY (296.05)
BY (296.02)
x > − arcsinh
a,
b
BY (298.00)
136
Hyperbolic Functions
√
2b cosh x a + b sinh x
a + b sinh x dx =
+ [F (α, r) − 2 E (α, r)] + √
a2 + b2 + a + b sinh x
√
√
a + b sinh x
a2 + b 2 − a
4
2 + b2 E (α, r) −
√
a
F (α, r)
dx
=
4
2
2
cosh2 x
√ 2 a +b
√
√
a2 + b2 − a − b sinh x
a + b sinh x
a + a2 + b 2
·√
−
·
b
cosh x
a2 + b2 + a + b sinh x
17.
18.
2.464
√
4
a2
b2
BY (298.02)
BY (298.03)
19.
20.
2
1
cosh x dx
= √
E (α, r)
√
2 √
2 4 a2 + b 2
2
2
b
a + b + a + b sinh x
a + b sinh x
√
a + b sinh x dx
1
√
E (α, r)
√
2 = − √
4
2
2
a +b
a2 + b 2 − a
a2 + b2 − a − b sinh x
√
b
cosh x a + b sinh x
+√
·
2
2
2
a + b − a a + b2 − (a + b sinh x)2
21.
22.
23.
24.
25.11
27.
28.
29.
BY (298.04)
a−b
x
,r=
[0 0]:
Notation: In 2.464 21–2.464 31, we set α = arcsin tanh
2
a+b
dx
2
√
BY (297.25)
=√
F (α, r)
a + b cosh x
a+b
√
√
x√
a + b cosh x dx = 2 a + b [F (α, r) − E (α, r)] + 2 tanh
a + b cosh x
BY (297.29)
2
√
2
x√
cosh x dx
2
2 a+b
√
E (α, r) + tanh
a + b cosh x
BY (297.33)
=√
F (α, r) −
b
b
2
a + b cosh x
a+b
√
tanh2 x2
2 a+b
√
[F (α, r) − E (α, r)]
BY (297.28)
dx =
a−b
a + b cosh x
√
√
tanh4 x2
sinh x2 a + b cosh x
2
2 a+b
√
[(3a + b) F (α, r) − 4a E (α, r)] +
dx =
x
3(a − b)2
3(a − b)
a + b cosh x
cosh3
2
26.
BY (298.01)
BY (297.28)
2+
,
√
x √
tanh
a + b cosh x − a + b E (α, r)
2
cosh x − 1
√
dx =
b
a + b cosh x
√
2
(cosh x − 1)
4 a+b
√
[(a + 3b) E (α, r) − b F (α, r)]
dx =
3b2+
a + b cosh x
,
x√
x
4
+ 2 b cosh2 − (a + 3b) tanh
a + b cosh x
3b
2
2
√
√
a + b cosh x
dx = a + b E (α, r)
cosh x + 1
√
2b
a+b
dx
√
√
E (α, r) −
=
F (α, r)
a−b
(cosh x + 1) a + b cosh x
(a − b) a + b
BY (297.31)
BY (297.31)
BY (297.26)
BY (297.30)
2.464
Algebraic functions of hyperbolic functions
30.
137
1
dx
√
=
b(5b − a) F (α, r)
2√
(cosh x + 1) a + b cosh x 3(a − b)2 a + b
sinh x2 √
1
·
+ (a − 3b)(a + b) E (α, r) +
a + b cosh x
6(a − b) cosh3 x2
297.30)
31.
(1 + cosh x) dx
2
√
=√
Π α, p2 , r
2
[1 + + (1 − p ) cosh x] a + b cosh x
a+b
Notation: In 2.464 32–2.464 40, we set:
a − b cosh x
α = arcsin
a−b
+
a,
a−b
r=
0 < b < a, 0 < x < arccosh
a+b
b
p2
BY (297.27)
32.
33.
dx
2
√
=√
F (α, r)
a − b cosh x
a+b
√
√
a − b cosh x dx = 2 a + b [F (α, r) − E (α, r)]
34.
35.
36.
37.
38.
39.
BY (297.54)
√
2
cosh x dx
2 a+b
√
E (α, r) − √
=
F (α, r)
b
a − b cosh x
a+b
√
√
cosh2 x dx
2
2(b − 2a)
4a a + b
√
E (α, r) + sinh x a − b cosh x
= √
F (α, r) +
2
3b
3b
a − b cosh x
3b a + b
√
(1 + cosh x) dx
2 a+b
√
E (α, r)
=
b
a − b cosh x
dx
2b
a−b
√
,r
= √
Π α,
a
cosh x a − b cosh x
a a+b
x√
dx
1
1
√
tanh
a − b cosh x
=√
E (α, r) −
a+b
2
(1 + cosh x) a − b cosh x
a+b
BY (297.56)
BY (297.56)
BY (297.51)
BY (297.57)
BY (297.58)
1
dx
= [(a + 3b) E (α, r) − b F (α, r)]
√
(1 + cosh x) a − b cosh x 3 (a + b)3
√
tanh x2 a − b cosh x
1
[2a + 4b + (a + 3b) cosh x]
−
3(a + b)2
cosh x + 1
2
40.
BY (297.50)
BY (297.58)
dx
2
√
√
=
Π α, p2 , r
(a − b − ap2 + bp2 cosh x) a − b cosh x
(a − b) a + b
Notation: In 2.464 41 –2.464 47, we set:
b (cosh x − 1)
,
α = arcsin
b cosh x − a
a+b
[0 < a < b, x > 0]
r=
2b
BY (297.52)
138
Hyperbolic Functions
2
dx
√
F (α, r)
=
b
b cosh −a
√
√
2
2b sinh x
F (α, r) − 2 2b E (α, r) + √
b cosh x − a dx = (b − a)
b
b cosh x − a
2
1
dx
<
[2b E (α, r) − (b − a) F (α, r)]
= 2
·
2
b
−
a
b
3
(b cosh x − a)
2
1
dx
<
[(b − 3a)(b − a) F (α, r) + 8ab E (α, r)]
=
2
2
2
b
5
3 (b − a )
(b cosh x − a)
sinh x
2b
·<
+
2
2
3 (b − a )
3
(b cosh x − a)
2.464
41.
42.
43.
44.
45.
46.
47.
48.
49.
50.
51.
52.
53.
BY (297.05)
BY (297.06)
BY (297.06)
2 sinh x
2
[F (α, r) − 2 E (α, r)] + √
BY (297.03)
b
b cosh x − a
2
(cosh x + 1) dx
2
<
E (α, r)
=
BY (297.01)
b−a b
3
(b cosh x − a)
√
b cosh x − a dx
2 =
Π α, p2 , r
BY (297.02)
2
2
p b − a + b (1 − p ) cosh x
b
.
b cosh x − a
2b
and r =
for
Notation: In 2.464 48–2.464 55, we set α = arcsin
b (cosh x − 1)
a+b
+
a,
0 arccosh :
b
dx
2
√
BY (297.75)
=√
F (α, r)
b cosh x − a
a+b
√
√
x√
b cosh x − a dx = −2 a + b E (α, r) + 2 coth
b cosh x − a
BY (297.79)
2
√
coth2 x2 dx
2 a+b
√
E (α, r)
BY (297.76)
=
a−b
b cosh x − a
√
√
b cosh x − a
dx = a + b [F (α, r) − E (α, r)]
BY (297.77)
cosh x − 1
√
1
a+b
dx
√
E (α, r) − √
BY (297.78)
=
F (α, r)
a−b
(cosh x − 1) b cosh x − a
a+b
1
dx
√
=
(a − 2b)(a − b) F (α, r)
2√
(cosh x − 1) b cosh x − a 3(a − b)2 a + b
cosh x2 √
a+b
·
+ (3a − b)(a + b) E (α, r) +
b cosh x − a
6b(a − b) sinh3 x2
54.
BY (297.00)
cosh x dx
√
=
b cosh x − a
√
dx
1
2 b cosh x − a
√
=√
[F (α, r) − E (α, r)] +
(a + b) sinh x
a+b
(cosh x + 1) b cosh x − a
BY (297.78)
BY (297.80)
2.471
Hyperbolic functions and powers
55.
1
dx
= (a + b) F (α, r)
2√
(cosh x + 1) b cosh x − a 3 (a + b)3
√
b cosh x − a
− (a + 3b) E (α, r) +
3(a + b) sinh x
139
2
x
a + 3b
− tanh2
a+b
2
BY (297.80)
56.
57.
Notation: In 2.464 56–2.464 60, we set
√
4 2
b − a2
α = arccos √
,
a sinh x + b cosh x
a
1
0 < a < b, − arcsinh √
<x
r= √
2
b 2 − a2
dx
4
√
F (α, r)
BY (299.00)
= 4 2
b − a2
a sinh x + b cosh x
√
2 (a cosh x + b sinh x)
a sinh x + b cosh x dx = 4 4 (b2 − a2 ) [F (α, r) − 2 E (α, r)] + √
a sinh x + b cosh x
58.
59.
60.
.
BY (299.02)
4
BY (299.03)
3 [2 E (α, r) − F (α, r)]
(b2 − a2 )
.
a cosh x + b sinh x
4
dx
14
2
<
·<
=
F (α, r) +
3 (b2 − a2 )5
3 (b2 − a2 )
5
3
(a sinh x + b cosh x)
(a sinh x + b cosh x)
dx
<
=
3
(a sinh x + b cosh x)
4
√ 2
b − a2 + a sinh x + b cosh x dx
4
<
=24 2
E (α, r)
b − a2
3
(a sinh x + b cosh x)
BY (299.03)
BY (299.01)
2.47 Combinations of hyperbolic functions and powers
2.471 1.
xr sinhp x coshq x dx
=
1
(p + q)xr sinhp−1 x coshq−1 x
(p + q)2
−rxr−1 sinhp x coshq x + r(r + 1) xr−2 sinhp x coshq x dx
+ rp xr−1 sinhp−1 x coshq−1 x dx + (q − 1)(p + q) xr sinhp x coshq−2 x dx
1
=
(p + q)xr sinhp−1 x coshq+1 x
(p + q)2
−rxr−1 sinhp x coshq x + r(r − 1) xr−2 sinhp x coshq x dx
− rq xr−1 sinhp−1 x coshq−1 x dx − (p − 1)(p + q) xr sinhp−2 x coshq x dx
GU (353)(1)
140
Hyperbolic Functions
n
2.
x sinh
2m
x dx = (−1)
xn sinh2m+1 x dx =
3.
n
4.
x cosh
2m
x dx =
xn cosh2m+1 x dx =
5.
2.472
1.
m
1 22m
m−1
xn+1
1
+
(−1)k
22m (n + 1) 22m−1
2m
m
m
(−1)k
k=0
m
1 22m
2m + 1
k
k=0
= x cosh x − nx
n−1
= x sinh x − nx
2n
5.11
6.
7.
8.
9.
10.
2.473
1.
xn cosh(2m − 2k)x dx
xn cosh(2m − 2k + 1)x dx
sinh x + n(n − 1) xn−2 sinh x dx
cosh x + n(n − 1) xn−2 cosh x dx
n
n
x2k
x2k−1
cosh x −
sinh x
sinh x dx = (2n)!
(2k)!
(2k − 1)!
x
k=0
k=1
x
x2k
cosh x −
sinh x
x
sinh x dx = (2n + 1)!
(2k + 1)!
(2k)!
k=0
n
n
x2k
x2k−1
2n
sinh x −
cosh x
x cosh x dx = (2n)!
(2k)!
(2k − 1)!
k=0
k=1
n
x2k+1
x2k
2n+1
x
sinh x −
cosh x
cosh x dx = (2n + 1)!
(2k + 1)!
(2k)!
k=0
x sinh x dx = x cosh x − sinh x
x2 sinh x dx = x2 + 2 cosh x − 2x sinh x
x cosh x dx = x sinh x − cosh x
x2 cosh x dx = x2 + 2 sinh x − 2x cosh x
4.
n−1
2n+1
n
2k+1
Notation: z1 = a + bx
1
b
z1 sinh kx dx = z1 cosh kx − 2 sinh kx
k
k
xn cosh(2m − 2k)x dx
3.
2m
k
xn cosh x dx = xn sinh x − n xn−1 sinh x dx
n
xn sinh(2m − 2k + 1)x dx
k=0
2m
k
xn sinh x dx = xn cosh x − n xn−1 cosh x dx
n
2.
2m + 1
k
k=0
m−1
xn+1
1
+
22m (n + 1) 22m−1
2m
m
2.472
2.474
Hyperbolic functions and powers
2.
z1 cosh kx dx =
1
b
z1 sinh kx − 2 cosh kx
k
k
z12 sinh kx dx =
1
k
z12 +
2b2
k2
cosh kx −
2bz1
sinh kx
k2
z12 cosh kx dx =
1
k
z12 +
2b2
k2
sinh kx −
2bz1
cosh kx
k2
z13 sinh kx dx =
z1
k
z12 +
6b2
k2
cosh kx −
3b
k2
z12 +
2b2
k2
sinh kx
z13 cosh kx dx =
z1
k
z12 +
6b2
k2
sinh kx −
3b
k2
z13 +
2b2
k2
cosh kx
z14 sinh kx dx =
1
k
z14 +
12b2 2 24b4
z + 4
k2 1
k
cosh kx −
4bz1
k2
z12 +
6b2
k2
sinh kx
z14 cosh kx dx =
1
k
z14 +
12b2 2 24b4
z + 4
k2 1
k
sinh kx −
4bz1
k2
z12 +
6b2
k2
cosh kx
z15 sinh kx dx =
z1
k
z14 +
20b2 2
b4
z1 + 120 4
2
k
k
cosh kx −
5b
k2
z14 + 12
z15 cosh kx dx =
z1
k
z14 + 20
b2 2
b4
z
+
120
k2 1
k4
sinh kx −
5b
k2
z14 + 12
z16 sinh kx dx =
1
k
3.
4.
5.
6.
7.
8.
9.
10.
11.
−
z16 cosh kx dx =
12.
1
k
−
2.474
b2 2
b4
z1 + 24 4
2
k
k
sinh kx
b2 2
b4
z
+
24
k2 1
k4
cosh kx
b2 4
b4
b6
z1 + 360 4 z12 + 720 6 cosh kx
2
k
k
k
2
4
b
b
z14 + 20 2 z12 + 120 4 sinh kx
k
k
z16 + 30
6bz1
k2
b2 4
b4 2
b6
z
+
360
z
+
720
sinh kx
1
1
k2
k4
k6
2
4
b
b
z14 + 20 2 z1 + 120 4 cosh kx
k
k
z16 + 30
6bz1
k2
xn−2k
xn−2k−1
sinh 2x − 2k+1
cosh 2x
22k (n − 2k)!
2
(n − 2k − 1)!
n/2
n! xn+1
+
x sinh x dx = −
2(n + 1)
4
n
1.
141
2
k=0
GU (353)(2b)
n! xn+1
+
x cosh x dx =
2(n + 1)
4
n/2
n
2.
2
k=0
GU (353)(3e)
x sinh2 x dx =
3.
4.
xn−2k
xn−2k−1
sinh 2x − 2k+1
cosh 2x
22k (n − 2k)!
2
(n − 2k − 1)!
1
x2
1
x sinh 2x − cosh 2x −
4
8
4
x2 sinh2 x dx =
1
4
x2 +
1
2
sinh 2x −
x3
x
cosh 2x −
4
6
MZ 257
142
Hyperbolic Functions
x cosh2 x dx =
5.
1
x2
x
sinh 2x − cosh 2x +
4
8
4
x2 cosh2 x dx =
6.
2.475
1
4
x2 +
1
2
sinh 2x −
x3
x
cosh 2x +
4
6
MZ 261
xn sinh3 x dx
7.
n/2
n! =
4
xn−2k
(n − 2k)!
k=0
cosh 3x
xn−2k−1
−
3
cosh
x
−
32k+1
(n − 2k − 1)!
sinh 3x
− 3 sinh x
32k+2
GU (353)(2f)
xn cosh3 x dx
8.
=
n/2
n! 4
xn−2k
(n − 2k)!
k=0
x sinh3 x dx =
9.
11.
1.
3x2
3
+
4
2
1
x2
+
12 54
cosh x +
cosh 3x +
x
3x
sinh x −
sinh 3x.
2
18
MZ 257
1
3
x
3
cosh 3x + x sinh x +
sinh 3x
x cosh3 x dx = − cosh x −
4
36
4
12
x2 cosh3 x dx =
12.
cosh 3x
+ 3 cosh x
32k+2
1
3
x
3
sinh x −
sinh 3x − x cosh x −
cosh 3x
4
36
4
12
x2 sinh3 x dx = −
10.
sinh 3x
xn−2k−1
+ 3 sinh x −
2k+1
3
(n − 2k − 1)!
GU (353)(3f)
2.475
3 2 3
x +
4
2
sinh x +
1
x2
+
12 54
x
3
cosh 3x
sinh 3x − x cosh x −
2
18
(p − 2) sinhq x + qx sinhq−1 x cosh x
sinhq x
dx
=
−
xp
(p − 1)(p − 2)xp−1
sinhq−2 x
sinhq x
q(q − 1)
q2
+
dx
+
dx
p−2
(p − 1)(p − 2)
x
(p − 1)(p − 2)
xp−2
MZ 262
[p > 2]
GU (353)(6a)
2.
3.
4.
coshq x
(p − 2) coshq x + qx coshq−1 x sinh x
dx = −
p
x
(p − 1)(p − 2)xp−1
q(q − 1)
q2
coshq−2 x
coshq x
−
dx
+
dx
(p − 1)(p − 2)
xp−2
(p − 1)(p − 2)
xp−2
sinh x
1
dx = −
x2n
x(2n − 1)!
sinh x
1
dx = −
2n+1
x
x(2n)!
n−2
(2k + 1)!
k=0
n−1
(2k)!
k=0
x2k
x2k+1
cosh x +
cosh x +
n−1
k=0
n−1
k=0
GU (353)(7a)
(2k)!
sinh x
x2k
[p > 2]
+
1
chi(x)
(2n − 1)!
(2k + 1)!
1
shi(x)
sinh x +
2k+1
x
(2n)!
GU (353)(6b)
GU (353)(6b)
2.476
Hyperbolic functions and powers
5.
6.
7.
8.
n−2
(2k + 1)!
cosh x
1
dx = −
x2n
x(2n − 1)!
k=0
n−1
(2k)!
1
cosh x
dx = −
2n+1
x
(2n)!x
k=0
x2k
x2k+1
m−1
1
sinh2m x
dx = 2m−1
(−1)k
x
2
k=0
m
1 sinh2m+1 x
dx = 2m
(−1)k
x
2
10.
11.
12.
m−1
1
cosh2m x
dx = 2m−1
x
2
m
1 cosh2m+1 x
dx = 2m
x
2
sinh
x
1
dx =
22m
14.
2m+1
cosh
x2
1
22m x
x
1.
(2k + 1)!
1
chi(x)
cosh x +
2k+1
x
(2n)!
(−1)m
22m
2m
m
ln x
shi(2m − 2k + 1)x
chi(2m − 2k)x +
1
22m
2m
m
GU (353)(6c)
GU (353)(6d)
ln x
chi(2m − 2k + 1)x
2m
k
GU (353)(7b)
GU (353)(7c)
GU (353)(7c)
cosh(2m − 2k)x
− (2m − 2k) shi(2m − 2k)x
x
2m + 1
k
k=0
sinh(2m − 2k + 1)x
− (2m − 2k + 1) chi(2m − 2k + 1)x
x
2m
m
−
1
22m−1
m−1
k=0
2m
k
cosh(2m − 2k)x
− (2m − 2k) shi(2m − 2k)x
x
dx
=−
2.476
GU (353)(7b)
(−1)k+1
cosh2m x
dx
x2
=−
m
(2k)!
1
shi(x)
cosh x +
x2k
(2n − 1)!
chi(2m − 2k)x +
2m + 1
k
k=0
×
2m
k
sinh2m x
(−1)m−1 2m
dx
=
x2
22m x
m
m−1
1
+ 2m−1
(−1)k+1
2
2m+1
n−1
k=0
2m + 1
k
k=0
x2
13.
2m
k
k=0
k=0
sinh x +
k=0
9.
sinh x +
n−1
143
m
1 22m
k=0
2m + 1
k
cosh(2m − 2k + 1)x
− (2m − 2k + 1) shi(2m − 2k + 1)x
x
sinh kx
1
ka
ka
dx =
shi(u) − sinh
chi(u)
cosh
a + bx
b b
b
1
ka
ka
=
exp −
Ei(u) − exp
Ei(−u)
2b
b
b
k
u = (a + bx)
b
144
Hyperbolic Functions
2.477
1
ka
cosh kx
ka
dx =
chi(u) − sinh
shi(u)
cosh
a + bx
b b
b
1
ka
ka
k
=
exp −
Ei(u) + exp
Ei(−u)
u = (a + bx)
2b
b
b
b
sinh kx
1 sinh kx k cosh kx
+
dx
(see 2.476 2)
dx = − ·
(a + bx)2
b a + bx
b
a + bx
cosh kx
1 cosh kx k sinh kx
·
+
dx
(see 2.476 1)
dx
=
−
(a + bx)2
b a + bx
b a + bx
k 2 sinh kx
sinh kx
sinh kx
k cosh kx
+
dx
dx = −
− 2
(a + bx)3
2b(a + bx)2
2b (a + bx) 2b2 a + bx
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
2
cosh kx
k
cosh kx
k sinh kx
+ 2
dx = −
− 2
3
2
(a + bx)
2b(a + bx)
2b (a + bx) 2b
(see 2.476 1)
cosh kx
dx
a + bx
(see 2.476 2)
sinh kx
k 3 cosh kx
sinh kx
k cosh kx
k sinh kx
+
dx
dx
=
−
−
−
(a + bx)4
3b(a + bx)3
6b2 (a + bx)2
6b3 (a + bx) 6b3
a + bx
2
(see 2.476 2)
cosh kx
k 3 sinh kx
cosh kx
k sinh kx
k cosh kx
+
dx
dx = −
− 2
− 3
(a + bx)4
3b(a + bx)3
6b (a + bx)2
6b (a + bx) 6b3 a + bx
2
(see 2.476 1)
2
sinh kx
sinh kx
k cosh kx
k sinh kx
dx = −
−
−
5
4
2
3
(a + bx)
4b(a + bx)
12b (a+ bx)
24b3 (a + bx)2
3
4
k
k cosh kx
sinh kx
+
dx
−
24b4 (a + bx) 24b4 a + bx
(see 2.476 1)
cosh kx
cosh kx
k sinh kx
k 2 cosh kx
dx = −
−
−
5
4
2
3
(a + bx)
4b(a + bx)
12b (a+ bx)
24b3 (a + bx)2
3
4
k
k sinh kx
cosh kx
+
dx
−
24b4 (a + bx) 24b4
a + bx
(see 2.476 2)
sinh kx
sinh kx
k cosh kx
k 2 sinh kx
k 3 cosh kx
dx
=
−
−
−
−
(a + bx)6
5b(a + bx)5
20b2 (a +bx)4
60b3 (a + bx)3
120b4 (a + bx)2
k5
k 4 sinh kx
cosh kx
+
dx
−
120b5 (a + bx) 120b5
a + bx
(see 2.476 2)
cosh kx
cosh kx
k sinh kx
k 2 cosh kx
k 3 sinh kx
dx
=
−
−
−
−
(a + bx)6
5b(a + bx)5
20b2 (a +bx)4
60b3 (a + bx)3
120b4 (a + bx)2
4
5
k
k cosh kx
sinh kx
+
dx
−
120b5 (a + bx) 120b5 a + bx
(see 2.476 1)
2.477
2.477
1.
2.
3.
Hyperbolic functions and powers
145
p(p − 1)
xp dx
−pxp−1 sinh x − (q − 2)xp cosh x
xp−2
+
dx
=
q
q−1
sinh x
(q − 1)(q − 2) sinhq−2 x
(q − 1)(q − 2) sinh
x
p
q−2
x dx
−
q − 1 sinhq−2 x
[q > 2]
pxp−1 cosh x + (q − 2)xp sinh x
xp−2 dx
xp dx
p(p − 1)
=
−
q
q−1
cosh x
(q − 1)(q − 2) coshq−2 x
(q −1)(q − 2) cosh
x
p
q−2
x dx
+
q − 1 coshq−2 x
[q > 2]
∞
2 − 22k B2k n+2k
xn
dx =
x
[|x| < π, n > 0]
sinh x
(n + 2k)(2k)!
4.
5.
6.11
7.
8.
9.
GU (353)(10a)
GU(353)(8b)
k=0
∞
E2k xn+2k+1
xn
dx =
cosh x
(n + 2k + 1)(2k)!
k=0
+
π
|x| < ,
2
n≥0
[|x| < π,
n ≥ 1]
,
GU (353)(10b)
n−1
dx
−1
n 2
=
−
[1
+
(−1)
Bn ln x
]
n
x sinh x
n!
∞
2 − 22k
B2k x2k−n
+
(2k − n)(2k)!
k=0
k= n
2
GU (353)(8a)
dx
=
xn cosh x
∞
k=0
k= n−1
2
GU (353)(9b)
E2k
1
En−1
x2k−n+1 + [1 + (−1)n ] +
ln x
(2k − n + 1)(2k)!
2
(n − 1)!
+
π,
|x| <
2
GU (353)(11b)
∞
22k B2k
xn
n
xn+2k−1
coth
x
+
n
dx
=
−x
2
(n + 2k − 1)(2k)!
sinh x
k=0
∞
2k
[n > 1,
22k − 1 B2k n+2k−1
2
xn
x
dx = xn tanh x − n
2
(n + 2k − 1)(2k)!
cosh x
k=1
+
n > 1,
|x| < π]
|x| <
π,
2
GU (353)(8c)
GU (353)(10c)
dx
2n n
coth x
n
Bn+1 ln x
−
[1
−
(−1)
]
=
−
2
n
x
(n + 1)!
xn sinh x
∞
k
n B2k
− n+1
(2x)2
x
(2k − n − 1)(2k)!
k=0
k= n+1
2
[|x| < π]
GU (353)(9c)
146
Hyperbolic Functions
10.
2n 2n+1 − 1 n
dx
tanh x
n
Bn+1 ln x
+ [1 − (−1) ] −
=
xn
(n + 1)!
xn cosh2 x
∞
22k − 1 B2k
k
n + n+1
(2x)2
x
(2k − n − 1)(2k)!
k=1
k= n+1
2
11.
12.
x
2n−1
x
15.
16.
17.
18.
19.
x
cosh
n−1
(2n − 3)(2n − 5) . . . (2n − 2k + 1)
(2n − 2)(2n − 4) . . . (2n − 2k)
k=1
x cosh x
1
x dx
n−1 (2n − 3)!!
+
×
+
(−1)
2n−2k
2n−2k−1
(2n − 2)!! sinh x
sinh
x (2n − 2k − 1) sinh
x
GU (353)(8e)
(see 2.477 15)
(−1)k
n−1
(2n − 2)(2n − 4) . . . (2n − 2k + 2)
x
dx
=
2n
(2n − 1)(2n − 3) . . . (2n − 2k + 1)
cosh x
k=1
x dx
x sinh x
1
(2n − 2)!!
+
+
×
2n−2k+1
2n−2k
(2n − 1)!! cosh2 x
cosh
x (2n − 2k) cosh
x
GU (353)(10e)
(see 2.477 18)
14.
GU (353)(11c)
dx
=
+
π,
|x| <
2
n−1
(2n − 2)(2n − 4) . . . (2n − 2k + 2)
x
(−1)k
dx
=
2n
(2n − 1)(2n − 3) . . . (2n − 2k + 1)
sinh x
k=1
x cosh x
1
x dx
n−1 (2n − 2)!!
+
×
+
(−1)
(2n − 1)!! sinh2 x
sinh2n−2k+1 x (2n − 2k) sinh2n−2k x
GU (353)(8e)
(see 2.477 17)
sinh
13.
2.477
2n−1
x
dx =
n−1
(2n − 3)(2n − 5) . . . (2n − 2k + 1)
(2n − 2)(2n − 4) . . . (2n − 2k)
k=1
x sinh x
1
x dx
(2n − 3)!!
+
×
+
2n−2k
2n−2k−1
(2n − 2)!! cosh x
cosh
x (2n − 2k − 1) cosh
x
GU (353)(10e)
(see 2.477 16)
∞
2 − 22k
x dx
=
B2k x2k+1
sinh x
(2k + 1)(2k)!
|x| < π
GU (353)(8b)a
π
2
GU (353)(10b)a
k=0
∞
E2k x2k+2
x dx
=
cosh x
(2k + 2)(2k)!
k=0
x dx
= −x coth x + ln sinh x
sinh2 x
x dx
= x tanh x − ln cosh x
cosh2 x
1
x dx
x cosh x
1
x dx
−
=−
−
3
2
2
sinh
x
2
sinh
x
sinh x
2 sinh x
|x| <
MZ 257
MZ 262
(see 2.477 15)
MZ 257
2.478
Hyperbolic functions and powers
20.
21.
22.
23.
24.
147
1
x dx
x sinh x
1
x dx
+
(see 2.477 16)
=
+
3
2
2
cosh
x
2
cosh
x
cosh x
2 cosh x
x cosh x
1
2
2
x dx
=−
−
+ x coth x − ln sinh x
4
3
2
3
3
sinh x
3 sinh x 6 sinh x
2
x dx
x sinh x
1
2
=
+
+ x tanh x − ln cosh x
4
3
2
3
3
cosh x
3 cosh x 6 cosh x
3
x dx
x cosh x
1
3x cosh x
3
x dx
+
=−
−
+
+
5
4
3
2
8
sinh
x
8
sinh
x
sinh x
4 sinh x 12 sinh x
8 sinh x
(see 2.477 15)
3
x dx
x sinh x
1
3x sinh x
3
x dx
+
=
+
+
+
5
4
3
2
cosh x
4 cosh x 12 cosh x 8 cosh x 8 cosh x 8 cosh x
MZ 262
MZ 258
MZ 262
MZ 258
(see 2.477 16)
2.478
1.
2.
xn cosh x dx
xn
n
m =−
m−1 + (m − 1)b
(a + b sinh x)
(m − 1)b (a + b sinh x)
xn sinh x dx
xn
m =−
m−1
(a + b cosh x)
(m − 1)b (a + b cosh x)
4.
5.
6.
7.
(a + b sinh x)
m−1
[m = 1]
n
xn−1 dx
+
(m − 1)b (a + b cosh x)m−1
MZ 263
MZ 263
x
x
x dx
= x tanh − 2 ln cosh
1 + cosh x
2
2
x
x
x dx
= x coth − 2 ln sinh
1 − cosh x
2
2
x
x sinh x dx
x
2 = − 1 + cosh x + tanh 2
(1 + cosh x)
x
x
x sinh x dx
2 = 1 − cosh x − coth 2
(1 − cosh x)
1
x dx
=
[L(u + t) − L(u − t) − 2 L(t)]
cosh 2x − cos 2t
2 sin 2t
[u = arctan (tanh x cot t) ,
8.
xn−1 dx
[m = 1]
3.
MZ 262
MZ 262-264
t = ±nπ]
LO III 402
1
u+t
u−t
x cosh x dx
υ+t
=
L
−L
+L π−
cosh 2x − cos 2t 2 sin t
2
2
2
υ−t
t
π−t
+ L
− 2L
− 2L
2
2
2
t
t
x
x
u = 2 arctan tanh · cot
, υ = 2 arctan coth · cot
2
2
2
2
;
t = ±nπ
LO III 403
148
2.479
Hyperbolic Functions
1.
xp
m
sinh2m x
dx =
(−1)m+k
n
k
cosh x
xp
sinh
x
dx =
(−1)m+k
n
cosh x
m
k=0
2.
m
2m+1
k=0
xp
3.
2.479
xp dx
(see 4.477 2)
coshn−2k x
m
sinh x
dx
xp
k
coshn−2k x
xp
sinh x
p
dx = −
+
n
n−1
cosh x
(n − 1) cosh
x n−1
[n > 1]
xp−1 dx
coshn−1 x
[n > 1]
4.
5.
6.
cosh2m x
dx =
sinhn x
xp tanh x dx =
7.
xp coth x dx =
∞
k=1
8.
10.
(see 2.477 2)
∞
k=0
m
k
GU (353)(13c)
2k
2 − 1 B2k p+2k
2
x
(2k + p)(2k)!
+
p > −1,
|x| <
22k B2k
xp+2k
(p + 2k)(2k)!
[p ≥ +1,
|x| < π]
2k
π,
2
x
x cosh x
x
dx = ln tanh −
2
2 sinh x
sinh x
x sinh x
x
+ arctan (sinh x)
dx = −
cosh x
cosh2 x
2.48 Combinations of hyperbolic functions, exponentials, and powers
2.481
eax sinh(bx + c) dx =
1.
2.
GU (353)(12)
xp cosh x
(see 2.477 1)
sinhn−2k x
k=0
p
m
m
cosh2m+1 x
x cosh x
dx
=
dx
(see 2.479 6)
xp
n−2k
k
sinhn x
sinh
x
k=0
xp
p
xp−1 dx
p cosh x
dx = −
x
+
n
sinh x
(n − 1) sinhn−1 x n − 1 sinhn−1 x
[n > 1]
(see 2.477 1)
xp
9.
m
(see 2.479 3)
eax cosh(bx + c) dx =
eax
[a sinh(bx + c) − b cosh(bx + c)]
a2 − b 2
2
a = b2
eax
[a cosh(bx + c) − b sinh(bx + c)]
a2 − b 2
2
a = b2
GU (353)(12d)
GU (353)(13d)
MZ 263
2.483
3.
4.
5.
6.
2.482
Hyperbolic functions, exponentials, and powers
For a2 = b2 :
1
1
eax sinh(ax + c) dx = − xe−c + e2ax+c
2
4a
1
1
e−ax sinh(ax + c) dx = xec + e−(2ax+c)
2
4a
1
1
eax cosh(ax + c) dx = xe−c + e2ax+c
2
4a
1
1
e−ax cosh(ax + c) dx = xec − e−(2ax+c)
2
4a
1
2
xp eax sinh bx dx =
1.
xp e(a+b)x dx −
MZ 275-277
xp e(a−b)x dx
xp eax cosh bx dx =
2.
1
2
a2 = b 2
xp e(a+b)x dx + xp e(a−b)x dx
3.
4.
5.
2.483
For a2 = b2 :
1
xp+1
xp eax sinh ax dx =
xp e2ax dx −
2
2(p + 1)
p+1
1
x
xp e−ax sinh ax dx =
−
xp e−2ax dx
2(p + 1) 2
1
xp+1
+
xp eax cosh ax dx =
xp e2ax dx
2(p + 1) 2
xeax sinh bx dx =
1.
3.
eax
2
a − b2
ax −
xeax cosh bx dx =
eax
2
a − b2
a2 + b 2
a2 − b 2
a2 = b 2
(see 2.321)
(see 2.321)
(see 2.321)
2ab
a2 − b 2
2
a = b2
sinh bx − bx −
MZ 276, 278
cosh bx
2ab
sinh
bx
a2 − b 2
2
a = b2
%
&
2 a2 + b 2
2a a2 + 3b2
eax
2 ax
2
x e sinh bx dx = 2
x+
ax −
sinh bx
2
a − b2
a2 − b 2
(a2 − b2 )
&
%
2
2b 3a2 + b2
4ab
2
x+
cosh x
a = b2
− bx − 2
2
2
2
a − b2
(a − b )
2.
149
ax −
a2 + b 2
a2 − b 2
cosh bx − bx −
150
Hyperbolic Functions
2 ax
4.
5.
6.
7.
8.
9.
10.
11.
2.484
1.
2.
3.
4.
5.
6.
7.
x e
&
2 a2 + b 2
2a a2 + 3b2
x+
ax −
cosh bx
2
a2 − b 2
(a2 − b2 )
&
2b 3a2 + b2
4ab
2
x+
sinh x
− bx − 2
2
a − b2
(a2 − b2 )
eax
cosh bx dx = 2
a − b2
%
2.484
%
2
For a2 = b2 :
e2ax
1
x2
xeax sinh ax dx =
x−
−
4a
2a
4
e−2ax
1
x2
xe−ax sinh ax dx =
x+
+
4a
2a
4
2
2ax
e
x
1
+
xeax cosh ax dx =
x−
4
4a
2a
2
−2ax
e
x
1
−
xe−ax cosh ax dx =
x+
4
4a
2a
1
e2ax
x
x3
x2 eax sinh ax dx =
x2 − + 2 −
4a
a 2a
6
−2ax
1
e
x
x3
x2 e−ax sinh ax dx =
x2 + + 2 +
4a
a 2a
6
e2ax
1
x3
x
+
x2 eax cosh ax dx =
x2 − + 2
6
4a
a 2a
2
1
dx
= {Ei[(a + b)x] − Ei[(a − b)x]}
a = b2
x
2
2
1
dx
= {Ei[(a + b)x] + Ei[(a − b)x]}
a = b2
eax cosh bx
x
2
dx
eax sinh bx 1
+ {(a + b) Ei[(a + b)x] − (a − b) Ei[(a − b)x]}
eax sinh bx 2 = −
x
2x
2
2
a = b2
dx
eax cosh bx 1
+ {(a + b) Ei[(a + b)x] + (a − b) Ei[(a − b)x]}
eax cosh bx 2 = −
x
2x
2
2
a = b2
eax sinh bx
For a2 = b2 :
1
dx
= [Ei(2ax) − ln x]
eax sinh ax
x
2
1
dx
= [ln x − Ei(−2ax)]
e−ax sinh ax
x
2
1
dx
= [ln x + Ei(2ax)]
eax cosh ax
x
2
a2 = b 2
MZ 276, 278
2.510
Powers of trigonometric functions
151
8.
9.
10.
dx
1 2ax
e
=−
− 1 + a Ei(2ax)
x2
2x
dx
1 1 − e−2ax + a Ei(−2ax)
e−ax sinh ax 2 = −
x
2x
dx
1 2ax
e
eax cosh ax 2 = −
+ 1 + a Ei(2ax)
x
2x
eax sinh ax
MZ 276, 278
2.5–2.6 Trigonometric Functions
2.50 Introduction
2.501
Integrals of the form
R (sin x, cos x) dx can always be reduced to integrals of rational functions
x
by means of the substitution t = tan .
2
2.502 If R (sin x, cos x) satisﬁes the relation
R (sin x, cos x) = −R (− sin x, cos x) ,
it is convenient to make the substitution t = cos x.
2.503 If this function satisﬁes the relation
R (sin x, cos x) = −R (sin x, − cos x) ,
it is convenient to make the substitution t = sin x.
2.504 If this function satisﬁes the relation
R (sin x, cos x) = R (− sin x, − cos x) ,
it is convenient to make the substitution t = tan x.
2.51–2.52 Powers of trigonometric functions
2.510
sinp−1 x cosq+1 x p − 1
+
sinp−2 x cosq+2 x dx
q+1
q + 1
sinp−1 x cosq+1 x p − 1
=−
+
sinp−2 x cosq x dx
p+q
p+q sinp+1 x cosq+1 x p + q + 2
+
=
sinp+2 x cosq x dx
p+1
p +1
sinp+1 x cosq−1 x q − 1
=
+
sinp+2 x cosq−2 x dx
p+1
p + 1
sinp+1 x cosq−1 x q − 1
+
=
sinp x cosq−2 x dx
p+q
p+q
sinp+1 x cosq+1 x p + q + 2
=−
+
sinp x cosq+2 x dx
q+1
q+1
q−1
sinp−1 x cosq−1 x
=
sin2 x −
p+q
p+q−2
(p − 1)(q − 1)
p−2
sin
x cosq−2 x dx
+
(p + q)(p + q − 2)
sinp x cosq x dx = −
FI II 89, TI 214
152
2.511
Trigonometric Functions
2.511
sinp x cos2n x dx
1.
n−1
(2n − 1)(2n − 3) . . . (2n − 2k + 1) cos2n−2k−1 x
cos
x+
(2n + p − 2)(2n + p − 4) . . . (2n + p − 2k)
k=1
(2n − 1)!!
+
sinp x dx
(2n + p)(2n + p − 2) . . . (p + 2)
sinp+1 x
=
2n + p
2.
2n−1
This formula is applicable for arbitrary real p, except for the following negative even integers:
−2, −4, . . . , −2n. If p is a natural number and n = 0, we have:
sin2l x dx
l−1
(2l − 1)(2l − 3) . . . (2l − 2k + 1)
cos x
2l−1
2l−2k−1
sin
sin
x+
x
=−
2l
2k (l − 1)(l − 2) . . . (l − k)
k=1
(2l − 1)!!
x
(see also 2.513 1)
+
2l l!
sin2l+1 x dx = −
3.
p
4.
sin x cos
2n+1
cos x
2l + 1
sin2l x +
l−1
k=0
sinp+1 x
x dx =
2n + p + 1
2k+1 l(l − 1) . . . (l − k)
sin2l−2k−2 x
(2l − 1)(2l − 3) . . . (2l − 2k − 1)
(see also 2.513 2)
cos
TI (232)
2n
x+
n
k=1
TI (233)
2k n(n − 1) . . . (n − k + 1) cos2n−2k x
(2n + p − 1)(2n + p − 3) . . . (2n + p − 2k + 1)
This formula is applicable for arbitrary real p, except for the negative odd integers: −1, −3, . . . ,
−(2n + 1).
2.512
1.
cosp x sin2n x dx
n−1
(2n − 1)(2n − 3) . . . (2n − 2k + 1) sin2n−2k−1 x
x+
sin
(2n + p − 2)(2n + p − 4) . . . (2n + p − 2k)
k=1
(2n − 1)!!
cosp x dx
+
(2n + p)(2n + p − 2) . . . (p + 2)
cosp+1 x
=−
2n + p
2.
2n−1
This formula is applicable for arbitrary real p, except for the following negative even integers:
−2, −4, . . . , −2n. If p is a natural number and n = 0, we have
l−1
(2l − 1)(2l − 3) . . . (2l − 2k + 1)
sin x
2l
2l−1
2l−2k−1
cos x dx =
cos
cos
x+
x
2l
2k (l − 1)(l − 2) . . . (l − k)
k=1
(2l − 1)!!
x
+
2l l!
(see also 2.513 3)
TI (230)
2.513
Powers of trigonometric functions
3.
sin x
cos2l+1 x dx =
2l + 1
cos2l x +
l−1
k=0
2k+1 l(l − 1) . . . (l − k)
cos2l−2k−2 x
(2l − 1)(2l − 3) . . . (2l − 2k − 1)
(see also 2.513 4)
cosp x sin2n+1 x dx
4.
153
cosp+1 x
=−
2n + p + 1
2n
sin
x+
n
k=1
TI (231)
2k n(n − 1) . . . (n − k + 1) sin2n−2k x
(2n + p − 1)(2n + p − 3) . . . (2n + p − 2k + 1)
This formula is applicable for arbitrary real p, except for the following negative odd integers: −1,
−3, . . . , −(2n + 1).
2.513
sin2n x dx =
1.
1
22n
2n
n
n−1
(−1)n (−1)k
22n−1
x+
k=0
2n
k
sin(2n − 2k)x
2n − 2k
(see also 2.511 2)
TI (226)
1
n+1
(−1)
(−1)k
22n
n
sin2n+1 x dx =
2.
k=0
2n + 1
k
cos(2n + 1 − 2k)x
2n + 1 − 2k
(see also 2.511 3)
TI (227)
cos2n x dx =
3.
1
22n
cos2n+1 x dx =
4.
5.
6.
7.
8.
9.
1
22n
2n
n
n
k=0
x+
1
22n−1
2n + 1
k
n−1
k=0
2n
k
sin(2n − 2k)x
2n − 2k
(see also 2.512 2)
TI (224)
(see also 2.512 3)
TI (225)
sin(2n − 2k + 1)x
2n − 2k + 1
1
1
1
1
sin2 x dx = − sin 2x + x = − sin x cos x + x
4
2
2
2
3
1
1
sin3 x dx =
cos 3x − cos x = cos3 x − cos x
12
4
3
3x sin 2x sin 4x
sin4 x dx =
−
+
8
4
32
1
3
3
= − sin x cos x − sin3 x cos x + x
8
4
8
5
1
5
sin5 x dx = − cos x +
cos 3x −
cos 5x
8
48
80
1
4
4
= − sin4 x cos x +
cos3 x − cos x
5
15
5
15
3
1
5
sin6 x dx =
x−
sin 2x +
sin 4x −
sin 6x
16
64
64
192
1
5
5
5
= − sin5 x cos x −
sin3 x cos x −
sin x cos x + x
6
24
16
16
154
Trigonometric Functions
7
7
1
35
cos x +
cos 3x −
cos 5x +
cos 7x
64
64
320
448
1
6
8
24
= − sin6 x cos x −
sin4 x cos x +
cos3 x −
cos x
7
35
35
35
x
1
1
1
cos2 x dx = sin 2x + = sin x cos x + x
4
2
2
2
3
1
1
sin 3x + sin x = sin x − sin3 x
cos3 x dx =
12
4
3
1
1
3
3
1
3
cos4 x dx = x + sin 2x +
sin 4x = x + sin x cos x + sin x cos3 x
8
4
32
8
8
4
5
1
4
4
5
1
cos5 x dx = sin x +
sin 3x +
sin 5x = sin x −
sin3 x + cos4 x sin x
8
48
80
5
15
5
15
3
1
5
cos6 x dx =
x+
sin 2x +
sin 4x +
sin 6x
16
64
64
192
5
5
5
1
=
x+
sin x cos x +
sin x cos3 x + sin x cos5 x
16
16
24
6
7
7
1
35
sin x +
sin 3x +
sin 5x +
sin 7x
cos7 x dx =
64
64
320
448
24
8
6
1
=
sin x −
sin3 x +
sin x cos4 x + sin x cos6 x
35
35
35
7
sin7 x dx = −
10.
11.
12.
13.
14.
15.
16.
17.
1
4
sin x cos3 x dx = −
cos4 x
4
sin x cos4 x dx = −
cos5 x
5
sin2 x cos x dx = −
1
4
18.
19.
20.
sin2 x cos2 x dx = −
21.
1
sin 3x − sin x
3
1
8
=−
=
cos3 x
3
sin3 x
3
1
sin 4x − x
4
1
1 1
sin 5x + sin 3x − 2 sin x
16 5
3
sin3 x
2
sin3 x 5
2
=
− sin2 x
cos x +
=
5
3
5
3
sin2 x cos3 x dx = −
22.
sin2 x cos4 x dx =
23.
sin3 x cos x dx =
24.
25.
1
cos 3x + cos x
3
sin x cos2 x dx = −
1
1
1
x
+
sin 2x −
sin 4x −
sin 6x
16 64
64
192
1
8
1
cos 4x − cos 2x
4
=
sin4 x
4
1
1 1
cos 5x − cos 3x − 2 cos x
16 5
3
1
1
= cos5 x − cos3 x
5
3
sin3 x cos2 x dx =
2.513
2.516
Powers of trigonometric functions
26.
1
3
cos 6x − cos 2x
6
2
sin3 x cos3 x dx =
1
32
sin3 x cos4 x dx =
1
2 3
cos3 x − − sin2 x + sin4 x
7
5 5
27.
155
sin5 x
5
1
1
1
1
x−
sin 2x −
sin 4x +
sin 6x
29.
sin4 x cos2 x dx =
16
64
64
192
2 3
1
+ cos2 x − cos4 x
30.
sin4 x cos3 x dx = sin3 x
7
5 5
1
1
3
31.
sin4 x cos4 x dx =
x−
sin 4x +
sin 8x
128
128
1024
sinp x
2.514
dx
cos2n x
n−1
(2n − p − 2)(2n − p − 4) . . . (2n − p − 2k)
sinp+1 x
2n−1
2n−2k−1
=
sec
sec
x+
x
2n − 1
(2n − 3)(2n − 5) . . . (2n − 2k − 1)
k=1
(2n − p − 2)(2n − p − 4) . . . (−p + 2)(−p)
sinp x dx
+
(2n − 1)!!
"
This formula is applicable for arbitrary real p. For sinp x dx, where p is a natural number, see 2.511 2,
3 and 2.513 1, 2. If n = 0 and p is a negative integer, we have for this integral:
2.515
l−1
2k (l − 1)(l − 2) . . . (l − k)
dx
cos x
2l−1
2l−2k−1
cosec
cosec
1.
=−
x+
x
2l − 1
(2l − 3)(2l − 5) . . . (2l − 2k − 1)
sin2l x
sin4 x cos x dx =
28.
k=1
2.
dx
sin2l+1 x
TI (242)
l−1
(2l − 1)(2l − 3) . . . (2l − 2k + 1)
cos x
cosec2l−2k x
cosec2l x +
2l
28k (l − 1)(l − 2) . . . (l − k)
k=1
x
(2l − 1)!!
ln tan
+
2l l!
2
=−
TI (243)
2.516
1.
2.
sinp x dx
cos2n+1 x
n−1
(2n − p − 1)(2n − p − 3) · · · (2n − p − 2k + 1)
sinp+1 x
2n
2n−2k
sec x +
sec
x
=
2n
2k (n − 1)(n − 2) · · · (n − k)
k=1
(2n − p − 1)(2n − p − 3) · · · (3 − p)(1 − p) sinp x
dx
+
2n n!
cos x
This formula is applicable for arbitrary real p. For n = 0 and p a natural number, we have
l
sin2k x
sin2l+1 x dx
=−
− ln cos x
cos x
2k
k=1
156
Trigonometric Functions
3.
2.517
sin2k−1 x
π x
sin2l x dx
=−
+ ln tan
+
cos x
2k − 1
4
2
l
k=1
2.517
1.
2.
2.518
1.
2.
m
1
dx
+ ln tan x
=
−
sin2m+1 x cos x
(2m
−
2k
+
2)
sin2m−2k+2 x
k=1
m
π x
1
dx
−
+ ln tan
=
−
2m
2m−2k+1
4
2
sin x cos x
(2m − 2k + 1) sin
x
k=1
sinp−1 x
sinp x
dx
=
−
(p
−
1)
sinp−2 x dx
cos2 x
cos x
⎧
cosp x dx cosp+1 x ⎨
=
cosec2n−1 x
2n − 1 ⎩
sin2n x
⎫
⎬
(2n − p − 2)(2n − p − 4) . . . (2n − p − 2k)
+
cosec2n−2k−1 x
⎭
(2n − 3)(2n − 5) . . . (2n − 2k − 1)
k=1
(2n − p − 2)(2n − p − 4) . . . (2 − p)(−p)
+
cosp x dx
(2n − 1)!!
"
This formula is applicable for arbitrary real p. For cosp x dx where p is a natural number, see
2.512 2, 3 and 2.513 3, 4. If n = 0 and p is a negative integer, we have for this integral:
n−1
2.519
1.
2.
2k (l − 1)(l − 2) . . . (l − k)
2l−2k−1
sec
sec
x+
x
(2l − 3)(2l − 5) . . . (2l − 2k − 1)
k=1
l−1
sin x
(2l − 1)(2l − 3) . . . (2l − 2k + 1)
dx
2l
2l−2k
=
sec
sec x +
x
cos2l+1 x
2l
2k (l − 1)(l − 2) . . . (l − k)
k=1
π x
(2l − 1)!!
ln tan
+
+
2l l!
4
2
sin x
dx
=
2l
cos x
2l − 1
2l−1
l−1
TI (240)
TI (241)
2.521
1.
⎧
cosp+1 x ⎨
cosp x dx
=−
cosec2n x
⎩
2n
sin2n+1 x
⎫
⎬
(2n − p − 1)(2n − p − 3) . . . (2n − p − 2k + 1)
2n−2k
+
cosec
x
⎭
2k (n − 1)(n − 2) . . . (n − k)
k=1
p
(2n − p − 1)(2n − p − 3) . . . (3 − p)(1 − p) cos x
dx
+
2n · n!
sin x
n−1
This formula is applicable for arbitrary real p. For n = 0 and p a natural number, we have
2.526
Powers of trigonometric functions
2.
157
cos2l+1 x dx cos2k x
=
+ ln sin x
sin x
2k
l
k=1
3.
2.522
1.
2.
2l
cos x dx
=
sin x
l
k=1
x
cos2k−1 x
+ ln tan
2k − 1
2
dx
1
=
+ ln tan x
2m+1
sin x cos
x
(2m − 2k + 2) cos2m−2k+2 x
m
k=1
dx
=
sin x cos2m x
m
k=1
x
1
+ ln tan
(2m − 2k + 1) cos2m−2k+1 x
2
GW (331)(15)
cosm x
cosm−1 x
−
(m
−
1)
cosm−2 x dx
dx
=
−
sin x
sin2 x
In formulas 2.524 1 and 2.524 2, s = 1 for m odd and m < 2n + 1; in other cases, s = 0.
2.523
2.524
1.
n
n
sin2n+1 x
dx
=
(−1)k+1
k
cosm x
m+1
cos2k−m+1 x
+ s(−1) 2
2k − m + 1
k=0
k= m−1
2
n
m−1
2
ln cos x
GU (331)(11d)
2.
n
n
cos2n+1 x
dx
=
(−1)k
m
k
sin x
k=0
k= m−1
2
2.525
1.
2.
m+n−1
dx
=
2m
2n
sin x cos x
k=0
m+n
dx
=
2m+1
2n+1
sin
x cos
x
k=0
m−1
sin2k−m+1 x
+ s(−1) 2
2k − m + 1
m+n−1
k
m+n
k
n
m−1
2
ln sin x
tan2k−2m+1 x
2k − 2m + 1
tan2k−2m x
+
2k − 2m
TI (267)
m+n
m
ln tan x
TI (268), GU (331)(15f)
2.526 1.
2.
3.
4.
5.
dx
x
= ln tan
sin x
2
dx
= − cot x
sin2 x
x
1 cos x
1
dx
=−
+ ln tan
3
2
2 sin x 2
2
sin x
1
dx
cos x
2
=−
− cot x = − cot3 x − cot x
4
3
3
sin x
3 sin x 3
dx
cos x
3 cos x
3
x
=−
−
+ ln tan
5
4
2
2
sin x
4 sin x 8 sin x 8
158
Trigonometric Functions
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
dx
4
cos x
4
cot3 x − cot x
=−
−
5
sin6 x
5 sin5 x 15
1
2
= − cot5 x − cot3 x − cot x
5
3
dx
cos x
1
5
15
x
5
=−
+
+
ln tan
+
7
2
4
2
8
16
2
sin x
6 sin x sin x 4 sin x
dx
1
3
cot7 x + cot5 x + cot3 x + cot x
=−
7
5
sin8 x
dx
π x
π x
1 + sin x
= ln tan
+
= ln cot
−
= ln
cos x
4
2
4
2
1 − sin x
dx
= tan x
cos2 x
1 sin x
1
π x
dx
=
+ ln tan
+
cos3 x
2 cos2 x 2
4
2
sin x
2
1
dx
=
+ tan x = tan3 x + tan x
cos4 x
3 cos3 x 3
3
sin x
3 sin x
3
x π
dx
=
+
+ ln tan
+
cos5 x
4 cos4 x 8 cos2 x 8
2
4
sin x
4
1
dx
4
2
=
+
tan3 x + tan x = tan5 x + tan3 x + tan x
cos6 x
5 cos5 x 15
5
5
3
sin x
5 sin x
5 sin x
5
x π
dx
=
+
+
+
ln tan
+
cos7 x
6 cos6 x 24 cos4 x 16 cos2 x 16
2
4
1
dx
3
= tan7 x + tan5 x + tan3 x + tan x
8
cos x
7
5
sin x
dx = − ln cos x
cos x
sin2 x
x
π
dx = − sin x + ln tan
=
cos x
4
2
3
2
sin x
1
sin x
dx = −
− ln cos x = cos2 x − ln cos x
cos x
2
2
4
sin x
1
x π
dx = − sin3 x − sin x + ln tan
+
cos x
3
2
4
2
1
sin x dx
=
2
cos x
cos x
2
sin x dx
= tan x − x
cos2 x
sin3 x dx
1
= cos x +
2
cos x
cos x
4
1
3
sin x dx
= tan x + sin x cos x − x
cos2 x
2
2
2.526
2.526
Powers of trigonometric functions
25.
26.
27.
28.
29.
30.
31.
32.
33.
34.
35.
36.
37.
38.
39.
40.
41.
42.
43.
44.
1
1
sin x dx
=
= tan2 x
cos3 x
2 cos2 x
2
sin x
1
π x
sin2 x dx
=
− ln tan
+
3
2
cos x
2 cos x 2
4
2
1 sin x
sin3 x dx
=
+ ln cos x
cos3 x
2 cos2 x
x π
1 sin x
3
sin4 x dx
=
+ sin x − ln tan
+
3
2
cos x
2 cos x
2
2
4
1
sin x dx
=
cos4 x
3 cos3 x
1
sin2 x dx
= tan3 x
cos4 x
3
1
1
sin3 x dx
=−
+
4
cos x
cos x 3 cos3 x
1
sin4 x dx
= tan3 x − tan x + x
cos4 x
3
cos x dx
= ln sin x
sin x
x
cos2 x dx
= cos x + ln tan
sin x
2
3
2
cos x
cos x dx
=
+ ln sin x
sin x
2
1
cos4 x dx
x
= cos3 x + cos x + ln tan
sin x
3
2
cos x
1
dx = −
sin x
sin2 x
cos2 x
dx = − cot x − x
sin2 x
1
cos3 x
dx = − sin x −
2
sin x
sin x
3
cos4 x
1
dx = − cot x − sin x cos x − x
2
2
2
sin x
1
cos x
dx = −
sin3 x
2 sin2 x
cos x
1
x
cos2 x
dx = −
− ln tan
2
sin3 x
2 sin2 x 2
3
1
cos x
dx = −
− ln sin x
3
sin x
2 sin2 x
1 cos x
3
x
cos4 x
dx = −
− cos x − ln tan
3
2
2 sin x
2
2
sin x
159
160
Trigonometric Functions
45.
46.
47.
48.
49.
50.
51.
52.
53.
54.
55.
56.
57.
58.
59.
60.
61.
62.
63.
64.
cos x
1
dx = −
sin4 x
3 sin3 x
cos2 x
1
dx = − cot3 x
4
3
sin x
1
1
cos3 x
dx =
−
4
sin x 3 sin3 x
sin x
1
cos4 x
dx = − cot3 x + cot x + x
3
sin4 x
dx
= ln tan x
sin x cos x
dx
1
x
=
+ ln tan
sin x cos2 x
cos x
2
dx
1
=
+ ln tan x
sin x cos3 x
2 cos2 x
dx
1
1
x
=
+
+ ln tan
sin x cos4 x
cos x 3 cos3 x
2
dx
π x
+
= ln tan
− cosec x
4
2
sin2 x cos x
dx
= −2 cot 2x
sin2 x cos2 x
1
3
3
1
π x
dx
−
+ ln tan
+
=
2
2x
3
2
cos
2
sin
x
2
4
2
sin x cos x
8
dx
1
− cot 2x
=
2
3
4
3 sin x cos x 3
sin x cos x
dx
1
=−
+ ln tan x
3
sin x cos x
2 sin2 x
1
x
dx
1
3
3
=−
−
+ ln tan
3
2
2
cos x 2 sin x 2
2
2
sin x cos x
dx
2 cos 2x
=−
+ 2 ln tan x
sin3 x cos3 x
sin2 2x
1
cos x
x
dx
2
5
+
−
=
+ ln tan
cos x 3 cos3 x 2 sin2 x 2
2
sin3 x cos4 x
dx
1
1
x π
−
+
=−
+ ln tan
sin x 3 sin3 x
2
4
sin4 x cos x
1
8
dx
=−
− cot 2x
sin4 x cos2 x
3 cos x sin3 x 3
dx
1
5
x π
2
sin x
−
+ ln tan
+
=−
+
sin x 3 sin3 x 2 cos2 x 2
2
4
sin4 x cos3 x
8
dx
= −8 cot 2x − cot3 2x
3
sin4 x cos4 x
2.527
2.532
2.527
1.
2.
Sines and cosines of multiple angles and functions of the argument
tanp−1 x
− tanp−2 x dx
[p = 1]
tan x dx =
p−1
n
n
1
− (−1)n ln cos x
tan2n+1 x dx =
(−1)n+k
k 2k cos2k x
k=1
n
(−1)k−1 tan2n−2k+2 x
− (−1)n ln cos x
=
2n − 2k + 2
p
k=1
3.
4.
5.
161
n
tan2n−2k+1 x
+ (−1)n x
2n − 2k + 1
k=1
cotp−1 x
p
− cotp−2 x dx
cot x dx = −
[p = 1]
p−1
n
n
1
+ (−1)n ln sin x
cot2n+1 x dx =
(−1)n+k+1
k 2k sin2k x
k=1
n
cot2n−2k+2 x
+ (−1)n ln sin x
=
(−1)k
2n − 2k + 2
tan2n x dx =
GU (331)(12)
k=1
cot2n x dx =
6.
(−1)k−1
n
k=1
(−1)k
cot2n−2k+1 x
+ (−1)n x
2n − 2k + 1
GU (331)(14)
For special formulas for p = 1, 2, 3, 4, see 2.526 17, 2.526 33, 2.526 22, 2.526 38, 2.526 27, 2.526 43,
2.526 32, and 2.526 48.
2.53–2.54 Sines and cosines of multiple angles and of linear and more complicated
functions of the argument
2.531
1.
2.
2.532
1
sin(ax + b) dx = − cos(ax + b)
a
1
cos(ax + b) dx = − sin(ax + b)
a
1.
sin(ax + b) sin(cx + d) dx =
2.8
sin[(a − c)x + b − d] sin[(a + c)x + b + d]
−
2(a − c)
2(a + c)
2
a = c2
sin(ax + b) cos(cx + d) dx = −
cos[(a − c)x + b − d] cos[(a + c)x + b + d]
−
2(a − c)
2(a + c)
2
a = c2
162
Trigonometric Functions
3.
cos(ax + b) cos(cx + d) dx =
5.
6.
2.533
1.
8
2.8
GU (332)(3)
2
cos(a + b)x cos(a − b)x
−
a = b2
2(a + b)
2(a − b)
⎧
cos(b + c − a)x
1 ⎨ cos(a − b + c)x
+
sin ax sin bx sin cx dx = −
4 ⎩ a−b+c
b+c−a
⎫
cos(a + b − c)x cos(a + b + c)x ⎬
−
+
a+b−c
a+b+c ⎭
sin ax cos bx dx = −
4.
⎧
1 ⎨ cos(a + b + c)x
PE (376)
cos(b + c − a)x
b+c−a
⎫
cos(a + b − c)x cos(a + c − b)x ⎬
+
+
a+b−c
a+c−b ⎭
sin ax cos bx cos cx dx = −
3.
cos ax sin bx sin cx dx =
5.
sin[(a − c)x + b − d] sin[(a + c)x + b + d]
+
2(a − c)
2(a + c)
2
a = c2
For c = a:
sin(2ax + b + d)
x
sin(ax + b) sin(ax + d) dx = cos(b − d) −
2
4a
cos(2ax + b + d)
x
sin(ax + b) cos(ax + d) dx = sin(b − d) −
2
4a
x
sin(2ax + b + d)
cos(ax + b) cos(ax + d) dx = cos(b − d) +
2
4a
4.
2.533
cos ax cos bx cos cx dx =
4⎩
a+b+c
⎧
1 ⎨ sin(a + b − c)x
−
PE (378)
sin(a + c − b)x
a+c−b
⎫
sin(a + b + c)x sin(b + c − a)x ⎬
−
−
a+b+c
b+c−a ⎭
4⎩
a+b−c
⎧
1 ⎨ sin(a + b + c)x
+
PE (379)
sin(b + c − a)x
b+c−a
⎫
sin(a + c − b)x sin(a + b − c)x ⎬
+
+
a+c−b
a+b−c ⎭
4⎩
a+b+c
+
PE (377)
2.535
2.534
Sines and cosines of multiple angles and functions of the argument
1.
2.
2.535
cos px + i sin px
dx = −2i
cos nx
⎧
⎪
⎪
⎪
⎨
= (2n + 1)
×
z p+n−1
dz
1 − z 2n
[z = cos x + i sin x]
Pe (374)
[z = cos x + i sin x]
Pe (373)
p
sinp x sin(2n + 1)x dx
z p+n−1
dz
1 − z 2n
p−1
x cos(a − 1)x dx
− sin x cos ax + p sin
1
sin x sin ax dx =
p+a
p
1.
2.
cos px + i sin px
dx = −2
sin nx
⎪
⎪
⎪
⎩
163
GU (332)(5a)
(2n + 1)2 − 12 (2n + 1)2 − 32 . . .
n
. . . (2n + 1)2 − (2k − 1)2
p+1
k
sin
x dx +
(−1)
(2k + 1)!
k=1
⎫
⎬
sin2k+p+1 x dx
⎭
TI (299)
=
Γ(p + 1)
p+3
+n
Γ
2
⎧
⎡
⎨ n−1
(−1)k−1 Γ p+1 + n − 2k
2
⎣
sinp−2k x cos(2n − 2k + 1)x
⎩
22k+1 Γ(p − 2k + 1)
k=0
⎤
p−1
Γ
+
n
−
2k
2
+ (−1)k 2k+2
sinp−2k−1 x sin(2n − 2k)x⎦
2
Γ(p − 2k)
⎫
⎬
(−1) Γ 2 − n
+ 2n
sinp−2n+1 x dx
⎭
2 Γ(p − 2n + 1)
n
p+3
GU (332)(5c)
164
Trigonometric Functions
⎧
⎨ sinp+2 x
sinp x sin 2nx dx = 2n
3.
+
2.536
⎩ p+2
n−1
(−1)k
k=1
⎫
⎬
4n2 − 22 4n2 − 42 . . . 4n2 − (2k)2
sin2k+p+2 x
⎭
(2k + 1)!(2k + p + 2)
TI (303)
⎧
p
n−1
⎨
k−1
(−1)
Γ 2 + n − 2k
Γ(p + 1)
sinp−2k x cos(2n − 2k)x
=
p
2k+1
⎩
2
Γ(p
−
2k
+
1)
+n+1
Γ
k=0
2
⎫
p
⎬
k
(−1) Γ 2 + n − 2k − 1
p−2k−1
−
sin
x
sin(2n
−
2k
−
1)x
⎭
22k+2 Γ(p − 2k)
[p is not equal to − 2, −4, . . . , −2n]
GU (332)(5c)
2.536
1.
2.
1
sin x cos ax dx =
p+1
p
p−1
x sin(a − 1)x dx
sin x sin ax − p sin
p
GU (332)(6a)
sinp x cos(2n + 1)x dx
n
2
2
(2n + 1)2 − 32 . . . (2n + 1)2 − (2k − 1)2
sinp+1 x k (2n + 1) − 1
+
(−1)
=
p+1
(2k)!(2k + p + 1)
k=1
× sin2k+p+1 x
TI (301)
⎧
⎡
p+1
n−1
⎨
k
(−1) Γ 2 + n − 2k
Γ(p + 1)
⎣
sinp−2k x sin(2n − 2k + 1)x
= p+3
⎩
22k+1 Γ(p − 2k + 1)
Γ 2 +n
k=0
⎤
(−1)k Γ p−1
+
n
−
2k
2
sinp−2k−1 x cos(2n − 2k)x⎦
+
22k+2 Γ(p − 2k)
⎫
⎬
(−1) Γ 2 − n
sinp−2n x cos x dx
+ 2n
⎭
2 Γ(p − 2n + 1)
n
p+3
[p is not equal to −3, −5, . . . , −(2n + 1)]
GU (332)(6c)
2.537
Sines and cosines of multiple angles and functions of the argument
165
sinp x cos 2nx dx
3.
=
p
sin x dx +
n
2
k 4n
(−1)
k=1
· 4n2 − 22 . . . 4n2 − (2k − 2)2
sin2k+p x dx
(2k)!
⎧
⎡
p
n−1
Γ(p + 1) ⎨ ⎣ (−1)k Γ 2 + n − 2k
sinp−2k x sin(2n − 2k)x
= p
22k+1 Γ(p − 2k + 1)
Γ 2 +n+1 ⎩
TI (300)
k=0
⎤
(−1)k Γ p2 + n − 2k − 1
sinp−2k−1 x cos(2n − 2k − 1)x⎦
+
22k+2 Γ(p − 2k)
⎫
⎬
(−1)n Γ p2 − n + 1
p−2n
sin
x
dx
+
⎭
22n Γ(p − 2n + 1)
GU (332)(6c)
2.537
cosp x sin ax dx =
1.
2.
1
p+a
− cosp x cos ax + p cosp−1 x sin(a − 1)x dx
GU (332)(7a)
cosp x sin(2n + 1)x dx
⎧
⎨ cosp+1 x
= (−1)n+1
⎩ p+1
+
n
k=1
=
⎫
⎬
2
2
2
2
2
2
(2n + 1) − 1 (2n + 1) − 3 . . . (2n + 1) − (2k − 1)
cos2k+p+1 x
(−1)k
⎭
(2k)!(2k + p + 1)
Γ(p + 1)
p+3
+n
Γ
2
⎧
⎨
⎩
−
n−1
k=0
p+1
TI (295)
Γ 2 +n−k
cosp−k x cos(2n − k + 1)x
22k+1 Γ(p − 2k + 1)
⎫
⎬
Γ p+3
2
+ n
cosp−n x sin(n + 1)x dx
⎭
2 Γ(p − n + 1)
[p is not equal to − 3, −5, . . . , −(2n + 1)]
GU (332)(7b)a
166
Trigonometric Functions
⎧
⎨ cosp+2 x
cosp x sin 2nx dx = (−1)n
3.
+
⎩ p+2
n−1
(−1)k
k=1
⎫
⎬
4n2 − 22 4n2 − 42 . . . 4n2 − (2k)2
cos2k+p+2 x
⎭
(2k + 1)!(2k + p + 2)
⎧
⎨
=
2.538
n−1
p
TI (297)
Γ 2 +n−k
Γ(p + 1)
cosp−k x cos(2n − k)x
−
p
k+1 Γ(p − k + 1)
⎩
2
+n+1
Γ
k=0
2
⎫
p
⎬
Γ 2 +1
+ n
cosp−n x sin nx dx
⎭
2 Γ(p − n + 1)
[p is not equal to − 2, −4, . . . , −2n]
GU (332)(7b)a
2.538 1.
cosp x cos ax dx =
2.
1
p+a
cosp x sin ax + p cosp−1 x cos(a − 1)x dx
cosp x cos(2n + 1)x dx
= (−1)n (2n + 1)
+
n
k=1
×
(−1)k
⎧
⎨
⎩
GU (332)(8a)
cosp+1 x dx
(2n + 1)2 − 12 (2n + 1)2 − 32 . . . (2n + 1)2 − (2k − 1)2
(2k + 1)!
⎫
⎬
cos2k+p+1 x dx
⎭
⎧
p+1
n−1
Γ(p + 1) ⎨ Γ 2 + n − k
cosp−k x sin(2n − k + 1)x
= p+3
2k+1 Γ(p − k + 1)
Γ 2 +n ⎩
TI (293)
k=0
⎫
⎬
Γ p+3
2
cosp−n x cos(n + 1)x dx
+ n
⎭
2 Γ(p − n + 1)
GU (332)(8b)a
2.541
Sines and cosines of multiple angles and functions of the argument
167
cosp x cos 2nx dx
2
2
n
2
2
2 n
p
k 4n 4n − 2 . . . 4n − (2k − 2)
2k+p
= (−1)
(−1)
x dx
cos x dx +
cos
(2k)!
3.
k=1
=
⎧
⎨ n−1
TI (294)
p
Γ 2 +n−k
Γ(p + 1)
p
cosp−k x sin(2n − k)x
k+1
Γ(p − k + 1)
Γ 2 + n + 1 ⎩ k=0 2
p
+
Γ 2 +1
2n Γ(p − n + 1)
⎫
⎬
cosp−n x cos nx dx
⎭
GU (332)(8b)a
2.539
1.
2.
3.
4.
5.
6.
7.
8.
2.541
1.
2.
sin 2kx
sin(2n + 1)x
dx = 2
+x
sin x
2k
n
k=1
sin 2nx
dx = 2
sin x
n
sin(2k − 1)x
2k − 1
k=1
GU (332)(5e)
cos 2kx
cos(2n + 1)x
dx = 2
+ ln sin x
sin x
2k
n
k=1
cos 2nx
dx = 2
sin x
n
k=1
x
cos(2k − 1)x
+ ln tan
2k − 1
2
GI (332)(6e)
cos 2kx
sin(2n + 1)x
dx = 2
+ (−1)n+1 ln cos x
(−1)n−k+1
cos x
2k
n
k=1
sin 2nx
dx = 2
cos x
n
(−1)n−k+1
k=1
cos(2k − 1)x
2k − 1
GU (332)(7d)
sin 2kx
cos(2n + 1)x
dx = 2
+ (−1)n x
(−1)n−k
cos x
2k
n
k=1
cos 2nx
dx = 2
cos x
n
(−1)n−k
k=1
sin(2k − 1)x
π x
+ (−1)n ln tan
+
.
2k − 1
4
2
GU (332)(8d)
1
sinn x sin nx
n
1
sin(n + 1)x cosn−1 x dx = − cosn x cos nx
n
sin(n + 1)x sinn−1 x dx =
BI (71)(1)a
BI (71)(2)a
168
Trigonometric Functions
2.542
3.
4.
5.
6.
2.542
1
sinn x cos nx
n
1
cos(n + 1)x cosn−1 x dx = cosn x sin nx
n
,
+
1
π
π
− x sinn−1 x dx = sinn x cos n
−x
sin (n + 1)
2
n
2
,
+
π
1
π
− x sinn−1 x dx = − sinn x sin n
−x
cos (n + 1)
2
n
2
cos(n + 1)x sinn−1 x dx =
1.
2.
2.543
1.
2.
2.544
For n = 2:
sin 2x
dx = 2 ln sin x
sin2 x
sin 2x dx
2
=
cosn x
(n − 2) cosn−2 x
For n = 2:
sin 2x
dx = −2 ln cos x
cos2 x
1.
2.
3.
4.
5.
6.
7.
8.
9.
2
sin 2x
dx = −
n
sin x
(n − 2) sinn−2 x
cos 2x dx
x
= 2 cos x + ln tan
sin x
2
cos 2x dx
= − cot x − 2x
sin2 x
x
cos x
3
cos 2x dx
=−
− ln tan
2
sin3 x
2 sin2 x 2
π x
cos 2x dx
= 2 sin x − ln tan
+
cos x
4
2
cos 2x dx
= 2x − tan x
cos2 x
sin x
3
π x
cos 2x dx
=−
+ ln tan
+
cos3 x
2 cos2 x 2
4
2
sin 3x dx
= x + sin 2x
sin x
sin 3x
x
dx = 3 ln tan + 4 cos x
2
2
sin x
sin 3x
dx = −3 cot x − 4x
sin3 x
BI (71)(3)a
BI (71)(4)a
BI (71)(5)a
BI (71)(6)a
2.548
Sines and cosines of multiple angles and functions of the argument
169
2.545 sin 3x
4
1
1.
dx =
−
n
n−3
cos x
(n − 3) cos
x (n − 1) cosn−1 x
For n = 1 and n = 3:
sin 3x
2.
dx = 2 sin2 x + ln cos x
cos x
sin 3x
1
3.
dx = −
− 4 ln cos x
3
cos x
2 cos2 x
2.546 cos 3x
1
4
1.
−
dx =
n−3
sinn x
(n − 3) sin
x (n − 1) sinn−1 x
For n = 1 and n = 3:
cos 3x
dx = −2 sin2 x + ln sin x
2.
sin x
cos 3x
1
3.
dx = −
− 4 ln sin x
3
sin x
2 sin2 x
2.547 1.
2.
3.
4.
2.548
1.
(k − n)π
x
+
sin
2n
2k + 1
sinm x dx
1 2(2n + 1) 2
=
π ln
(−1)n+k cosm
k+n+1
x
sin(2n + 1)x
2n + 1
2(2n + 1)
k=0
π−
sin
(2n + 1)
2
2.
3.
sin nx
sin(n − 1)x dx
sin(n − 2)x dx
dx = 2
−
p
p−1
cos x
cos
x
cosp x
cos 3x
dx = sin 2x − x
cos x
cos 3x
π x
dx = 4 sin x − 3 ln tan
+
2
cos x
4
2
cos 3x
dx = 4x − 3 tan x
cos3 x
(−1)n
sin2m x dx
=
sin 2nx
2n
ln cos x +
n−1
(−1)k cos2m
k=1
⎧
(−1)n ⎨
π x
sin2m+1 x
dx =
−
ln tan
⎩
sin 2nx
2n
4
2
+
n−1
(−1)k cos2m+1
k=1
kπ
ln tan
2n
[m a natural number ≤ 2n]
kπ
2 kπ
2
ln cos x − sin
2n
2n
TI (378)
[m a natural number ≤ n]
n+k
x
π−
4n
2
tan
n−k
x
π−
4n
2
[m a natural number < n]
TI (379)
⎫
⎬
⎭
170
Trigonometric Functions
4.
(−1)
sin x dx
π x
=
−
+
(−1)k
ln tan
cos(2n + 1)x
2n + 1 ⎩
4
2
2m
n+1
× cos2m
5.
⎧
⎨
2.549
n
kπ
2n + 1
(−1)n+1
sin2m+1 x dx
=
cos(2n + 1)x
2n + 1
k=1
ln tan
2n + 2k + 1
x
π−
4(2n + 1)
2
tan
⎫
⎬
2n − 2k + 1
x
π−
2(2n + 1)
2 ⎭
[m a natural number ≤ n]
ln cos x +
n
k
(−1) cos
2m+1
k=1
TI (381)
[m a natural number ≤ n]
2k − 2n + 1
x
π+
sin
2n−1
2k + 1
1 sinm x dx
8n
2
=
π ln
(−1)n+k cosm
2k + 2n + 1
x
cos 2nx
2n
4n
k=0
π−
sin
8n
2
TI (382)a
6.
7.
8.
1
cos2m+1 x dx
=
sin(2n + 1)x
2n + 1
[m a natural number < 2n]
ln sin x +
k=1
10.
kπ
kπ
ln sin2 x − sin2
2n + 1
2n + 1
1
cos2m x dx
x
=
ln tan
sin(2n + 1)x 2n + 1 ⎩
2
n
(−1)k cos2m
1
cos2m+1 x
dx =
sin 2nx
2n
1
cos2m x
dx =
sin 2nx
2n
ln tan
x
+
2
ln sin x +
kπ
2n + 1
ln tan
n−1
(−1)k cos2m+1
k=1
x
kπ
+
2 4n + 2
tan
kπ
x
−
2 4n + 2
k=0
⎭
TI (374)
[m a natural number ≤ n]
2k + 1
x
π+
sin
2k + 1
4n
2
π ln
2k + 1
x
2n
π−
sin
4n
2
TI (373)
(−1)k cos2m
k=1
[m is a natural number ≤ n]
2.549 π
2
S (x)
1.
sin x dx =
2
⎫
⎬
[m a natural number < n]
kπ
kπ
ln sin2 x − sin2
2n
2n
n−1
n−1
1
cosm x
dx =
(−1)k cosm
cos nx
n
TI (376)
TI (375)
[m a natural number ≤ n]
x kπ
x kπ
kπ
ln tan
+
−
tan
2n
2
4
2
4n
11.
TI (377)
[m a natural number ≤ n]
k=1
9.
(−1)k cos2m+1
⎧
⎨
+
n
kπ
kπ
ln cos2 x − sin2
2n + 1
2n + 1
TI (372)
2.552
Rational functions of sine and cosine
2
2.
cos x dx =
3.
11
π
C (x)
2
sin ax2 + 2bx + c dx =
cos ax2 + 2bx + c dx =
4.11
π
2a
ac − b2
S
cos
a
π
2a
ac − b
C
a
ax + b
√
a
6.
ac − b2
C
+ sin
a
ax + b
√
a
[a > 0]
2
cos
ax + b
√
a
− sin
ac − b2
S
a
ax + b
√
a
[a > 0]
5.
171
x
(sin ln x − cos ln x)
2
x
cos ln x dx = (sin ln x + cos ln x)
2
sin ln x dx =
PE (444)
PE (445)
2.55–2.56 Rational functions of the sine and cosine
2.551
1.
⎡
A + B sin x
1
⎣ (Ab − aB) cos x
n dx =
(n − 1) (a2 − b2 ) (a + b sin x)n−1
(a + b sin x)
+
⎤
(Aa − Bb)(n − 1) + (aB − bA)(n − 2) sin x
n−1
(a + b sin x)
dx⎦
TI (358)a
2.
3.
2.552
1.
2.
For n = 1:
B
Ab − aB
A + B sin x
dx
dx = x +
a + b sin x
b
b
a + b sin x
a tan x2 + b
2
dx
=√
arctan √
a + b sin x
a2 − b 2
a2 − b2√
x
a tan 2 + b − b2 − a2
1
=√
ln
b2 − a2 a tan x + b + b2 − a2
2
(see 2.551 3)
a2 > b 2
a2 < b 2
TI (342)
B
dx
A + B cos x
n dx = −
n
n−1 + A
(a + b sin x)
(a
+
b
sin x)
(n − 1)b (a + b sin x)
(see 2.552 3)
TI (361)
(see 2.551 3)
TI (344)
For n = 1:
B
dx
A + B cos x
dx = ln (a + b sin x) + A
a + b sin x
b
a + b sin x
172
Trigonometric Functions
3.
2.553
⎡
b cos x
dx
1
⎣
n =
(n − 1) (a2 − b2 ) (a + b sin x)n−1
(a + b sin x)
⎤
(n − 1)a − (n − 2)b sin x ⎦
+
dx
n−1
(a + b sin x)
(see 2.551 1)
TI (359)
2.553
1.
B
dx
A + B sin x
dx
=
+
A
n
n
n−1
(a + b cos x)
(a + b cos x)
(n − 1)b (a + b cos x)
(see 2.554 3)
2.
3.∗
For n = 1:
B
dx
A + B sin x
dx = − ln (a + b cos x) + A
a + b cos x
b
a + b cos x
2.554
1.
TI (355)
(see 2.553 3∗ )
dx
a + b cos x
(a − b) tan x2
2
√
arctan
=√
a2 − b 2
a2 − b 2
!
!
! (b − a) tan x + √b2 − ba !
2
!
!
x2 √
=√
ln !
!
2
2
2
a
!
a −b
(b − a) tan 2 − b − b !
(a − b) tan x2
2
√
=√
arctanh
b 2 − a2
b 2 − a2
(a − b) tan x2
2
√
=√
arccoth
b 2 − a2
b 2 − a2
a2 > b 2
b 2 > a2
!
,
x !! 2
!
! a2 , !(b − a) tan
! > b − a2
2
(compare with 2.551 3)
+
b 2 > a2 ,
⎡
A + B cos x
1
⎣ (aB − Ab) sin x
n dx =
(n − 1) (a2 − b2 ) (a + b cos x)n−1
(a + b cos x)
+
TI (343)
(Aa − bB)(n − 1) + (n − 2)(aB − bA) cos x
n−1
(a + b cos x)
⎤
dx⎦
TI (353)
2.557
2.
3.
Rational functions of sine and cosine
173
For n = 1:
B
Ab − aB
A + B cos x
dx
dx = x +
(see 2.553 3)
a + b cos x
b
b
a + b cos x
⎧
⎨
b sin x
1
dx
n =−
2
2
(n − 1) (a − b ) ⎩ (a + b cos x)n−1
(a + b cos x)
⎫
(n − 1)a − (n − 2)b cos x ⎬
−
dx
(see 2.554 1)
n−1
⎭
(a + b cos x)
TI (341)
TI (354)
In integrating the functions in formulas 2.551 3 and 2.553 3, we may not take the integration over points
a
in formula 2.551 3 or
at which the integrand becomes inﬁnite, that is, over the points x = arcsin −
b
a
over the points x = arccos −
in formula 2.553 3.
b
2.555 Formulas 2.551 3 and 2.553 3 are not applicable for a2 = b2 . Instead, we may use the following
formulas in these cases:
n−2
n − 2 tan2k+1 π ∓ x
A + B sin x
1
4
2
1.
2B
n dx = − n−1
k
2
2k + 1
(1 ± sin x)
k=0
π x
n−1
2k+1
n − 1 tan
4 ∓ 2
± (A ∓ B)
k
2k + 1
k=0
2.
A + B cos x
1
n dx = n−1
2
(1 ± cos x)
2B
± (A ∓ B)
TI (361)a
n−2
n−2
k
k=0
n−1
n−1
k
k=0
2k+1 π
π
x
∓ 4−2
2k + 1
tan2k+1 π4 ∓ π4 − x2
2k + 1
tan
4
TI (356)
3.11
4.
For n = 1 :
π x
A + B sin x
dx = ±Bx + (B ∓ A) tan
∓
1 ± sin x
4
2
+
π
π x ,
A + B cos x
dx = ±Bx ± (A ∓ B) tan
∓
−
1 ± cos x
4
4
2
2.556 1.
2.
2.557 1.
1 − a2 dx
= 2 arctan
1 − 2a cos x + a2
1+a
x
tan
1−a
2
(1 − a cos x) dx
x
= + arctan
1 − 2a cos x + a2
2
1+a
x
tan
1−a
2
1
dx
n = n
2
(a cos x + b sin x)
(a + b2 )
TI (250)
TI (248)
[0 < a < 1,
|x| < π]
FI II 93
[0 < a < 1,
|x| < π]
FI II 93
dx
sin
n
a
b
(see 2.515)
x + arctan
MZ 173a
174
Trigonometric Functions
2.558
ax − b ln sin x + arctan ab
sin x dx
=
a sin x + b cos x
a2 + b 2
ax + b ln sin x + arctan ab
cos x dx
=
a cos x + b sin x
a2 + b 2
1 ln tan 2 x + arctan ab
dx
√
=
a cos x + b sin x
a2 + b 2
cot x + arctan ab
1
a sin x − b cos x
dx
=+ 2
·
2 =−
2 + b2
2 a cos x + b sin x
a
a
+
b
(a cos x + b sin x)
2.
6
3.
4.
5.
MZ 174a
MZ 174a
2.558 A + B cos x + C sin x
1.
n dx
(a + b cos x + c sin x)
(Bc − Cb) + (Ac − Ca) cos x − (Ab − Ba) sin x
1
=
n−1 + (n − 1) (a2 − b2 − c2 )
2
2
2
(n − 1) (a − b − c ) (a + b cos x + c sin x)
(n − 1)(Aa − Bb − Cc) − (n − 2) [(Ab − Ba) cos x − (Ac − Ca) sin x]
×
dx
n−1
(a + b cos x + c sin x)
n = 1, a2 = b2 + c2
A n(Bb + Cc)
Cb − Bc + Ca cos x − Ba sin x
+
=
(−c cos x + b sin x)
n +
a
(n − 1)a2
(n − 1)a (a + b cos x + c sin x)
n−1
1
(n − 1)! (2n − 2k − 3)!!
·
×
n−k
k
(2n − 1)!!
(n − k − 1)!a
(a + b cos x + c sin x)
k=0
n = 1, a2 = b2 + c2
11
2.
For n = 1 :
A + B cos x + C sin x
Bc − Cb
Bb + Cc
dx = 2
ln (a + b cos x + c sin x) + 2
x
a + b cos x + c sin x
b + c2
b + c2
dx
Bb + Cc
a
+ A− 2
2
b +c
a + b cos x + c sin x
3.
4.
(see 2.558 4)
GU (331)(18)
dx
d(x − α)
n =
n,
(a + b cos x + c sin x)
[a + r cos(x − α)]
where b = r cos α, c = r sin α (see 2.554 3)
dx
a + b cos x + c sin x
(a − b) tan x2 + c
2
arctan √
a2 − b 2 − c 2
a2 − b 2 − c 2 √
(a − b) tan x2 + c − b2 + c2 − a2
1
=√
ln
b2 + c2 − a2 (a − b) tan x + c + b2 + c2 − a2
2
x
1
= ln a + c · tan
c
2
−2
=
x
c + (a − b) tan
2
=√
a2 > b 2 + c 2
a2 < b 2 + c 2
TI (253), FI II 94
TI (253)a
[a = b]
a2 = b 2 + c 2
TI (253)a
2.561
Rational functions of sine and cosine
c (a sin x − c cos x)
x
1
2 = c3 a (1 + cos x) + c sin x − a ln a + c tan 2
[a (1 + cos x) + c sin x]
A + B cos x + C sin x
dx
(a1 + b1 cos x + c1 sin x) (a2 + b2 cos x+ c2 sin x)
a1 + b1 cos x + c1 sin x
dx
dx
= A0 ln
+ A1
+ A2
a2 + b2 cos x + c + 2 sin x
a1 + b1 cos x + c1 sin x
a2 + b2 cos x + c2 sin x
GU (331)(19)
(see 2.558 4)
2.559 1.
2.
175
dx
where
!
!
!A B C!
!
!
!a1 b1 c1 !
!
!
!a2 b2 c2 !
A0 = !
!
!
! ,
!
!
!a1 b1 !2 !b1 c1 !2 !c1 a1 !2
!
! −!
! +!
!
!a2 b2 !
!b2 c2 !
!c2 a2 !
! !
! !
!!
!!
!! C B ! ! C A ! ! A B !!
! !
! !
!!
!!
!!c2 b2 ! !c2 a2 ! !a2 b2 !!
!
!
! a1
b1
c1 !!
!
! a2
b2
c2 !
A2 = !
!2 !
!2 !
!2 ,
!a1 b1 !
!
!
!
!
!
! − !b1 c1 ! + !c1 a1 !
!a2 b2 !
!b2 c2 !
!c2 a2 !
3.
! !
! !
!!
!!
!! B C ! ! A C ! ! B A !!
! !
! !
!!
!!
!!b1 c1 ! !a1 c1 ! !b1 a1 !!
!
!
! a1
b1
c1 !!
!
! a2
b2
c2 !
A1 = !
!2 !
!2 !
!2 ,
! a1 b 1 !
!
!
!
!
!
! − ! b 1 c 1 ! + ! c 1 a1 !
! a2 b 2 !
! b 2 c2 !
! c 2 a2 !
%!
! a1
!
! a2
!2 !
!c
b1 !!
+ !! 1
b2 !
c2
!2 !
!b
a1 !!
= !! 1
a2 !
b2
!2 &
c1 !!
c2 !
A cos2 x + 2B sin x cos x + C sin2 x
dx
a cos2 x + 2b sin x cos x + c sin2 x
⎧
⎨
1
= 2
[4Bb + (A − C)(a − c)]x + [(A − C)b − B(a − c)]
4b + (a − c)2 ⎩
× ln a cos2 x + 2b sin x cos x + c sin2 x
⎫
⎬
+ 2(A + C)b2 − 2Bb(a + c) + (aC − Ac)(a − c) f (x)
⎭
where
√
1
c tan x + b − b2 − ac
√
f (x) = √
ln
2 b2 − ac c tan x + b + b2 − ac
c tan x + b
1
arctan √
=√
2
ac − b
ac − b2
1
=−
c tan x + b
GU (331)(24)
b2 > ac
b2 < ac
b2 = ac
2.561 A
x Ba − Ab
dx
(A + B sin x) dx
= ln tan +
1.
sin x (a + b sin x)
a
2
a
a + b sin x
(see 2.551 3)
TI (348)
176
Trigonometric Functions
2.
2.561
A
a + b cos x
(A + B sin x) dx
x
=
a ln tan + b ln
2
sin x (a + b cos x) a2 −
b
2
sin x
dx
+B
a + b cos x
(see 2.553 3)
TI (349)
3.
4.
5.
2
2
2
2
For a = b (= 1) :
A
1
(A + B sin x) dx
x
x
=
ln tan +
+ B tan
sin x (a + b cos x)
2
2 1 + cos x
2
(A + B sin x) dx
A
1
x
x
=
ln tan −
− B cot
sin x (1 − cos x)
2
2 1 − cos x
2
(A + B sin x) dx
1
a + b sin x
π x
= 2
+
− (Ab − aB) ln
(Aa − Bb) ln tan
cos x (a + b sin x)
a − b2
4
2
cos x
TI (346)
6.
7.
For a = b (= 1):
π x
A±B
(A + B sin x) dx
A∓B
=
ln tan
+
∓
cos x (1 ± sin x)
2
4
2
2 (1 ± sin x)
A
B a + b cos x
π x
(A + B sin x) dx
= ln tan
+
+ ln
cos x (a + b cos x)
a 4
2
a
cos x
Ab
dx
−
a
a + b cos x
8.
9.
10.
11.
A
x B a + b sin x Ab
(A + B cos x) dx
= ln tan − ln
−
sin x (a + b sin x)
a
2
a
sin x
a
(see 2.553 3)
TI (351)a
dx
a + b sin x
(see 2.551 3)
1
(A + B cos x) dx
= 2
sin x (a + b cos x)
a − b2
(Aa − Bb) ln tan
a + b cos x
x
+ (Ab − Ba) ln
2
sin x
(see 2.551 3)
13.
TI (352)
TI (345)
For a2 = b2 (= 1) :
A∓B
A±B
x
(A + B cos x) dx
=±
+
ln tan
sin x (1 ± cos x)
2 (1 ± cos x)
2
2
A
a + b sin x
(A + B cos x) dx
π x
dx
= 2
+
−
b
ln
a
ln
tan
+
B
2
cos x (a + b sin x)
a −b
4
2
cos x
a + b sin x
2
12.
TI (350)
2
For a = b (= 1):
A±B
π x
A∓B
(A + B sin x) dx
=
ln tan
+
∓
cos x (1 ± sin x)
2
4
2
2 (1 ± sin x)
π x
A
(A + B cos x) dx
Ba − Ab
dx
= ln tan
+
+
cos x (a + b cos x)
a
4
2
a
a + b cos x
(see 2.553 3)
TI (347)
2.563
2.562
Rational functions of sine and cosine
1.
2.
a+b
tan x
a
sign a
a+b
tan x
=
arctanh
−
a
−a(a + b)
a+b
sign a
tan x
arccoth
−
=
a
−a(a + b)
dx
sign a
arctan
=
a + b sin2 x
a(a + b)
− sign a
dx
=
arctan
2
a + b cos x
a(a + b)
4.
5.
6.
2.563
b
> −1
a
b
< −1,
a
a
sin x < −
b
b
< −1,
a
a
sin x > −
b
2
2
MZ 155
b
> −1
a
b
< −1,
a
a
cos x < −
b
b
< −1,
a
cos2 x > −
2
a
b
MZ 162
√
dx
1
2 tan x
= √ arctan
1 + sin2 x
2
dx
= tan x
1 − sin2 x
√
dx
1
= − √ arctan
2 cot x
2
1 + cos x
2
dx
= − cot x
1 − cos2 x
1
b sin x cos x
dx
+
(2a + b)
2 =
2a(a + b)
a + b sin2 x a + b sin2 x
a + b sin2 x
(2a + b)
1.
2.
a+b
cot x
a
− sign a
a+b
=
cot x
arctanh
−
a
−a(a + b)
− sign a
a+b
cot x
=
arccoth
−
a
−a(a + b)
3.
177
dx
dx
2
(a + b cos2 x)
=
1
2a(a + b)
(see 2.562 1)
b sin x cos x
dx
−
a + b cos2 x a + b cos2 x
MZ 155
(see 2.562 2)
MZ 163
178
Trigonometric Functions
⎡
3.
2
1 ⎣
3
3+ 2 + 4
3 =
3
8pa
p
p
a + b sin x
dx
2
3
2
+ 3+ 2 − 4
p
p
+
b
q 2 = −1 − > 0,
a
3
2
3− 2 − 4
q
q
a
sin x < − ;
b
2
arctan (p tan x)
⎤
p tan x
+
1 + p2 tan2 x
⎡
1 ⎣
2
3
=
3− 2 + 4
3
8qa
q
q
4.
2.564
1
2
1 − 2 − 2 tan2 x
p
p
2p tan x
2 ⎦
1 + p2 tan2 x
b
p2 = 1 + > 0
a
arctanh (q tan x)
⎤
q tan x
1
2
+ 1 + 2 + 2 tan2 x
q
q
1 − q 2 tan2 x
2q tan x
2 ⎦
1 − q 2 tan2 x
a
for sin x > − , change arctanh (q tan x) to arccoth (q tan x)
b
MZ 156
⎡
dx
3
(a + b cos2 x)
=−
+
2
1 ⎣
3
3+ 2 + 4
8pa3
p
p
3
2
3+ 2 − 4
p
p
arctan (p cot x)
⎤
p cot x
1
2
+ 1 − 2 − 2 cot2 x
2
2
p
p
1 + p cot x
⎡
=−
+
b
q 2 = −1 − > 0,
a
1 ⎣
2
3
3− 2 + 4
8qa3
q
q
3
2
3− 2 − 4
q
q
a
cos x < − ;
b
2
2
2p cot x
2 ⎦
1 + p2 cot2 x
b
2
p =1+ >0
a
arctanh (q cot x)
⎤
q cot x
2
1
+ 1 + 2 + 2 cot2 x
q
q
1 − q 2 cot2 x
2p cot x
2 ⎦
1 − q 2 cot2 x
a
for cos x > − , change arctanh (q cot x) to arccoth (q cot x)
b
2
MZ 163a
2.564 ln cos2 x + m2 sin2 x
tan x dx
=
1.
2 (m2 − 1)
1 + m2 tan2 x
tan α − tan x
dx = sin 2α ln sin(x + α) − x cos 2α
2.
tan α + tan x
1
tan x dx
= 2
3.
{bx − a ln (a cos x + b sin x)}
a + b tan x
a + b2
%
&
b
b
dx
1
x−
arctan
tan x
4.
=
a−b
a
a
a + b tan2 x
LA 210 (10)
LA 210 (11)a
PE (335)
PE (334)
2.571
Integrals with
2.57 Integrals containing
√
a ± b sin x or
√
a ± b cos x
179
√
√
a ± b sin x or a ± b cos x
Notation:
α = arcsin
γ = arcsin
2.571
1.
2.
3.
b (1 − sin x)
,
a+b
.
(a + b) (1 − cos x)
δ = arcsin
,
2 (a − b cos x)
β = arcsin
b (1 − cos x)
,
a+b
dx
−2
√
=√
F (α, r)
a + b sin x
a
+ b
1
2
F β,
=−
b
r
+
a > b > 0,
+
0 < |a| < b,
r=
2b
a+b
π,
π
≤x<
2
2
π,
a
− arcsin < x b > 0, − ≤ x <
BY (288.03)
E (α, r)
= √
F (α, r) −
2
2
b
b a+b
+
a
π,
2
1
1
0 < |a| < b, − arcsin < x <
BY (288.54)
=
F β,
− 2 E β,
b
2
b
r
r
√
√
2 2a2 + b2
sin2 x dx
4a a + b
2
√
√
cos x a + b sin x
E (α, r) −
=
F (α, r) −
2
2
3b
3b +
a + b sin x
3b a + b
π
π,
a > b > 0, − ≤ x <
2
2
√
1
1
2 4a
2a + b
2
E β,
F β,
cos x a + b sin x
=
−
−
b 3b
r
3b
r
+ 3b
π,
a
0 < |a| < b, − arcsin < x b > 0,
+
b ≥ |a| > 0,
0 ≤ x ≤ π]
0 ≤ x < arccos −
a ,
b
BY (289.00)
dx
2
√
=√
F (δ, r)
a − b cos x
a+b
[a > b > 0,
0 ≤ x ≤ π]
BY (291.00)
180
Trigonometric Functions
6.
cos x dx
2
√
= √
(a + b) E
a + b cos x b a + b
x
,r − aF
2
2.572
x
,r
2
0 ≤ x ≤ π]
[a > b > 0,
=
2
b
2E
1
γ,
r
−F
1
γ,
r
BY (289.03)
+
b > |a| > 0,
0 ≤ x < arccos −
a ,
b
BY (290.04)
7.6
8.
cos x dx
2
√
= √
(b − a) Π δ, r2 , r + a F (δ, r)
a − b cos x
b a+b
BY (291.03)
[a > b > 0, 0 ≤ x ≤ π]
2
√
2
x
2
x
cos x dx
2
√
2a + b2 F
, r − 2a(a + b) E
,r
sin x a + b cos x
+
= 2√
2
2
3b
a + b cos x 3b a + b
[a > b > 0, 0 ≤ x ≤ π]
1
=
3b
2
b
(2a + b) F
1
γ,
r
− 4a E
1
γ,
+r
√
2
sin x a + b cos x
+
3b
b ≥ |a| > 0,
BY (289.03)
0 ≤ x < arccos −
a ,
b
BY (290.04)
9.
2
cos2 x dx
2
√
= 2√
2a + b2 F (δ, r) − 2a(a + b) E (δ, r)
a − b cos x 3b a + b
a + b cos x
2
+ sin x √
[a > b > 0, ]
3b
a − b cos x
BY (291.04)a
2.572
tan2 x dx
√
a + b sin x
a
1
√
F (α, r) +
E (α, r)
=√
a+b
(a − b) a + b
b − a sin x √
− 2
a + b sin x
2
(a − b ) cos x
2a + b
1
2
1
ab
F β,
E β,
=
+ 2
b 2(a + b)
r
a − b2
r
√
b − a sin x
a + b sin x
− 2
(a − b2 ) cos x
+
π,
π
0 < b < a, − < x <
2
2
+
0 < |a| < b,
− arcsin
π,
a
<x<
b
2
BY(288.08, 288.58)
2.573
1.
1 − sin x
dx
2
·√
=
1 + sin x
a + b sin x a − b
√
⎫
⎬
√
π x
−
a + b E (α, r) − tan
a + b sin x
⎭
4
2
+
π
π,
0 < b < a, − ≤ x <
2
2
BY (288.07)
2.575
Integrals with
2.
2.574
(2 −
2.
a ± b sin x or
√
a ± b cos x
p2
+
p2
dx
1
√
Π α, p2 , r
=−
a+b
sin x) a + b sin x
+
0 < b < a,
dx
1
√
=−
a+b
(a + b − p2 b + p2 b sin x) a + b sin x
π,
π
≤x<
2
2
2
1
Π β, p2 ,
b +
r
− arcsin
π,
a
<x<
b
2
BY (288.52)
(2 −
p2
+
p2
dx
1
x 2
√
,p ,r
=√
Π
2
cos x) a + b cos x
a+b
4.
−
BY (289.07)
BY (288.02)
0 < |a| < b,
3.
181
√
1 − cos x
dx
x√
x
2 a+b
2
√
tan
E
,r
a + b cos x −
=
1 + cos x a + b cos x a − b
2
a−b
2
[a > b > 0, 0 ≤ x < π]
1.
√
√
dx
(a + b −
p2 b
+
p2 b cos x)
√
a + b cos x
=
[a > b > 0,
2
1
√ Π γ, p2 ,
r
(a + b) b
+
b ≥ |a| > 0,
0 ≤ x < π]
BY (289.02)
0 ≤ x < arccos −
a ,
b
BY (290.02)
2.575
1.
2b cos x
dx
2
<
√
√
= 2
E (α, r)
−
2 ) a + b sin x
3
a+b
(a
−
b
(a
−
b)
+
(a + b sin x)
0 < b < a,
=
2
b
2b
E
b 2 − a2
1
β,
r
1
F
−
a+b
1
β,
+r
−
π,
π
≤x<
2
2
BY (288.05)
cos x
2b
·√
+ 2
b − a2
a + b sin x
π,
a
0 < |a| < b, − arcsin < x <
b
2
BY (288.56)
182
Trigonometric Functions
2.
2.575
2
dx
2
<
a − b2 F (α, r) − 4a(a + b) E (α, r)
=
2√
2
2
5
3 (a − b ) a + b
(a + b sin x)
2b 5a2 − b2 + 4ab sin x
<
+
cos x
2
3
3 (a2 − b2 )
(a + b sin x)
+
π
π,
0 < b < a, − ≤ x <
2
2
BY (288.05)
2
1
1
1
=−
(3a − b)(a − b) F β,
+ 8ab E β,
2
2
2
b
r
r
3 (a − b )
2b a2 − b2 + 4a (a + b sin x)
<
+
cos x
2
3
3 (a2 − b2 )
(a + b sin x)
+
0 < |a| < b,
− arcsin
a
π,
<x b > 0,
1
γ,
r
0 ≤ x ≤ π]
BY (289.05)
sin x
2b
·√
+ 2b E
+ 2
2
b
−
a
a
+
b cos x ,
+
a
b ≥ |a| > 0, 0 ≤ x < arccos −
b
1
γ,
r
BY (290.06)
4.
dx
2
<
√
=
E (δ, r)
3
(a
−
b)
a
+
b
(a − b cos x)
[a > b > 0,
0 ≤ x ≤ π]
(291.01)
2.578
Integrals with
√
2 a+b
dx
−
a ± b sin x or
4a E
<
=
2
5
3 (a2 − b2 )
(a + b cos x)
5.
√
2b
2
3 (a2 − b2 )
·
√
a ± b cos x
183
x
,r
2
x
, r − (a − b) F
2
5a2 − b2 + 4ab cos x
<
sin x
3
(a + b cos x)
[a > b > 0,
=
1
3 (a2 − b2 )
+
2
2
b
(a − b)(3a − b) F
2b 5a2 − b2 + 4ab cos x sin x
<
2
3
3 (ab − b2 )
(a + b cos x)
γ,
1
r
+ 8ab E
+
b ≥ |a| > 0,
γ,
1
r
0 ≤ x ≤ π]
BY (289.05)
0 ≤ x < arccos −
a ,
b
BY (290.06)
2.576
1.
√
√
a + b cos x dx = 2 a + b E
=
2
b
x
,r
2
(a − b) F
[a > b > 0,
1
γ,
r
+ 2b E
1
γ,
r+
0 ≤ x ≤ π]
BY (289.01)
b ≥ |a| > 0,
0 ≤ x < arccos −
a ,
b
BY (290.03)
2.
2.577
1.
3
√
√
2b sin x
a − b cos x dx = 2 a + b E (δ, r) − √
a − b cos x
[a > b > 0,
0 ≤ x ≤ π]
√
2(a − b)
a − b cos x
2ap
√
dx =
,r
Π δ,
1 + p cos x
(a
+
b)(1
+ p)
(1 + p) a + b
[a > b > 0,
0 ≤ x ≤ π,
.
2.3
2(a − b)
a − b cos x
dx = Π δ, −r2 ,
1 + p cos x
(1 + p)(a + b)
.
2(ap + b)
(1 + p)(a + b)
2.578
tan x dx
1
√
arccos
=√
2
b
−a
a + b tan x
√
b−a
√
cos x
b
[b > a,
p = −1]
BY (291.02)
[a > b > 0,
BY (291.05)
0 ≤ x ≤ π,
b > 0]
p = −1]
PE (333)
184
Trigonometric Functions
2.580
2.58–2.62 Integrals reducible to elliptic and pseudo-elliptic integrals
2.580 ,
+
dϕ
dψ
c
√
1.
=2 ϕ = 2ψ + α, tan α = , p = b2 + c2
b
a + b cos ϕ + c sin ϕ
a − p + 2p cos2 ψ
dx
dϕ
=2 √
2.
2
2
A + Bx + Cx2 − Dx3 + Ex4
a + b cos ϕ + c sin ϕ + d cos ϕ + e sin ϕ cos ϕ + f sin ϕ
+
,
ϕ
tan = x, A = a + b + d, B = 2c + 2e, C = 2a − 2d + 4f, D = 2c − 2e, E = a − b + d
2
Forms containing 1 − k2 sin2 x
√
Notation: Δ = 1 − k 2 sin2 x, k = 1 − k 2
2.581 1.
sinm x cosn xΔr dx
⎧
⎨
1
sinm−3 x cosn+1 xΔr+2 + (m + n − 2) + (m + r − 1)k 2
2
(m + n + r)k ⎩
⎫
⎬
× sinm−2 x cosn xΔr dx − (m − 3) sinm−4 x cosn xΔr dx
⎭
⎧
⎨
,
+
1
m+1
n−3
r+2
2
2
=
x
cos
xΔ
+
(n
+
r
−
1)k
−
(m
+
n
−
2)k
sin
(m + n + r)k2 ⎩
⎫
⎬
2
sinm x cosn−4 xΔr dx
× sinm x cosn−2 xΔr dx + (n − 3)k ⎭
=
[m + n + r = 0]
2.
3.
4.
5.
For r = −3 and r = −5:
sinm x cosn x
sinm−1 x cosn−1 x
dx =
3
Δ
k 2Δ
n − 1 sinm x cosn−2 x
m − 1 sinm−2 x cosn x
dx +
dx
− 2
k
Δ
k2
Δ
sinm x cosn x
sinm−1 x cosn−1 x
dx
=
Δ5
3k 2Δ3
m − 1 sinm−2 x cosn x
n − 1 sinm x cosn−2 x
−
dx +
dx
3k 2
Δ3
3k 2
Δ3
For m = 1 or n = 1:
2 cosn−1 xΔr+2
(n − 1)k cosn−2 x sin xΔr dx
−
sin x cosn xΔr dx = −
(n + r + 1)k 2
(n + r + 1)k 2
sinm−1 xΔr+2
m−1
m
r
sin x cos xΔ dx = −
+
sinm−2 x cos xΔr dx
(m + r + 1)k 2
(m + r + 1)k 2
For m = 3 or n = 3:
2.583
Elliptic and pseudo-elliptic integrals
(n + r + 1)k 2 cos2 x − (r + 2)k2 + n + 1
sin x cos xΔ dx =
cosn−1 xΔr+2
4
(n
+
r
+
1)(n
+
r
+
3)k
2
(r + 2)k 2 + n + 1 (n − 1)k −
cosn−2 x sin xΔr dx
(n + r + 1)(n + r + 3)k 4
3
6.
185
n
r
sinm x cos3 xΔr dx
7.
=
,
+
2
(m + r + 1)k 2 sin2 x − (r + 2)k2 − (m + 1)k (m + r + 1)(m++ r + 3)k 4
,
2
(r + 2)k2 − (m − 1)k (m − 1) × sinm−1 xnm−1 xΔr+2 +
sinm−2 x cos xΔr dx
(m + r + 1)(m + r + 3)k 4
2.582 n−1 n−2
2 − k2
1 − k2
Δn−2 dx −
Δn−4 dx
1.
Δn dx =
n
n
k2
+ sin x cos x · Δn−2
n
2.
3.
n−3 1
dx
−
Δn−1
n − 1 k 2
sinn−3 x
sinn x
n − 2 1 + k 2 sinn−2 x
dx =
dx
cos
x
·
Δ
+
Δ
(n − 1)k 2
n − 1 k2
Δ
n−3
sinn−4 x
dx
−
(n − 1)k2
Δ
4.
5.
6.
dx
k2 sin x cos x
n−22−k
=−
+
n+1
Δ
n − 1 k 2
(n − 1)k 2 Δn−1
2
cosn−3 x
cosn x
n − 2 2k 2 − 1
dx =
sin x · Δ +
2
Δ
(n − 1)k
n − 1 k2
2 n−4
cos
x
n−3k
dx
+
2
n−1 k
Δ
(n − 2) 2 − k
tann−3 x Δ
tann x
dx =
2 cos2 x −
Δ
(n − 1)k
(n − 1)k 2
n−3
tann−4 x
−
dx
2
Δ
(n − 1)k
cotn−1 x Δ
n−2 cotn x
dx = −
−
2 − k2
2
Δ
n − 1 cos x n − 1
n − 3 2 cotn−4 x
k
dx
−
n−1
Δ
LA (316)(1)a
dx
Δn−3
LA 317(8)a
LA 316(1)a
cosn−2 x
dx
Δ
2 LA 316(2)a
tann−2 x
dx
Δ
LA 317(3)
cotn−2 x
dx
Δ
LA 317(6)
2.583 1.
Δ dx = E (x, k)
186
Trigonometric Functions
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
Δ cos x k −
ln (k cos x + Δ)
2
2k
Δ sin x
1
Δ cos x dx =
+
arcsin (k sin x)
2
2k
2
k
Δ
2k 2 − 1
Δ sin2 x dx = − sin x cos x + 2 F (x, k) +
E (x, k)
3
3k
3k 2
Δ3
Δ sin x cos x dx = − 2
3k
2
k
Δ
k2 + 1
sin x cos x − 2 F (x, k) +
Δ cos2 x dx =
E (x, k)
3
3k
3k 2
2k 2 sin2 x + 3k 2 − 1
3k 4 − 2k 2 − 1
Δ sin3 x dx = −
Δ cos x +
ln (k cos x + Δ)
2
8k
8k 3
2k 2 sin2 x − 1
1
Δ sin2 x cos x dx =
Δ sin x + 3 arcsin (k sin x)
2
8k
8k
4
2
2
2
2k cos x + k
k
Δ sin x cos2 x dx = −
Δ cos x + 3 ln (k cos x + Δ)
2
8k
8k
2
2
2
2
2k cos x + 2k + 1
4k − 1
Δ cos3 x dx =
Δ sin x +
arcsin (k sin x)
2
8k
8k 3
3k 2 sin2 x + 4k 2 − 1
Δ sin4 x dx = −
Δ sin x cos x
4 15k2 2 2 2k − k − 1
8k 4 − 3k 2 − 2
−
F (x, k) +
E (x, k)
4
15k
15k 4
3k 4 sin4 x − k 2 sin2 x − 2
Δ
Δ sin3 x cos x dx =
15k 4
3k 2 cos2 x − 2k 2 + 1
Δ sin2 x cos2 x dx = −
Δ sin x cos x
15k 2
2
2
k 1 + k
2 k4 − k2 + 1
F (x, k) +
E (x, k)
−
15k 4
15k 4
3k 4 sin4 x − k 2 5k 2 + 1 sin2 x + 5k 2 − 2
3
Δ sin x cos x dx = −
Δ
15k 4
3k2 cos2 x + 3k 2 + 1
Δ cos4 x dx =
Δ sin x cos x
15k 2
2
2
2k k − 2k 2
3k 4 + 7k 2 − 2
F
(x,
k)
+
E (x, k)
+
15k 4
15k 4
−8k 4 sin4 x − 2k 2 5k 2 − 1 sin2 x − 15k 4 + 4k 2 + 3
Δ sin5 x dx =
Δ cos x
48k 4
6
4
2
5k − 3k − k − 1
ln (k cos x + Δ)
+
16k 5
8k 4 sin4 x − 2k 2 sin2 x − 3
1
Δ sin x +
arcsin (k sin x)
Δ sin4 x cos x dx =
4
48k
16k 5
2
Δ sin x dx = −
2.583
2.583
Elliptic and pseudo-elliptic integrals
8k4 sin4 x − 2k 2 k 2 + 1 sin2 x − 3k 4 + 2k 2 − 3
Δ sin x cos x dx =
Δ cos x
48k 4
4
k k2 + 1
+
ln (k cos x + Δ)
16k 5
−8k 4 sin4 x + 2k 2 6k 2 + 1 sin2 x − 6k 2 + 3
2
3
Δ sin x
Δ sin x cos x dx =
48k 4
2
2k − 1
arcsin (k sin x)
+
16k 5
−8k 4 sin4 x + 2k 2 7k 2 + 1 sin2 x − 3k 4 − 8k 2 + 3
4
Δ cos x
Δ sin x cos x dx =
48k 4
6
k
ln (k cos x + Δ)
−
16k 5
8k4 sin4 x − 2k 2 12k 2 + 1 sin2 x + 24k 4 + 12k 2 − 3
5
Δ sin x
Δ cos x dx =
48k 4
4
2
8k − 4k + 1
arcsin (k sin x)
+
16k 5
2
k
k2
2
2
Δ3 dx =
F (x, F ) + Δ sin x cos x
1 + k E (x, k) −
3
3
3
4
2
2
2
3k
2k sin x + 3k − 5
Δ cos x −
ln (k cos x + Δ)
Δ3 sin x dx =
8
8k
3
−2k 2 sin2 x + 5
Δ3 cos x dx =
Δ sin x +
arcsin (k sin x)
8
8k
2
k 3 − 4k2
3k2 sin2 x + 4k 2 − 6
2
3
Δ sin x dx =
Δ sin x cos x +
F (x, k)
15
15k 2
4
2
8k − 13k + 3
E (x, k)
−
15k 2
Δ5
Δ3 sin x cos x dx = − 2
5k
2
2
k k2 + 3
−3k sin2 x + k 2 + 5
3
2
Δ sin x cos x −
Δ cos x dx =
F (x, k)
15
15k 2
4
2
2k − 7k − 3
E (x, k)
−
15k 2
8k4 sin4 x + 2k 2 5k 2 − 7 sin2 x + 15k 4 − 22k 2 + 3
3
3
Δ cos x
Δ sin x dx =
48k 2
6
4
2
5k − 9k + 3k + 1
ln (k cos x + Δ)
−
16k 3
−8k 4 sin4 x + 14k 2 sin2 x − 3
Δ3 sin2 x cos x dx =
Δ sin x
48k 2
1
+
arcsin (k sin x)
16k 3
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
29.
3
2
187
188
Trigonometric Functions
−8k 4 sin4 x + 2k 2 k 2 + 7 sin2 x + 3k 4 − 8k2 − 3
Δ sin x cos x dx =
48k 2
6
k
×Δ cos x +
ln (k cos x + Δ)
16k 3
8k4 sin4 x − 2k 2 6k 2 + 7 sin2 x + 30k 2 + 3
3
3
Δ sin x
Δ cos x dx =
48k 2
2
6k − 1
arcsin (k sin x)
+
16k 3
1 Δ + cos x
Δ dx
= − ln
+ k ln k (k cos x + Δ)
sin x
2 Δ − cos x
Δ dx
k Δ + k sin x
=
ln
+ k arcsin (k sin x)
cos x
2
Δ − k sin x
Δ dx
2
= k F (x, k) − E (x, k) − Δ cot x
sin2 x
1 1 − Δ k Δ + k
Δ dx
= ln
+ ln
sin x cos x
2 1+Δ
2
Δ − k
Δ dx
= F (x, k) − E (x, k) + Δ tan x
cos2 x
sin x
k Δ + k
Δ dx = Δ tan x dx = −Δ + ln
cos x
2
Δ − k
1 1−Δ
cos x
Δ dx = Δ cot x dx = Δ + ln
sin x
2 1+Δ
2
Δ + cos x
Δ dx
Δ cos x k
ln
=−
+
3
2
4
Δ
− cos x
sin x
2 sin x
1 + k 2 Δ − k sin x
Δ dx
−Δ
−
ln
=
2
sin x
2k Δ + k sin x
sin x cos x
Δ
1 Δ + cos x
Δ dx
=
+ ln
2
sin x cos x
cos x 2 Δ − cos x
Δ dx
Δ sin x
1
Δ + k sin x
=
+ ln
3
2
cos x
2 cos x 4k
Δ − k sin x
Δ
Δ sin x dx
=
− k ln (k cos x + Δ)
cos2 x
cos x
Δ cos x dx
Δ
− k arcsin (k sin x)
=−
sin x
sin2 x
Δ sin2 x dx
Δ sin x 2k2 − 1
k Δ + k sin x
=−
+
arcsin (k sin x) + ln
cos x
2
2k
2
Δ − k sin x
Δ cos x k 2 + 1
1 Δ + cos x
Δ cos2 x dx
=
+
ln (k cos x + Δ) + ln
sin x
2
2k
2 Δ − cos x
Δ dx
1
2
=
−Δ cot3 x + k 2 − 3 Δ cot x + 2k F (x, k) + k 2 − 2 E (x, k)
4
3
sin x
30.
31.
32.
33.
34.
35.
36.
37.
38.
39.
40.
41.
42.
43.
44.
45.
46.
47.
3
2
2.583
2.583
Elliptic and pseudo-elliptic integrals
48.
49.
51.
52.
53.
54.
55.
56.
57.
58.
59.
60.
61.
62.
63.
64.
65.
66.
Δ dx
=
sin x cos2 x
2
50.
k2 − 2 1 + Δ
Δ dx
Δ
k Δ + k
ln
ln
+
=
−
+
2
Δ − k
4
1−Δ
sin3 x cos x
2 sin2 x
1
1 + k
tan
x
−
cot
x
Δ
+
2
F
(x,
k)
−
E (x, k)
2
k
k 2
2
Δ
1 1 + Δ 2 − k2 Δ + k
Δ dx
=
− ln
+
ln
3
2
sin x cos x
2 cos x 2 1 − Δ
4k Δ − k
+
,
2
2
1
Δ dx
2
2
2
k
=
tan
x
−
2k
−
3
tan
x
Δ
+
2k
F
(x,
k)
+
k
−
2
E
(x,
k)
cos4 x
3k 2
Δ
k2
Δ + k
sin x
Δ
dx
=
+
ln
cos3 x
2 cos2 x 4k Δ − k cos x
Δ
k2 1 + Δ
ln
Δ
dx
=
−
+
4
1−Δ
sin3 x
2 sin2 x
2
sin x
Δ dx = tan2 xΔ dx = Δ tan x + F (x, k) − 2 E (x, k)
cos2 x
cos2 x
2
cot2 xΔ dx = −Δ cot x + k F (x, k) − 2 E (x, k)
2 Δ dx =
sin x
sin3 x
k 2 sin2 x + 3k 2 − 1
k Δ + k
Δ dx = −
Δ + ln
2
cos x
3k
2
Δ − k
k 2 sin2 x − 3k 2 − 1
cos3 x
1 1−Δ
Δ dx = −
Δ + ln
sin x
3k 2
2 1+Δ
2
2
2
k − 3 sin x + 2
k k2 + 3
Δ dx
Δ + cos x
ln
=
cos xΔ +
16
Δ − cos x
sin5 x
8 sin4 x
2
2
3 − k sin x + 1
Δ dx
k Δ − k sin x
=−
Δ − ln
4
3
2
Δ + k sin x
sin x cos x
3 sin x
Δ dx
3 sin2 x − 1
k 2 − 3 Δ − cos x
ln
=
Δ
+
4
Δ + cos x
sin3 x cos2 x
2 sin2 x cos x
2
2k2 − 3 Δ + k sin x
Δ dx
3 sin x − 2
Δ−
ln
=
2
2
3
2 sin x cos x
4k Δ − k sin x
sin x cos x
2
2
2k − 3 sin x − 3k 2 + 4
Δ dx
1 Δ + cos x
=
Δ + ln
2
4
3
sin x cos x
2 Δ − cos x
3k cos x
2
2
2
2k − 3 sin x − 4k + 5
Δ dx
4k 2 − 3 Δ + k sin x
=
ln
sin
xΔ
−
2
cos5 x
Δ − k sin x
8k cos4 x
16k 3
2
2
2
4
2
− 2k + 1 k sin x + 3k − k + 1
sin x
Δ dx =
Δ
4
cos x
3k 2 cos3 x
cos x
Δ3
4 Δ dx = −
sin x
3 sin3 x
sin x
2k 2 − 1 Δ + k sin x
sin2 x
Δ
dx
=
Δ
+
− k arcsin (k sin x)
ln
cos3 x
2 cos2 x
4k Δ − k sin x
189
190
Trigonometric Functions
67.
68.
69.
70.
71.
2.584
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
cos2 x
cos x
k2 + 1 Δ + cos x
ln
− k ln (k cos x + Δ)
Δ
dx
=
−
Δ
−
4
Δ − cos x
sin3 x
2 sin2 x
sin3 x
sin2 x − 3
3k2 − 1
Δ dx = −
Δ−
ln (k cos x + Δ)
2
cos x
2 cos x
2k
cos3 x
2k2 + 1
sin2 x + 2
Δ
−
arcsin (k sin x)
Δ
dx
=
−
2 sin x
2k
sin2 x
sin4 x
2k 2 sin2 x + 4k 2 − 1
Δ dx = −
Δ sin x
cos x
8k 2
4
2
k Δ + k sin x
8k − 4k − 1
ln
arcsin
(k
sin
x)
+
+
8k 3
2
Δ − k sin x
−2k 2 sin2 x + 5k 2 + 1
cos4 x
Δ dx =
Δ cos x
sin x
8k 2
4
1 Δ + cos x 3k + 6k 2 − 1
+
ln (k cos x + Δ)
+ ln
2 Δ − cos x
8k 3
dx
= F (x, k)
Δ
1
Δ − k cos x
1
sin x dx
=
ln
= − ln (k cos x + Δ)
Δ
2k Δ + k cos x
k
1
1
k sin x
cos x dx
= arcsin (k sin x) = arctan
Δ
k
k
Δ
2
1
1
sin x dx
= 2 F (x, k) − 2 E (x, k)
Δ
k
k
Δ
sin x cos x dx
=− 2
Δ
k
2
1
cos2 x dx
k
= 2 E (x, k) − 2 F (x, k)
Δ
k
k
3
sin x dx
cos xΔ 1 + k 2
=
−
ln (k cos x + Δ)
Δ
2k 2
2k 3
sin2 x cos x dx
sin xΔ arcsin (k sin x)
=−
+
Δ
2k 2
2k 3
2
cos xΔ
sin x cos2 x dx
k
=−
+
ln (k cos x + Δ)
Δ
2k 2
2k 3
cos3 x dx
sin xΔ 2k 2 − 1
=
+
arcsin (k sin x)
Δ
2k 2
2k 3
2 1 + k2
sin x cos xΔ 2 + k2
sin4 x dx
=
+
F (x, k) −
E (x, k)
Δ
3k 2
3k 4
3k 4
sin3 x cos x dx
1 = − 4 2 + k 2 sin2 x Δ
Δ
3k
2.584
2.584
Elliptic and pseudo-elliptic integrals
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
sin x cos xΔ 2 − k 2
sin2 x cos2 x dx
2k 2 − 2
=−
+
E
(x,
k)
+
F (x, k)
Δ
3k 2
3k 4
3k 4
1
sin x cos3 x dx
2
= − 4 k 2 cos2 x − 2k Δ
Δ
3k
cos4 x dx
sin x cos xΔ 4k 2 − 2
3k 4 − 5k 2 + 2
=
+
E
(x,
k)
+
F (x, k)
Δ
3k 2
3k 4
3k 4
sin5 x dx
2k 2 sin2 x + 3k 2 + 3
3 + 2k 2 + 3k 4
=
cos xΔ −
ln (k cos x + Δ)
4
Δ
8k
8k 5
sin4 x cos x dx
2k 2 sin2 x + 3
3
=−
sin xΔ + 5 arcsin (k sin x)
Δ
8k 4
8k
sin3 x cos x dx
2k 2 cos2 x − k 2 − 3
k 4 + 2k 2 − 3
=
cos xΔ −
ln (k cos x + Δ)
4
Δ
8k
8k 5
sin2 x cos3 x dx
2k 2 cos2 x + 2k 2 − 3
4k 2 − 3
=−
sin
xΔ
+
arcsin (k sin x)
Δ
8k 4
8k 5
sin x cos4 x dx
3 − 5k 2 + 2k 2 sin2 x
3k 4 − 6k 2 + 3
=
cos xΔ −
ln (k cos x + Δ)
4
Δ
8k
8k 5
cos5 x dx
2k 2 cos2 x + 6k 2 − 3
8k 4 − 8k 2 + 3
=
sin
xΔ
+
arcsin (k sin x)
Δ
8k 4
8k 5
sin6 x dx 3k 2 sin2 x + 4k 2 + 4
=
sin x cos xΔ
Δ
15k 4
4
2
8k 4 + 7k 2 + 8
4k + 3k + 8
F
(x,
k)
−
E (x, k)
+
15k 6
15k 6
3k 4 sin4 x + 4k 2 sin2 x + 8
sin5 x cos x dx
=−
Δ
Δ
15k 6
sin4 x cos x dx 3k 2 cos2 x − 2k 2 − 4
=
sin x cos xΔ
Δ
15k 4
4
2
2k 4 + 3k 2 − 8
k + 7k − 8
F (x, k) −
E (x, k)
+
6
15k
15k 6
3k 4 sin4 x − 5k 4 − 4k 2 sin2 x − 10k 2 + 8
sin3 x cos3 x dx
=
Δ
Δ
15k 6
sin2 x cos4 x dx
3k 2 cos2 x + 3k 2 − 4
=−
sin x cos xΔ
Δ
15k 4
4
2
3k 4 − 13k 2 + 8
9k − 17k + 8
F
(x,
k)
−
E (x, k)
+
15k 6
15k 6
2
sin x cos5 x dx
−3k 4 cos4 x + 4k 2 k cos2 x − 8k 4 + 16k 2 − 8
=
Δ
Δ
15k 6
cos6 x dx 3k 2 cos2 x + 8k 2 − 4
=
sin x cos xΔ
Δ
15k 4
6
4
2
23k 4 − 23k 2 + 8
15k − 34k + 27k − 8
F
(x,
k)
+
E (x, k)
+
15k 6
15k 6
191
192
Trigonometric Functions
29.
30.
31.
32.
33.
34.
35.
36.
37.
38.
39.
40.11
41.
42.
sin7 x dx 8k 4 sin4 x + 10k 2 k 2 + 1 sin2 x + 15k 4 + 14k 2 + 15
=
cos xΔ
Δ
4
2 48k
6
2
5k − 2k + 5 k + 1
−
ln (k cos x + Δ)
16k 7
8k 4 sin4 x + 10k 2 sin2 x + 15
sin6 x cos x dx
5
=−
sin xΔ +
arcsin (k sin x)
Δ
48k 6
16k 7
sin5 x cos2 x dx −8k 4 sin4 x + 2k 2 k 2 − 5 sin2 x + 3k 4 + 4k 2 − 15
=
cos xΔ
Δ
48k 6
6
4
2
k + k + 3k − 5
ln (k cos x + Δ)
−
16k 7
sin4 x cos3 x dx 8k 4 sin4 x − 2k 2 6k 2 − 5 sin2 x − 18k 2 + 15
=
sin xΔ
Δ
48k 6
2
6k − 5
arcsin (k sin x)
+
16k 7
sin3 x cos4 x dx 8k 4 sin4 x − 2k 2 6k 2 − 5 sin2 x + 3k 4 − 22k 2 + 15
=
cos xΔ
Δ
48k 6
6
4
2
k + 3k − 9k + 5
ln (k cos x + Δ)
−
16k 7
sin2 x cos5 x dx −8k 4 sin4 x + 2k 2 12k 2 − 5 sin2 x − 24k 4 + 36k 2 − 15
=
sin xΔ
Δ
48k 6
4
2
8k − 12k + 5
arcsin (k sin x)
+
16k 7
sin x cos6 x dx −8k 4 sin4 x + 2k 2 13k 2 − 5 sin2 x − 33k 4 + 40k 2 − 15
=
cos xΔ
Δ
48k 6
6
5k
ln (k cos x + Δ)
+
16k 7
cos7 x dx 8k 4 sin4 x − 2k 2 18k 2 − 5 sin2 x + 72k 4 − 54k 2 + 15
=
sin xΔ
Δ
48k 6
16k 6 − 24k 4 + 18k 2 − 5
arcsin (k sin x)
+
16k 7
dx
1
k2 sin x cos x
= 2 E (x, k) − 2
3
Δ
Δ
k
k
sin x dx
cos x
=− 2
Δ3
k Δ
cos x dx
sin x
=
Δ3
Δ
2
sin x dx
1
1
1 sin x cos x
= 2 2 E (x, k) − 2 F (x, k) − 2
3
Δ
k
Δ
k k
k
sin x cos x dx
1
= 2
Δ3
k Δ
2
1
1
sin x cos x
cos x dx
= 2 F (x, k) − 2 E (x, k) +
Δ3
k
k
Δ
2.584
2.584
Elliptic and pseudo-elliptic integrals
43.
44.
45.
46.
47.
48.
49.
50.
51.
52.9
53.
54.
55.
56.
57.
58.
59.
60.
sin3 x dx
cos x
1
= − 2 2 + 3 ln (k cos x + Δ)
3
Δ
k k Δ k
sin2 x cos x dx
1
sin x
= 2 − 3 arcsin (k sin x)
Δ3
k Δ
k
sin x cos2 x dx
1
cos x
= 2 − 3 ln (k cos x + Δ)
3
Δ
k Δ
k
1
k sin x
cos3 x dx
+ 3 arcsin (k sin x)
=
−
Δ3
k2 Δ
k
2
k + 1
2
sin x cos x
sin4 x dx
=
E (x, k) − 4 F (x, k) −
3
Δ
k
k 2 k4
k2 k 2 Δ
2
sin3 x cos x dx
2 − k 2 sin2 x
=
Δ3
k4 δ
2 − k2
2
sin x cos x
sin2 x cos2 x dx
=
F (x, k) − 4 E (x, k) +
3
Δ
k4
k
k2 Δ
k 2 sin2 x + k2 − 2
sin x cos3 x dx
=
Δ3
k4 Δ
cos4 x dx
k + 1
2k k sin x cos x
=
E (x, k) − 4 F (x, k) −
3
4
Δ
k
k
k2 Δ
2
2
2
k2 + 3
k 2 k sin2 x + k2 − 3
sin5 x dx
cos x +
=
ln (k cos x + Δ)
2
3
4
Δ
2k 5
2k k Δ
2
sin4 x cos x dx
3
−k 2 sin2 x + 3
sin x − 5 arcsin (k sin x)
=
Δ3
2k 4 Δ
2k
sin3 x cos2 x dx
−k 2 sin2 x + 3
k2 − 3
=
cos x +
ln (k cos x + Δ)
4
Δ
2k Δ
2k 5
sin2 x cos3 x dx
2k 2 − 3
k 2 sin2 x + 2k 2 − 3
sin
x
−
=
arcsin (k sin x)
Δ3
2k 4 Δ
2k 5
3k k 2 sin2 x + 2k 2 − 3
sin x cos4 x dx
cos x +
=
ln (k cos x + Δ)
3
4
Δ
2k Δ
2k 5
2
cos5 x dx
4k2 − 3
−k 2 sin2 x + 2k 4 − 4k 2 + 3
sin
x
+
=
arcsin (k sin x)
Δ3
2k 4 Δ
2k 5
2
dx −k 2 sin x cos x 2k k + 1 sin x cos x
1
=
−
− 2 F (x, k)
2
4
5
3
Δ
3k Δ
3k Δ
3k
2
2 k + 1
+
E (x, k)
3k 4
2
sin x dx
2k 2 sin2 x + k 2 − 3
=
cos x
Δ5
3k 4 Δ3
cos x dx
−2k 2 sin2 x + 3
=
sin x
Δ5
3Δ3
193
194
Trigonometric Functions
61.
62.
63.
64.
65.
66.
67.
68.
69.
70.
71.
72.
73.
74.
75.
76.
77.
78.
79.
sin2 x dx k 2 + 1
1
= 4 2 E (x, k) − 2 2 F (x, k)
5
Δ
3k k
3k k
k 2 k 2 + 1 sin2 x − 2
+
sin x cos x
3k 4 Δ3
1
sin x cos x dx
= 2 3
Δ5
3k Δ
k 2 2k 2 − 1 sin2 x − 3k 2 + 2
1
2k 2 − 1
cos2 x dx
= 2 F (x, k) +
E (x, k) +
sin x cos x
Δ5
3k
3k 2 k 2
2k 2 Δ
2
3k − 1 sin2 x − 2
sin3 x
dx =
cos x
Δ5
3k 4 Δ3
sin2 x cos x
sin3 x
dx
=
Δ5
3Δ3
sin x cos2 x
cos3 x
dx = − 2
5
Δ
3k Δ3
2
− 2k + 1 sin2 x + 3
cos3 x dx
=
sin x
Δ5
3Δ3
1 Δ + cos x
dx
= − ln
Δ sin x
2 Δ − cos x
dx
1
Δ − k sin x
= − ln
Δ cos x
2k
Δ + k sin x
dx
1 + cot2 x
dx = F (x, k) − E (x, k) − Δ cot x
=
Δ
Δ sin2 x
dx
dx
1 1−Δ
1
Δ + k
= (tan x + cot x)
= ln
+ ln
Δ sin x cos x
Δ
2 1 + Δ 2k
Δ − k
dx
1
dx
1
=
1 + tan2 x
= F (x, k) − 2 E (x, k) + 2 Δ tan x
2
Δ cos x
Δ
k
k
dx
1
sin x dx
Δ+k
= tan x
= ln
cos x Δ
Δ
2k
Δ − k
dx
1 1−Δ
cos x dx
= cot x
= ln
sin x Δ
Δ
2 1+Δ
dx
Δ cos x 1 + k 2 Δ + cos x
ln
=−
−
3
4
Δ − cos x
Δ sin x
2 sin2 x
Δ
dx
1
Δ − k sin x
=
−
−
ln
sin x 2k Δ + k sin x
Δ sin2 x cos x
Δ
dx
1 Δ − cos x
= 2
+ ln
2
Δ sin x cos x
k cos x 2 Δ + cos x
Δ sin x
dx
2k 2 − 1 Δ − k sin x
= 2
ln
+
3
2
Δ cos x
Δ + k sin x
2k cos x
4k 3
sin x dx
Δ
= 2
cos2 x Δ
k cos x
2.584
2.584
Elliptic and pseudo-elliptic integrals
80.
81.
82.
83.
84.
85.
86.
87.
88.
89.
90.
91.
92.
93.
94.
Δ
cos x dx
=−
sin x
sin2 x Δ
1
Δ + k sin x 1
sin2 x dx
= ln
− arcsin (k sin x)
cos x Δ
2k
Δ − k sin x k
cos2 x dx
1 Δ + cos x 1
= ln
+ ln (k cos x + Δ)
sin x Δ
2 Δ − cos x k
dx
1
−Δ cot3 x − Δ 2k 2 + 3 cot x + k 2 + 2 F (x, k) − 2 k 2 + 1 E (x, k)
=
4
3
Δ sin x
dx
dx
=
tan x + 2 cot x + cot3 x
3
Δ
Δ sin x cos x
Δ
Δ + k
k2 + 2 1 + Δ
1
ln
=−
−
+ ln
2
Δ − k
4
1−Δ
2 sin x 2k
dx
2
dx
=
tan x + 2 + cot2 x
2
2
Δ
Δ sin x cos x
tan x
k2 − 2
=
− cot x Δ +
E (x, k) + 2 F (x, k)
k 2
k 2
dx
dx
=
cot x + 2 tan x + tan3 x
3
Δ sin x cos x
Δ
Δ
1 1 + Δ 2 − 3k 2 Δ + k +
= − 2
ln
− ln
Δ − k
2k cos2 x 2 1 − Δ
4k 3
⎧
1 ⎨
dx
5k2 − 3
=
Δ tan x − 3k 2 − 2 F (x, k)
Δ tan3 x −
2
2
4
Δ cos x 3k ⎩
k
⎫
2
⎬
2 2k − 1
+
E (x, k)
2
⎭
k
dx
Δ
Δ + k
k2
tan x 1 + tan2 x
= 2
ln
−
Δ
2k cos2 x 4k 3 Δ − k Δ
cos x dx
k2 1 + Δ
=−
ln
−
3
2
4
1−Δ
sin x Δ
2 sin x
2
2
tan x
Δ
1
sin x dx
=
dx = 2 tan x − 2 E (x, k)
cos2 x Δ
Δ
k
k
cos2 x dx
cot2 x
=
dx = −Δ cot x − E (x, k)
Δ
sin2 x Δ
sin3 x dx
Δ
1
Δ + k
= 2 + ln
cos x Δ
k
2k
Δ − k
Δ
cos3 x dx
1 1+Δ
= 2 − ln
sin x Δ
k
2 1−Δ
3 1 + k 2 sin2 x + 2
dx
3k 4 + 2k 2 + 3 Δ + cos x
ln
=
−
Δ cos x +
5
2
16
Δ − cos x
Δ sin x
8 sin x
sin x dx
=
cos3 x Δ
195
196
Trigonometric Functions
95.
96.
97.
98.
99.
100.
101.
102.
103.
104.
105.
106.
107.
2.585
1.
2.585
3 + 2k 2 sin2 x + 1
1
Δ − k sin x
dx
=−
Δ − ln
4
3
2k
Δ + k sin x
Δ sin x cos x
3 sin x
2
3 − k 2 sin2 x − k dx
k 2 + 3 Δ − cos x
ln
=
Δ
+
4
Δ + cos x
Δ sin3 x cos2 x
2k 2 sin2 x cos x
2
2
2
2
3 − 2k sin x − 2k
dx
4k − 3 Δ + k sin x
ln
Δ
−
=
Δ − k sin x
Δ sin2 x cos3 x
2k 2 sin x cos2 x
4k 3
2
2
2
5k − 3 sin x − 6k + 4
1 Δ + cos x
dx
Δ − ln
=
4
Δ sin x cos x
2 Δ − cos x
3k 4 cos3 x
3 2k 2 − 1 sin2 x − 8k 2 + 5
dx
8k 4 − 8k 2 + 3 Δ + k sin x
=
ln
Δ
sin
x
+
Δ cos5 x
Δ − k sin x
8k 4 cos4 x
16k 5
sin x dx
2k 2 cos2 x − k
=−
Δ
4
cos x Δ
2k 4 cos3 x
2
2k 2 sin2 x + 1
cos x dx
=
−
Δ
sin4 x Δ
3 sin3 x
Δ sin x
Δ + k sin x
sin2 x dx
1
=
ln
−
2
3
3
2
cos x Δ
Δ − k sin x
2k cos x 4k
Δ cos x k Δ + cos x
cos3 x dx
=
−
ln
+
3
2
4
Δ − cos x
sin x Δ
2 sin x
2
sin3 x dx
Δ
1
= 2
+ ln (k cos x + Δ)
cos2 x Δ
k cos x k
−Δ
1
cos3 x dx
=
− arcsin (k sin x)
sin x k
sin2 x Δ
sin4 x dx
Δ sin x
1
Δ + k sin x 2k 2 + 1
=
−
+
ln
arcsin (k sin x)
cos x Δ
2k 2
2k Δ − k sin x
2k 3
Δ cos x 1 Δ + cos x 3k 2 − 1
cos4 x dx
=
+
+ ln
ln (k cos x + Δ)
sin x Δ
2k 2
2 Δ − cos x
2k 3
(a + sin x)
Δ
=
p+3
dx
1
(p + 2)k2
p
(a + sin x) cos xΔ
p+2
(a + sin x)p+1 dx
dx
(a + sin x)
+ (p + 1) 1 + k 2 − 6a2 k 2
+2(2p + 3)ak2
Δ Δ
(a + b sin x)p dx
2
2 2
−a(2p + 1) 1 + k − 2a k
Δ &
p−1
(a
+
sin
x)
dx
− p 1 − a2 1 − a 2 k 2
Δ
1
p = −2, a = ±1, a = ±
k
For p = n a natural number, this integral can be reduced to the following three integrals:
2.586
Elliptic and pseudo-elliptic integrals
2.
3.
4.6
5.
2.586
1.
2.
3.
197
1
Δ − k cos x
a + sin x
dx = a F (x, k) +
ln
Δ
2k Δ + k cos x
2
1 + k 2 a2
1
a Δ − k cos x
(a + sin x)
dx =
F (x, k) − 2 E (x, k) + ln
Δ
k2
k
k Δ + k cos x
1
1
sin x dx
dx
,
= Π x, 2 , k − 2
(a + sin x) Δ
a
a
a − sin2 x Δ
where
√
√
−1
1 − a2 Δ − 1 − k 2 a2 cos x
sin x dx
= √
ln √
a2 − sin2 x Δ
1 − a2 Δ + 1 − k 2 a2 cos x
2 (1 − a2 ) (1 − a2 k 2 )
%
dx
cos xΔ
1
−
=
n
n−1
(a + sin x) Δ (n − 1) (1 − a2 ) (1 − a2 k 2 )
(a + sin x)
dx
−(2n − 3) 1 + k 2 − 2a2 k 2 a
n−1
(a
+
sin
x)
Δ
2 2
dx
−(n − 2) 6a k − k 2 − 1
n−2
(a + sin x)
Δ
&
dx
dx
2
2
− (10 − 4n)ak
− (n − 3)k
n−3
n−4
(a + sin x)
Δ
(a + sin x)
Δ
1
n = 1, a = ±1, a = ±
k
This integral can be reduced to the integrals:
dx
1
dx
cos xΔ
2
2 2
=
−
a
1
+
k
−
2a
k
−
2
2
2
2
a + sin x
(a + sin x) Δ
(a + sin x) Δ (1 − a ) (1 − a k )
&
2
dx
(a
+
sin
x)
dx
(a
+
sin
x)
+ k2
− 2ak2
Δ
Δ
(see 2.585 2, 3, 4)
⎡
1
dx
dx
⎣ − cos xΔ − 3a 1 + k 2 − 2a2 k 2
=
3
2
2
(a + sin x) Δ 2 (1 − a2 ) (1 − a2 k 2 )
(a + sin x)
(a + sin x) Δ
⎤
2 2
dx
+ 2ak2 F (x, k)⎦
− 6a k − k 2 − 1
(a + sin x) Δ
(see 2.585 4 and 2.586 2)
4.
For a = ±1, we have:
%
dx
dx
1
cos xΔ
2
=
∓
n
n + (n − 1) 1 − 5k
n−1
2
(1 ± sin x) Δ (2n − 1)k
(1 ± sin x)
(1 ± sin x)
Δ
&
dx
dx
− (n − 2)k2
+ 2(2n − 3)k2
n−2
n−3
(1 ± sin x)
Δ
(1 ± sin x)
Δ
GU (241)(6a)
This integral can be reduced to the following integrals:
198
Trigonometric Functions
2.587
5.
6.
∓ cos xΔ
dx
1
= 2
+ F (x, k) − 2 E (x, k)
(1 ± sin x) Δ
k (1 ± sin x)
k
2
1 − 5k 2 cos xΔ
1
dx
k cos xΔ
=
∓
2
2 ∓
4
1 ± sin x
(1 ± sin x) Δ 3k (1 ± sin x)
2
+ 1 − 3k 2 k F (x, k) − 1 − 5k 2 E (x, k)
GU (241)(6c)
GU (241)(6b)
7.
1
For a = ± , we have
k
dx
dx
1
k cos xΔ
2
=
+
(n
−
1)
5
−
k
±
n
n
n−1
(1 ± k sin x) Δ (2n − 1)k 2
(1 ± k sin x)
(1 ± k sin x)& Δ
dx
dx
− 2(2n − 3)
+ (n − 2)
n−2
n−3
(1 ± k sin x)
Δ
(1 ± k sin x)
Δ
GU (241)(7a)
8.
9.
This integral can be reduced to the following integrals:
k cos xΔ
dx
1
= ± 2
+ 2 E (x, k)
(1 ± k sin x) Δ
k (1 ± k sin x) k
%
2
k 5 − k2 cos xΔ
1
dx
kk cos xΔ
=
±
2
2 ±
4
1 ± k sin x
(1 ± k sin x) Δ 3k (1 ± k sin x)
2
− 2k F (x, k) + 5 − k 2 E (x, k)
GU (241)(7b)
GU(241)(7c)
2.587
1.
(b + cos x)
Δ
p+3
dx
1
(p + 2)k2
%
p
(b + cos x) sin xΔ + 2(2p + 3)bk 2
(b + cos x)
Δ
dx
p+1
dx
(b + cos x)
Δ
p
(b + cos x) dx
2
2
2 2
+(2p + 1)b k − k + b k
Δ
&
p−1
dx
(b
+
cos
x)
2
+ p 1 − b2 k + k 2 b2
Δ
p = −2, b = ±1,
2
p+2
−(p + 1) k − k + 6b k
2.
3.
4.
2
2 2
ik
b=
k
For p = n a natural number, this integral can be reduced to the following three integrals:
1
b + cos x
dx = b F (x, k) + arcsin (k sin x)
Δ
k
2
2 2
2
(b + cos x)
b k −k
1
2b
dx =
arcsin (k sin x)
F (x, k) + 2 E (x, k) +
Δ
k2
k
k
b
1
cos x dx
dx
,
= 2
Π x, 2
,k + (b + cos x) Δ
b −1
b −1
1 − b2 − sin2 x Δ
where
2.589
Elliptic and pseudo-elliptic integrals
5.
2.588
1.
2.
√
1 − b2 Δ + k k 2 + k 2 b2 sin x
cos x dx
1
= <
ln √
1 − b2 − sin2 x Δ
1 − b2 Δ − k k 2 + k 2 b2 sin x
2 (1 − b2 ) k 2 + k 2 b2
%
2
dx
1
−k sin xΔ
2
=
n
−1
(b + cos x) Δ (n − 1) (1 − b2 ) k + b2 k 2
(b + cos x)
dx
−(2n − 3) 1 − 2k 2 + 2b2 k 2 b
n−1
(b
+
cos
x)
Δ
2
dx
−(n − 2) 2k − 1 − 6b2 k 2
n−2
(b + cos x)
Δ
&
dx
dx
2
2
− (4n − 10)bk
+ (n − 3)k
n−3
n−4
(b + cos x)
Δ
(b + cos x)
Δ
ik
n = 1, b = ±1, b = ±
k
This integral can be reduced to the following integrals:
⎡
2
1
dx
dx
⎣ −k sin xΔ − 1 − 2k 2 + 2b2 k 2 b
=
2
2
2
2
2
b + cos x
(b + cos x) Δ
(b + cos x) Δ (1 − b ) k + b k
⎤
2
b + cos x
(b + cos x)
dx − k 2
dx⎦
+ 2bk 2
Δ
Δ
3.
⎡
dx
3
(b + cos x) Δ
=
(see 2.587 2, 3, 4)
1
−k sin xΔ
⎣
2
2
2 (1 − b2 ) k + b2 k 2
(b + cos x)
dx
−3b 1 − 2k2 + 2k 2 b2
2
(b + cos x) Δ
⎤
2
dx
− 2bk 2 F (x, k)⎦
− 2k − 1 − 6b2 k 2
(b + cos x) Δ
2
(see 2.588 2 and 2.587 4)
2.589
1.
p+3
(c + tan x)
Δ
⎡
dx
=
p
p+2
1
dx
⎣ (c + tan x) Δ + 2(2n + 3)ck 2 (c + tan x)
2
2
cos x
Δ
(p + 2)k
p+1
dx
(c + tan x)
2
2 2
−(p + 1) 1 + k + 6c k
Δ
p
(c + tan x) dx
2
2 2
+(2p + 1)c 1 + k + 2c k
Δ
⎤
p−1
dx ⎦
(c + tan x)
2
− p 1 + c2 1 + k c 2
Δ
[p = −2]
For p = n a natural number, this integral can be reduced to the following three integrals:
199
200
Trigonometric Functions
2.
3.
4.
5.
2
where
√
1 + c2 k 2 + 1 + c2 Δ
sin x cos x dx
1
= <
√
ln c2 − (1 + c2 ) sin2 x Δ
1 + c2 k 2 − 1 + c2 Δ
2 (1 + c2 ) 1 + c2 k 2
1.
⎡
dx
1
Δ
⎣−
=
n
n−1
(c + tan x) Δ (n − 1) (1 + c2 ) 1 + k 2 c2
(c + tan x)
cos2 x
dx
2
2
+(2n − 3)c 1 + k + 2c2 k n−1
Δ
(c + tan x)
dx
2
2
−(n − 2) 1 + k + 6c2 k n−2
(c + tan x)
Δ
⎤
dx
dx
2
2
⎦
− (n − 3)k + (4n − 10)ck
n−3
n−4
(c + tan x)
Δ
(c + tan x)
Δ
This integral can be reduced to the integrals:
⎡
dx
1
−Δ
⎣
=
2
2 2
2
(c
+
tan
x) cos2 x
(c + tan x) Δ (1 + c ) 1 + k c
dx
2
2
+c 1 + k + 2c2 k (c + tan x) Δ
⎤
2
c
+
tan
x
(c
+
tan
x)
2
2
dx + k dx⎦
− 2ck
Δ
Δ
3.
1
c + tan x
Δ + k
dx = c F (x, k) + ln
Δ
2k
Δ − k
(c + tan x)
1
1
c
Δ + k
dx = 2 tan xΔ + c2 F (x, k) − 2 E (x, k) + ln
Δ
k
Δ − k
k
k
c
1 + c2
dx
1
=
Π
x,
−
F
(x,
k)
+
,k
(c + tan x) Δ 1 + c2
c (1 + c2 )
c2
sin x cos x dx
,
− 2
c − (1 + c2 ) sin2 x Δ
2.591
2.
2.591
⎡
dx
3
(c + tan x) Δ
=
(see 2.589 2, 3, 4)
1
−Δ
⎣
2
2 2
2 (1 +
1+k c
(c + tan x) cos2 x
dx
2
2
+3c 1 + k + 2c2 k 2
(c + tan x) Δ
⎤
dx
2
2
2
+ 2ck F (x, k)⎦
− 1 + k + 6c2 k (c + tan x) Δ
c2 )
(see 2.591 2 and 2.589 4)
2.593
2.592
1.
Elliptic and pseudo-elliptic integrals
201
n
a + sin2 x
dx
Pn =
Δ
The recursion formula
Pn+1 =
n
a + sin2 x sin x cos xΔ + (2n + 2) 1 + k 2 + 3ak2 Pn+1
− (2n + 1) 1 + 2a 1 + k 2 + 3a2 k 2 Pn + 2na(1 + a) 1 + k 2 a Pn−1
1
(2n + 3)k 2
reduces this integral (for n an integer) to the integrals:
2.
P1
3.
P0
4.
P−1 =
5.
6.
(see 2.584 1 and 2.584 4)
(see 2.584 1)
1
dx
1
= Π x, , k
2
a
a
a + sin x Δ
For a = 0
dx
sin2 xΔ
dx
Tn = n
h + g sin2 x Δ
(see 2.584 70)
H (124)a
can be calculated by means of the recursion formula:
1
−g 2 sin x cos xΔ
2
2
Tn−3 =
n−1 + 2(n − 2) g 1 + k + 3hk Tn−2
2
(2n − 5)k 2
h + g sin x
− (2n − 3) g + 2hg 1 + k
2.593
1.
2
2
2 2
+ 3h k
Tn−1 + 2(n − 1)h(g + h) g + hk
2
Tn
n
b + cos2 x
dx
Δ
The recursion formula
⎧
⎨
n
1
b + cos2 x sin x sin xΔ − (2n + 2) 1 − 2k 2 − 3bk 2 Qn+1
Qn+2 =
2
(2n + 3)k ⎩
Qn =
⎫
⎬
+ (2n + 1) k2 + 2b k 2 − k 2 − 3b2 k 2 n− 2nb(1 − b) k 2 − k 2 b Qn−1
⎭
reduces this integral (for n an integer) to the integrals:
2.
Q1
(see 2.584 1 and 2.584 6)
3.
Q0
(see 2.584 1)
202
Trigonometric Functions
4.
5.
2.594
1.
Q−1 =
2.594
1
1
dx
=
Π x, −
,k
(b + cos2 x) Δ
b+1
b+1
For b = 0
dx
cos2 xΔ
(see 2.584 72)
H (123)
n
c + tan2 x dx
Rn =
Δ
The recursion formula
⎧
⎨ c + tan2 xn tan xΔ
1
− (2n + 2) 1 + k 2 − 3ck 2 Rn+1
Rn+2 =
2
2
(2n + 3)k ⎩
cos x
⎫
⎬
,
+
+ (2n − 1) 1 − 2c 1 + k 2 + 3c2 k 2 Rn + 2nc(1 − c) 1 − k 2 c Rn−1
⎭
reduces this integral (for n an integer) to the integrals:
2.
R1
3.
R0
4.
R−1 =
(see 2.584 1 and 2.584 90)
(see 2.584 1)
1
1−c
1
dx
=
F (x, k) +
Π x,
,k
2
c
−
1
c(1
−
c)
c
c + tan x Δ
For c = 0, see 2.582 5.
2.595
Integrals of the type
R sin x, cos x,
<
1 − p2 sin2 x
dx for p2 > 1.
Notation: α = arcsin (p sin x).
Basic formulas
1.
2.
dx
1
1
= F α,
2
2
p
p
1 − p sin x
<
1
1 − p2 sin2 x dx = p E α,
p
3.
−
p2 − 1
F
p
α,
r2 1
dx
1
= Π α, 2 ,
2
p
p p
1 − r2 sin x
1 − p2 sin x
2
To evaluate integrals of the form
<
R sin x, cos x,
1
p
p2 > 1
p2 > 1
p2 > 1
1 − p2 sin2 x
k with p;
(2)
k
2
with 1 − p2 ;
BY (283.03)
BY (283.02)
dx for p2 > 1, we may use formulas
2.583 and 2.584, making the following modiﬁcations in them. We replace
(1)
BY (283.00)
2.597
Elliptic and pseudo-elliptic integrals
1
F α, p1 ;
p
(3)
F (x, k) with
(4)
E (x, k) with p E
α,
1
p
−
p2 − 1
F
p
α,
1
.
p
For example (see 2.584 15):
2.596
⎡
4
2
2 sin2 x
x
dx
1
−
p
−
2
4p
cos
sin
x
cos
x
⎣p E α, 1
+
=
1.10
2
4
2
2
3p
3p
p
1 − p sin x
⎤
1 ⎦ 2 − 5p2 + 3p4 1
1
p2 − 1
F α,
· F α,
−
+
4
p
p
3p
p
p
1
sin x cos x 1 − p2 sin2 x p2 − 1
4p2 − 2
−
F α,
E
=
+
2
3
3p
3p
p
3p3
2.
3.
203
α,
1
p
For example (see 2.583 36):
<
1
1 − p2 sin2 x
1
1
p2 − 1
2 sin2 x +
dx
=
tan
x
F
F
α,
1
−
p
−
p E α,
−
2
cos x
p p
p
p
<
1
1
= p F α,
− E α,
+ tan x 1 − p2 sin2 x
p
p
2
p >1
α,
p2 > 1
1
p
For example (see 2.584 37):
dx
1
sin x cos x
−1
1
p2 − 1
p2
<
F
α,
=
p
E
α,
−
−
2
2
3
p
−
1
p
p
p
1
−
p
2
1
− p2 sin2 x
1 − p2 sin x
2
sin x cos x
1
1
1
p
p
E α,
+ F α,
= 2
− 2
2
p − 1 1 − p2 sin x p
p
p −1
p
2
p >1
<
Integrals of the form R sin x, cos x, 1 + p2 sin2 x dx
1 + p2 sin x
Notation: α = arcsin 1 + p2 sin2 x
Basic formulas
dx
p
1
1.
F α, =
1 + p2
1 + p2
1 + p2 sin2 x
<
sin x cos x
p
2
2
2
2.
1 + p sin x dx = 1 + p E α, − p2 2
1+p
1 + p2 sin2 x
2.597
BY (282.00)
BY (282.03)
204
Trigonometric Functions
1 + p2 sin2 x dx
p
1
2
Π α, r , =
1 + (p2 − r2 p2 − r2 ) sin2 x
1 + p2
1 + p2
p cos x
sin x dx
1
= − arcsin 2
2
p
1 + p2
1 + p sin x
<
cos x dx
1
= ln p sin x + 1 + p2 sin2 x
p
1 + p2 sin2 x
1 + p2 sin2 x − cos x
dx
1
ln
=
2
sin x 1 + p2 sin2 x
1 + p2 sin2 x + cos x
1 + p2 sin2 x + 1 + p2 sin x
dx
1
ln = 2 1 + p2
cos x 1 + p2 sin2 x
1 + p2 sin2 x − 1 + p2 sin x
1 + p2 sin2 x + 1 + p2
1
tan x dx
ln
=
2 1 + p2
1 + p2 sin2 x
1 + p2 sin2 x − 1 + p2
cot x dx
1 1 − 1 + p2 sin2 x
= ln
2 1 + 1 + p2 sin2 x
1 + p2 sin2 x
3.
4.
5.
6.
7.
8.
9.
2.598
BY (282.02)
"
2.598 To calculate integrals of the form R sin x, cos x, 1 + p2 sin2 x dx, we may use formulas
2.583 and 2.584, making the following modiﬁcations in them. We replace
(1)
k2 with −p2 ;
(2)
k with 1 + p2 ;
(3)
F (x, k) with √ 1
(4)
E (x, k) with
(5)
1
k
ln (k cos x + Δ) with
(6)
1
k
arcsin (k sin x) with
2
1+p2
α, √ p
F
;
1+p2
1 + p2 E
α, √ p
1
p
1
p
− p2 √sin x2cos x2 ;
1+p2
1+p sin x
p cos x
arcsin √
;
2
1+p
ln p sin x +
1 + p2 sin2 x .
For example (see 2.584 90):
1.
⎡
<
1
⎣
tan x 1 + p2 sin2 x
=
2)
2
2
(1
+
p
1 + p sin x
⎤
sin
x
cos
x
p
⎦
− 1 + p2 E α, + p2 1 + p2
1 + p2 sin2 x
p
tan x
1
E α, +
= −
1 + p2
1 + p2
1 + p2 sin2 x
tan2 x dx
For example (see 2.584 37):
2.611
Elliptic and pseudo-elliptic integrals
2.
<
dx
1+
p2
1
3 = 1 + p2 E
sin x
2
p
α, 1 + p2
Integrals of the form R sin x, cos x, a2 sin2 x − 1 dx
a cos x
.
Notation: α = arcsin a2 − 1
2.599
205
a2 > 1
Basic formulas:
√
2
a2 − 1
dx
1
a >1
1.
= − F α,
2
2
a
a
a sin x − 1
√
√
2−1
2−1
a
a
1
− a E α,
2.
a2 sin2 x − 1 dx = F α,
a
a
a
2
a >1
√
r 2 a2 − 1
a2 − 1
dx
1
Π α, 2 2
,
3.
=
a (r2 − 1)
a (r − 1)
a
1 − r2 sin2 x
a2 sin2 x − 1
2
a > 1, r2 > 1
2
sin x dx
α
a >1
4.
=−
a
a2 sin2 x − 1
2
cos x dx
1
5.
a >1
= ln a sin x + a2 sin2 x − 1
2
2
a
a sin x − 1
2
dx
cos x
a >1
6.
= − arctan sin x a2 sin2 x − 1
a2 sin2 x − 1
√
a2 − 1 sin x + a2 sin2 x − 1
1
dx
ln √
= √
7.
2 a2 − 2
cos x a2 sin2 x − 1
a2 − 1 sin x − a2 sin2 x − 1
2
a >1
√
a2 − 1 + a2 sin2 x − 1
tan x dx
1
8.
ln √
= √
2 a2 − 1
a2 sin2 x − 1
a2 − 1 − a2 sin2 x − 1
2
a >1
2
1
cot x dx
a >1
= − arcsin
9.
a sin x
a2 sin2 x − 1
2.611 To calculate integrals of the type R sin x, cos x, a2 sin2 x − 1 dx for a2 > 1,
BY (285.00)a
BY (285.06)a
BY (285.02)a
we may use
formulas 2.583 and 2.584. In doing so, we should follow the procedure outlined below:
(1)
In the right members of these formulas, the following functions should be replaced with integrals
equal to them:
206
Trigonometric Functions
2.611
F (x, k)
should be replaced with
E (x, k)
should be replaced with
1
− ln (k cos x + Δ)
k
should be replaced with
1
arcsin (k sin x)
k
should be replaced with
1 Δ − cos x
ln
2 Δ + cos x
should be replaced with
1
Δ + k sin x
ln
2k Δ − k sin x
should be replaced with
1
Δ + k
ln
2k Δ − k should be replaced with
1 1−Δ
ln
2 1+Δ
should be replaced with
(2)
dx
Δ
Δdx
sin x dx
Δ
cos x dx
Δ
dx
Δ sin x
dx
Δ cos x
tan x
dx
Δ
cot x
dx
Δ
2
Then, on both sides of the equations, we should replace Δ with i a2 sin2 x − 1, k with a and k
2
with 1 − a .
(3)
Both sides of theresulting equations should be multiplied by i, as a result of which only real
functions a2 > 1 should appear on both sides of the equations.
(4)
The integrals on the right sides of the equations should be replaced with their values found from
formulas 2.599.
Examples:
1.
We rewrite equation 2.584 4 in the form
sin2 x
dx
1
1
dx = 2
− 2 i a2 sin2 x − 1 dx,
a
i a2 sin2 x − 1
i a2 sin2 x − 1 a
from which we get
√
2−1
a
sin2 x dx
1
dx
1
a2 sin2 x − 1 dx = − E α,
= 2
+
a
a
a
a2 sin2 x − 1
a2 sin2 x − 1
2
a >1
2.
We rewrite equation 2.584 58 as follows:
2a4 a2 − 2 sin2 x − 3a2 − 5 a2
dx
<
<
5 = −
3 sin x cos x
2
a2 sin2 x − 1
a2 sin2 x − 1
i5
3 (1 − a2 ) i3
1
2a2 − 4
dx
−
i a2 sin2 x − 1 dx
−
3 (1 − a2 ) i a2 sin2 x − 1 3 (1 − a2 )2
2.613
Elliptic and pseudo-elliptic integrals
207
from which we obtain
2a4 a2 − 2 sin2 x − 3a2 − 5 a2
1
dx
<
<
=
sin x cos x +
2
5
3
2
3 (1 − a2 ) a
a2 sin2 x − 1
a2 sin2 x − 1
3 (1 − a2 )
√
√
2
2
a2 − 1
a2 − 1
2
− 2a a − 2 E α,
× a − 3 F α,
a
a
2
a >1
3.
We rewrite equation 2.584 71 in the form
dx
cot x dx
tan x dx
=
+
,
2
2
2
2
sin x cos xi a sin x − 1
i a sin x − 1
i a2 sin2 x − 1
from which we obtain
√
a2 − 1 + a2 sin2 x − 1
dx
1
ln √
= √
− arcsin
2 a2 − 1
sin x cos x a2 sin2 x − 1
a2 − 1 − a2 sin2 x − 1
2
a >1
1
a sin x
√
" − k 2 cos2 x dx.
Integrals of the form R
sin x, cos x, 1
π
To ﬁnd integrals of the form R sin x, cos x, 1 − k 2 cos2 x dx, we make the substitution x = −y,
2
which yields
<
R sin x, cos x, 1 − k 2 cos2 x dx = − R cos y, sin y, 1 − k 2 sin2 y dy.
<
The integrals R cos y, sin y, 1 − k 2 sin2 y dy are found from formulas 2.583 and 2.584. As a
2.612
result of the use of these formulas (where it is assumed that the original integral can be reduced only to
integrals of the ﬁrst and second Legendre forms), when we replace the functions F (x, k) and E (x, k) with
the corresponding integrals, we obtain an expression of the form
<
dy
1 − k 2 sin2 y dy
−
B
−g (cos y, sin y) − A 1 − k 2 sin2 y
Returning now to the original variable x, we obtain
dx
R sin x, cos x, 1 − k 2 cos2 x dx = −g (sin x, cos x) − A √
1 − k 2 cos2 x dx
−B
1 − k 2 cos2 x
The integrals appearing in this expression are found from the formulas
sin x
dx
√
1.
= F arcsin √
,k
2
2
1 − k cos x
1 − k2 cos2 x
sin x
k 2 sin x cos x
2.
1 − k 2 cos2 x dx = E arcsin √
,k − √
1 − k 2 cos2 x
1 − k 2 cos2 x
[p > 1].
R sin x, cos x, 1 − p2 cos2 x dx
To ﬁnd integrals of the type R sin x, cos x, 1 − p2 cos2 x dx, where [p > 1], we proceed as in
2.613
Integrals of the form
section 2.612. Here, we use the formulas
208
Trigonometric Functions
2.614
1.
2.
1
1
dx
= − F arcsin (p cos x) ,
2
2
p
p
1 − p cos x
1
p2 − 1
F arcsin (p cos x) ,
1 − p2 cos2 x dx =
p
p
[p > 1]
− pE
arcsin (p cos x) ,
1
p
R sin x, cos x, 1 + p2 cos2 x dx.
To ﬁnd integrals of the type R sin x, cos x, 1 + p2 cos2 x dx, we need to make the substitution
π
x = − y. This yields
2
<
R sin x, cos x, 1 + p2 cos2 x dx = − R cos y, sin y, 1 + p2 sin2 y dy.
<
To calculate the integrals − R cos y, sin y, 1 + p2 sin2 y dy, we need to use ﬁrst what was said
2.614
Integrals of the form
in 2.598 and 2.612 and then, after returning to the variable x, the formulas
1
dx
p
=
1.
F x, 1 + p2 cos2 x
1 + p2
1 + p2
p
2.
1 + p2 cos2 x dx = 1 + p2 E x, 1 + p2
R sin x, cos x, a2 cos2 x − 1 dx
[a > 1] .
To ﬁnd integrals of the type R sin x, cos x, a2 cos2 x − 1 dx, we need to make the substitution
π
x = − y. This yields
2
<
2
2
R sin x, cos x, a cos x − 1 dx = − R cos y, sin y, a2 sin2 y − 1 dy
"
To calculate the integrals − R cos y, sin y, a2 sin2 y − 1 dy, we use what was said in 2.611 and
then, after returning to the variable x, we use the formulas
√
a sin x
a2 − 1
1
dx
√
= F arcsin √
,
1.
a
a
a2 cos2 x − 1
a2 − 1
2.615
2.
Integrals of the form
[a > 1]
√
a sin x
a2 − 1
2
2
a cos x − 1 dx = a E arcsin √
,
a
a2 − 1
√
a sin x
1
a2 − 1
[a > 1]
− F arcsin √
,
a
a
a2 − 1
2.616
Elliptic and pseudo-elliptic integrals
11
2.616
Integrals of the form R sin x, cos x,
1 − p2 sin x
.
Notation: α = arcsin 1 − p2 sin2 x
1.
<
1 − p2 sin2 x,
dx
1
<
F
=
1 − p2
1 − p2 sin2 x 1 − q 2 sin2 x
α,
.
<
1 − q 2 sin2 x
209
dx.
q 2 − p2
1 − p2
+
0 < p2 < q 2 < 1,
0<x≤
π,
2
BY (284.00)
2.
3.
4.
5.
tan2 x dx
tan x 1 − q 2 sin2 x
<
=
(1 − q 2 ) 1 − p2 sin2 x .
1 − p2 sin2 x 1 − q 2 sin2 x
q 2 − p2
1
−
E α,
1 − p2
(1 − q 2 ) 1 − p2
+
π,
0 < p2 < q 2 < 1, 0 < x ≤
BY (284.07)
2
tan4 x dx
<
1 − p2 sin2 x 1 − q 2 sin2 x
.
.
%
&
2 − p2
2 − p2
q
q
1
=
2 2 − p2 − q 2 E α,
− 1 − q 2 F α,
3 ×
2
1 − p2
1 − p2
3 (1 − q 2 ) (1 − p2 ) 2
.
2p2 + q 2 − 3 + sin2 x 4 − 3p2 − 2q 2 + p2 q 2 sin x
1 − q 2 sin2 x
+
2
cos2 x 1 − p2 sin2 x
3 (1 − p2 ) (1 − q 2 )
+
π,
0 < p2 < q 2 < 1, 0 < x ≤
BY (284.07)
2
sin2 x dx
<
3
1 − p2 sin2 x 1 − q 2 sin2 x
.
.
1 − p2
q 2 − p2
q 2 − p2
1
E α,
=
F α,
−
(1 − q 2 ) (q 2 − p2 )
1 − p2
1 − p2
(q 2 − p2 ) 1 − p2
sin x cos x
<
−
1 − p2 sin2 x 1 − q 2 sin2 x
(1 − q 2 )
+
π,
BY (284.06)
0 < p2 < q 2 < 1, 0 < x ≤
2
cos2 x dx
<
3 1 − p2 sin2 x
1 − q 2 sin2 x
.
.
1 − p2
q 2 − p2
q 2 − p2
1 − q2
= 2
E α,
F α,
−
q − p2
1 − p2
1 − p2
(q 2 − p2 ) 1 − p2
,
+
π
0 < p2 < q 2 < 1, 0 < x ≤
BY (284.05)
2
210
Trigonometric Functions
2.617
6.
7.
cos4 x dx
5 1 − p2 sin2 x
1 − q 2 sin2 x
3 ⎡
.
2 2
2 − p2
2 + p2 − 3q 2 1 − q 2
1−p
q
⎣
F α,
=
2
2
1 − p2
3 (q 2 − p2 )
(1 − p2 )
.
⎤ 1 − p2 sin x cos x 1 − q 2 sin2 x
2q 2 − p2 − 1
q 2 − p2 ⎦
<
+2
E α,
+
3
1 − p2
1 − p2
1 − p2 sin2 x
3 (q 2 − p2 )
+
,
π
BY (284.05)
0 < p2 < q 2 < 1, 0 < x ≤
2
.
.
1 − q 2 sin2 x
q 2 − p2
dx
1
E α,
=
1 − p2
1 − p2 sin2 x 1 − p2 sin2 x
1 − p2
+
π,
0 < p2 < q 2 < 1, 0 < x ≤
2
<
BY (284.01)
8.
/
.
0
2
2 − p2
0 1 − p2 sin2 x
1
−
p
q
sin x cos x
q 2 − p2
1
<
dx =
E α,
−
3
2
2
2
2
2
1
−
q
1
−
p
1
−
q
2
1 − q sin x
1 − p sin2 x 1 − q 2 sin2 x
+
π,
.
0 < p2 < q 2 < 1, 0 < x ≤
2
9.
.
dx
2
2
1 + (p r − p2 − r2 ) sin2 x
.
BY (284.04)
1 − p2 sin2 x
q 2 − p2
1
2
Π
α,
r
,
=
1 − p2
1 − q 2 sin2 x
1 − p2
+
π,
0 < p2 < q 2 < 1, 0 < x ≤
.
2
BY (284.02)
.√
2.617
Notation: α = arcsin
1.
b2 + c2 − b sin x − c cos x
√
,
2 b 2 + c2
dx
√
a + b sin x + c cos x
2
F (α, r)
= −
√
a + b 2 + c2 0 < b2 + c2 < a,
.
r=
√
2 b 2 + c2
√
.
a + b 2 + c2
b
b
arcsin √
− π ≤ x < arcsin √
2
2
2
b +c
b + c2
√
2
= −√
F (α, r)
4 2
b + c2
b
a
− arccos − √
0 < |a| < b2 + c2 , arcsin √
2
2
2
b +c
b + c2
BY (294.00)
b
≤ x < arcsin √
2
b + c2
BY (293.00)
2.618
Elliptic and pseudo-elliptic integrals
2.
√
2b
2c √
sin x dx
√
{2 E (α, r) − F (α, r)} + 2
a + b sin x + c cos x
= −<
b + c2
3
4
a + b sin x + c cos x
(b2 + c2 )
b
a
b
− arccos − √
≤ x < arcsin √
0 < |a| < b2 + c2 , arcsin √
b 2 + c2
b 2 + c2
b 2 + c2
BY (293.05)
3.
4.
211
√
(b cos x − c sin x) dx
√
= 2 a + b sin x + c cos x
a + b sin x + c cos x
√ 2
b + c2 + b sin x + c cos x
√
dx
a + b sin x + c cos x
√
<
2 + c2
2
a
−
b
= −2 a + b2 + c2 E (α, r) + F (α, r)
√
a + b 2 + c2
b
b
2
2
0 < b + c < a, arcsin √
− π ≤ x < arcsin √
b 2 + c2
b 2 + c2
√ 4
= −2 2 b2 + c2 E (α, r)
b
a
0 < |a| < b2 + c2 , arcsin √
− arccos − √
2
2
2
b +c
b + c2
BY (294.04)
b
≤ x < arcsin √
2
b + c2
BY (293.01)
5.
√
a + b sin x + c cos x dx
<
= −2 a + b2 + c2 E (α, r)
0 < b2 + c2 < a,
b
b
arcsin √
− π ≤ x < arcsin √
b 2 + c2
b 2 + c2
BY
(294.01)
√ √
2 + c2 − a
√ 2
b
4
√
= −2 2 b2 + c2 E (α, r) +
F (α, r)
4 2
b + c2
−a
b
b
− arccos √
0 < |a| < b2 + c2 , arcsin √
≤ x < arcsin √
b 2 + c2
b 2 + c2
b 2 + c2
BY (293.03)
√
1
R sin ax, cos ax, cos 2ax dx =
R sin t, cos t, 1 − 2 sin2 t dt
a
where the substitution √
t = ax has been used.
2 sin ax
Notation: α = arcsin
√
The integrals R sin ax, cos ax, cos 2ax dx are special cases of the integrals 2.595. for (p = 2).
2.618
Integrals of the form
We give some formulas:
dx
1
1
√
1.
= √ F α, √
a 2
cos 2ax
2
2
cos ax
1
1
√
2.
dx = √ E α, √
cos 2ax
a 2
2
+
π,
0 < ax ≤
4
+
π,
0 < ax ≤
4
212
Trigonometric Functions
3.
√
2
dx
√
E
=
2
a
cos ax cos 2ax
1
α, √
2
−
tan x √
cos 2ax
a
2.619
+
π,
0 < ax ≤
4
√
√
2
ax + 1 sin ax √
6
cos
2
1
1
2 2
dx
√
E α, √
F α, √
=
cos 2ax
4.
−
−
a
3a
3a,cos3 ax
cos4 ax cos 2ax
2
2+
π
0<x≤
4
√
2
√
1
2
1
tan ax dx
1
1
√
E α, √
=
5.
− √ F α, √
− tan ax cos 2ax
a
a
cos 2ax
2
a 2
2
+
π,
0<x≤
2
tan4 ax dx
1
1
sin ax √
√
6.
= √ F α, √
cos 2ax
−
3a cos3 ax
cos 2ax
3a 2
2
+
π,
0 < ax ≤
4
+
1
π,
1
dx
√
= √ Π α, r2 , √
0 < ax ≤
7.
2
4
1 − 2r2 sin ax cos 2ax
a 2
2
√
1
2
1
1
dx
sin 2ax
√
= √ F α, √
E α, √
8.
−
+ √
3
a
a 2
2
2
a cos 2ax
cos 2ax
+
π,
0 < ax ≤
4
+
sin 2ax
sin2 ax dx
1
1
π,
√
= √
9.
− √ E α, √
0 < ax ≤
4
2a cos 2ax a 2
2
cos3 2ax
+
1
1
dx
π,
sin 2ax
√
= √ F α, √
10.
0 < ax ≤
+ √
4
2
3a 2
cos5 2ax
3a cos3 2ax
√
√
1
2
1
1
E α, √
cos 2ax dx =
11.
− √ F α, √
a
2
a 2
2
+
π,
0 < ax ≤
4
√
√
√
cos 2ax
2
1
1
1
dx =
12.
F α, √
+ tan ax cos 2ax
− E α, √
cos2 ax
a
a
2
2
+
π,
0<x≤
4
√
1
2.619 Integrals of the form R sin ax, cos ax, − cos 2ax dx =
R sin x, cos x, 2 sin2 x − 1 dx
a
√
2
cos
ax
Notation: α = arcsin
The integrals R sin x, cos x, 2 sin2 x − 1 dx are special cases of the integrals 2.599 and 2.611
for (a = 2). We give some formulas:
2.621
Elliptic and pseudo-elliptic integrals
213
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
1
dx
1
√
= − √ F α, √
− cos 2ax
a 2
2
2
1
cos ax dx
1
1
√
= √ E α, √
− F α, √
− cos 2ax
2
2
a 2
4
√
1
1
cos ax dx
1
5
1
√
sin 2ax − cos 2ax
= √ 3 F α, √
− E α, √
−
2
12a
− cos 2ax
2
2
3a 2
√
√
1
2
dx
1
√
E α, √
= cot ax − cos 2ax −
2
a
a
sin ax − cos 2ax
2
dx
2
1
1
√
√
√
√
=
F
α,
−
6
E
α,
sin4 ax − cos 2ax 3a 2
2
2
√
1 cos ax 2
6 sin ax + 1 − cos 2ax
+
3a sin3 ax
√
cot2 ax dx
1
1
1
1
√
= √ F α, √
− 2 E α, √
+ cot ax − cos 2ax
a
− cos 2ax
a 2
2
2
dx
1
1
√
= − √ Π α, r2 , √
2
2
(1 − 2r cos ax) − cos 2ax
a 2
2
dx
1
1
1
sin 2ax
√
= √ F α, √
− 2 E α, √
+ √
3
a
− cos 2ax
a 2
2
2
− cos 2ax
2
cos ax dx
1
sin 2ax
1
√
= √
− √ E α, √
3
2a − cos 2ax a 2
2
− cos 2ax
1
1
dx
sin 2ax
√
= − √ F α, √
− √
5
3a 2
2
− cos 2ax
3a − cos3 2ax
√
1
1
1
− cos 2ax dx = √ F α, √
− 2 E α, √
a 2
2
2
2.621
Integrals of the form
Notation: α = arcsin
√
R sin ax, cos ax, sin 2ax dx.
2 sin ax
.
1 + sin ax + cos ax
√
1
dx
2
√
F α, √
=
a
2
sin 2ax
⎡
√
1+i 1
2 ⎣1 + i
sin ax dx
√
Π α,
,√
=
a
2
2
2
sin 2ax
1.
2.
1−i 1
1−i
Π α,
,√
+
2
2
2
3.
BY (287.50)
+F
√ 2
sin ax dx
√
=
F
a
(1 + sin ax + cos ax) sin 2ax
1
α, √
2
1
α, √
2
− 2E
−E
⎤
1 ⎦
α, √
2
1
α, √
2
BY (287.57)
BY (287.54)
214
Trigonometric Functions
4.
5.
6.
√
2
sin ax dx
√
=
a
(1 − sin ax + cos ax) sin 2ax
√
tan ax − E
√
1
2
(1 + cos ax) dx
√
E α, √
=
a
2
(1 + sin ax + cos ax) sin 2ax
√
2
(1 + cos ax) dx
1
√
=
F α, √
a
2
(1 − sin ax + cos ax) sin 2ax
7.
8.
√
2
(1 − sin ax + cos ax) dx
√
=
a
(1 + sin ax + cos ax) sin 2ax
2E
2.631
1
α, √
2
+
π,
ax =
2
BY (287.55)
BY (287.51)
√
1
− E α, √
+ tan ax
2
+
π,
ax =
2
1
1
− F α, √
α, √
2
2
BY (287.56)
.
BY (287.52)
√
2
(1 + sin ax + cos ax) dx
1
√
Π α, r2 , √
=
2
a
2
[1 + cos ax + (1 − 2r ) sin ax] sin 2ax
BY (287.53)
2.63–2.65 Products of trigonometric functions and powers
2.631
⎡
xr sinp x cosq x dx =
1.
1
⎣(p + q)xr sinp+1 x cosq−1 x
(p + q)2
p
r−1
q
+rx
sin x cos x − r(r − 1) xr−2 sinp x cosq x dx
⎤
− rp xr−1 sinp−1 x cosq−1 x dx + (q − 1)(p + q) xr sinp x cosq−2 x dx⎦
⎡
1
⎣ − (p + q)xr sinp−1 x cosq+1 x
=
(p + q)2
+rxr−1 sinp x cosq x − r(r − 1) xr−2 sinp x cosq x dx
+ rq
r−1
x
p−1
sin
x cos
q−1
⎤
p−2
r
q
x dx + (p − 1)(p + q) x sin
x cos x dx⎦
GU (331)(1)
xm sinn x dx =
2.
3.
xm cosn x dx =
m−1
n−1
sin
x
{m sin x − nx cos x}
2
n
n−1
m(m − 1)
+
xm−2 sinn x dx
xm sinn−2 x dx −
n
n2
x
xm−1 cosn−1 x
{m cos x + nx sin x}
n2
n−1
m(m − 1)
m
n−2
x cos
+
x dx −
xm−2 cosn x dx
n
n2
2.633
Trigonometric functions and powers
xn sin2m x dx =
4.
2m
xn+1
m 22m (n + 1)
m−1
(−1)m (−1)k
+ 2m−1
2
k=0
2m
k
215
xn cos(2m − 2k)x dx
(see 2.633 2)
xn sin2m+1 x dx =
5.
(−1)
22m
m
(−1)k
k=0
2m + 1
k
TI 333
xn sin(2m − 2k + 1)x dx
(see 2.633 1)
xn cos2m x dx =
6.
m
TI 333
n+1
2m
x
m 22m (n + 1)
m−1
2m 1
+ 2m−1
xn cos(2m − 2k)x dx
k
2
k=0
(see 2.633 2)
xn cos2m+1 x dx =
7.
m
1 22m
k=0
2m + 1
k
TI 333
xn cos(2m − 2k + 1)x dx
(see 2.633 2)
2.632
xμ−1 sin βx dx =
1.
i
i
−μ
−μ
(iβ) γ(μ, iβx) − (−iβ) γ (μ, −iβx)
2
2
xμ−1 sin ax dx = −
2.
xμ−1 cos βx dx =
3.
4.
2.633
xn sin ax dx = −
1.
2.
1
2aμ
1
−μ
−μ
(iβ) γ(μ, iβx) + (−iβ) γ (μ, −iβx)
2
n
xn cos ax dx =
n
2n
x
n
k
k!
k=0
3.
n
k
k!
k=0
8
[Re μ > −1, x > 0]
ET I 317(2)
πi
πi
(μ − 1) Γ(μ, −iax) + exp
(1 − μ) Γ(μ, iax)
exp
2
2
[Re μ < 1, a > 0, x > 0] ET I 317(3)
[Re μ > 0, x > 0]
π
π
1
Γ(μ, −iax) + exp −iμ
Γ(μ, iax)
xμ−1 cos ax dx = − μ exp iμ
2a
2
2
TI 333
sin x dx = (2n)!
n
k=0
ET I 319(22)
ET I 319(23)
xn−k
1
cos ax + kπ
ak+1
2
TI (487)
xn−k
1
sin ax + kπ
ak+1
2
k+1
(−1)
TI (486)
n−1
x2n−2k
x2n−2k−1
k
cos x +
sin x
(−1)
(2n − 2k)!
(2n − 2k − 1)!
k=0
216
Trigonometric Functions
n
2n−2k
x2n−2k+1
k x
cos x +
sin x
x
sin x dx = (2n + 1)!
(−1)
(−1)
(2n − 2k + 1)!
(2n − 2k)!
k=0
k=0
n
n−1
2n−2k
2n−2k−1
x
x
x2n cos x dx = (2n)!
sin x +
cos x
(−1)k
(−1)k
(2n − 2k)!
(2n − 2k − 1)!
k=0
k=0
n
n
x2n−2k+1
x2n−2k
2n+1
k
x
sin x +
cos x
cos x dx = (2n + 1)!
(−1)
(2n − 2k + 1)!
(2n − 2k)!
4.
5.
6.
2.634
2n+1
n
k+1
k=0
k=0
2.634
1.
n/2
(2k)
Pn (x) sin mx
cos mx Pn (x) sin mx dx = −
(−1)k
+
m
m2k
m
k=0
2.
sin mx m
n/2
Pn (x) cos mx dx =
k=0
(2k)
Pn (x)
(−1)k
m2k
+
cos mx
m
(n+1)/2
(2k−1)
(−1)k−1
k=1
(n+1)/2
k=1
Pn
(x)
m2k−1
(2k−1)
(−1)k−1
Pn
(x)
m2k−1
(k)
In formulas 2.634, Pn (x) is any nth -degree polynomial, and Pn (x) is its k th derivative with respect to x.
2.635
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
Notation: z1 = a + bx.
1
b
z1 sin kx dx = − z1 cos kx + 2 sin kx
k
k
1
b
z1 cos kx dx = z1 sin kx + 2 cos kx
k
k
1 2b2
2bz1
z12 sin kx dx =
− z12 cos kx + 2 sin kx
k k2
k
1
2b2
2bz1
z12 cos kx dx =
z12 − 2 sin kx + 2 cos kx
k
k
k
2
z1 6b
2b2
3b
2
2
−
z
−
z13 sin kx dx =
cos
kx
+
z
sin kx
1
1
k
k2
k2
k2
z1
6b2
2b2
3b
z13 cos kx dx =
z12 − 2 sin kx + 2 z12 − 2 cos kx
k
k
k
k
2
4
1
12b
24b
6b2
4bz1
cos kx + 2 z12 − 2 sin kx
z14 sin kx dx = −
z14 − 2 z12 + 4
k
k
k
k
k
2
4
2
1
12b
24b
6b
4bz1
sin kx + 2 z12 − 2 cos kx
z14 cos kx dx =
z14 − 2 z12 + 4
k
k
k
k
k
5b
12b2
24b4
20b2
120b4
z1
sin kx −
cos kx
z15 sin kx dx = 2 z14 − 2 z12 + 4
z14 − 2 z12 +
k
k
k
k
k
k4
5b
12b2
24b4
20b2
120b4
z1
cos kx +
sin kx
z15 cos kx dx = 2 z14 − 2 z12 + 4
z14 − 2 z12 +
k
k
k
k
k
k4
2.637
Trigonometric functions and powers
6bz1
20b2 2 120b4
4
−
z +
z
sin kx
1
k2
k2 1
k4
30b2
360b4 2 720b6
1
z −
−
z16 − 2 z14 +
k
k
k4 1
k6
z16 sin kx dx =
11.
2.636
xn+1
2(n +
⎧1)
n/2
n! ⎨ (−1)k+1 xn−2k
sin 2x +
+
4 ⎩
22k (n − 2k)!
xn sin2 x dx =
1.
k=0
xn cos2 x dx =
xn+1
2(n +
⎧1)
n/2
n! ⎨ (−1)k+1 xn−2k
sin 2x +
−
4 ⎩
22k (n − 2k)!
k=0
x sin2 x dx =
x cos2 x dx =
5.
2.637
1.11
x
1
x3
− cos 2x −
6
4
4
(n−1)/2
k=0
⎫
⎬
(−1)k+1 xn−2k−1
cos
2x
⎭
22k+1 (n − 2k − 1)!
x2 −
1
2
sin 2x
MZ 241
1
2
sin 2x
MZ 245
x
1
x2
+ sin 2x + cos 2x
4
4
8
x2 cos2 x dx =
6.
k=0
⎫
⎬
(−1)k+1 xn−2k−1
cos
2x
⎭
22k+1 (n − 2k − 1)!
x
1
x2
− sin 2x − cos 2x
4
4
8
x2 sin2 x dx =
4.
(n−1)/2
GU (333)(3e)
3.
sin kx
GU (333)(2e)
2.
cos kx
6bz1
20b2 2 120b4
4
−
z +
z
cos kx
1
k2
k2 1
k4
30b2
360b4 2 720b6
1
z −
+
z16 − 2 z14 +
k
k
k4 1
k6
z16 cos kx dx =
12.
217
x
1
x3
+ cos 2x +
6
4
4
x2 −
⎧
n/2
⎨
(−1)k xn−2k
n!
3
n
x sin x dx =
4 ⎩
(n − 2k)!
k=0
(n−1)/2
−
k=0
cos 3x
− 3 cos x
32k+1
xn−2k−1
(−1)k
(n − 2k − 1)!
⎫
⎬
sin 3x
−
3
sin
x
⎭
32k+2
GU(333)(2f)
218
Trigonometric Functions
2.
⎧
n/2
n! ⎨ (−1)k xn−2k
xn cos3 x dx =
4 ⎩
(n − 2k)!
k=0
[(n−1)/2]
+
(−1)k
k=0
x sin3 x dx =
x cos3 x dx =
5.
2.638
1.
2.
3.6
4.6
cos 3x
+ 3 cos x
⎭
32k+2
3 2 3
x +
4
2
cos x +
x2
1
+
12 54
x
3
sin 3x
cos 3x + x sin x −
2
18
MZ 241
1
3
x
3
cos x +
cos 3x + x sin x +
sin 3x
4
36
4
12
x2 cos3 x dx =
6.
x
(n − 2k − 1)!
⎫
⎬
1
3
x
3
sin x −
sin 3x − x cos x +
cos 3x
4
36
4
12
x2 sin3 x dx = −
4.
n−2k−1
GU(333)(3f)
3.
sin 3x
+ 3 sin x
32k+1
2.638
3 2 3
x −
4
2
sin x +
1
x2
−
12 54
x
3
cos 3x
sin 3x + x cos x +
2
18
sinq x
sinq−1 x [(p − 2) sin x + qx cos x]
dx
=
−
xp
(p − 1)(p − 2)xp−1
q2
q(q − 1)
sinq x dx
sinq−2 x dx
−
+
p−2
(p − 1)(p − 2)
x
(p − 1)(p − 2)
xp−2
[p = 1, p = 2]
cosq x
cosq−1 x [(p − 2) cos x − qx sin x]
dx = −
p
p−1
x
(p − 1)(p
− 2)x
2
q
q
q(q − 1)
cos x dx
cosq−2 x dx
−
+
p−2
(p − 1)(p − 2)
x
(p − 1)(p − 2)
xp−2
[p = 1, p = 2]
cos x dx
sin x dx
sin x
1
=−
+
p
p−1
x
(p − 1)x
p−1
xp−1
sin x
cos x
1
sin x dx
=−
−
−
p−1
p−2
(p − 1)x
(p − 1)(p − 2)x
(p − 1)(p − 2)
xp−2
(p > 2)
cos x
1
cos x dx
sin x dx
=−
−
xp
(p − 1)xp−1
p−1
xp−1
cos x
sin x
1
cos x dx
=−
+
−
(p − 1)xp−1
(p − 1)(p − 2)xp−2
(p − 1)(p − 2)
xp−2
(p > 2)
MZ 245, 246
TI (496)
TI (495)
TI (492)
TI (491)
2.641
Trigonometric functions and powers
2.639
1.
219
⎧
n−2
sin x dx
(−1)n+1 ⎨ (−1)k (2k + 1)!
=
cos x
x2n
x(2n − 1)! ⎩
x2k+1
k=0
⎫
n−1
⎬ (−1)n+1
(−1)k+1 (2k)!
ci(x)
+
sin
x
+
⎭ (2n − 1)!
x2k
k=0
2.
⎧
⎨ n−1
(−1)k+1 (2k)!
sin x
(−1)n+1
dx
=
x2n+1
x(2n)! ⎩
+
n−1
k=0
3.
cos x dx
(−1)n+1
dx
=
x2n
x(2n − 1)! ⎩
n−2
k=0
4.
cos x
⎫
⎬ (−1)n
(−1)k+1 (2k + 1)!
si(x)
sin
x
+
⎭
x2k+1
(2n)!
k=0
⎧
⎨ n−1
(−1)k+1 (2k)!
−
x2k
GU (333)(6b)a
x2k
GU (333)(6b)a
cos x
⎫
⎬
(−1)k (2k + 1)!
(−1)n
si(x)
sin
x
+
⎭ (2n − 1)!
x2k+1
k=0
GU (333)(7b)
⎧
⎨ n−1
(−1)k+1 (2k + 1)!
cos x dx (−1)n+1
=
x2n+1
x(2n)! ⎩
cos x
x2k+1
⎫
n−1
⎬ (−1)n
(−1)k+1 (2k)!
−
ci(x)
sin
x
+
⎭
x2k
(2n)!
k=0
k=0
GU (333)(7b)
2.641
1.
2.
3.
4.
5.
1
ka
sin kx
ka
dx =
si(u) − sin
ci(u)
cos
a + bx
b
b
b
cos kx
1
ka
ka
dx =
ci(u) + sin
si(u)
cos
a + bx
b
b
b
sin kx
k cos kx
1 sin kx
+
dx
dx = −
(a + bx)2
b a + bx
b a + bx
k sin kx
1 cos kx
cos kx
−
dx
dx = −
(a + bx)2
b a + bx
b a + bx
sin kx
k2
sin kx
k cos kx
−
dx
=
−
−
(a + bx)3
2b(a + bx)2
2b2 (a + bx) 2b2
k
u = (a + bx)
b
k
u = (a + bx)
b
(see 2.641 2)
(see 2.641 1)
sin kx
dx
a + bx
(see 2.641 1)
220
Trigonometric Functions
6.
7.
8.
9.
10.
11.
12.
2.642
1.
2.
k2
cos kx
k sin kx
cos kx
−
dx
=
−
+
(a + bx)3
2b(a + bx)2
2b2 (a + bx) 2b2
4.
cos kx
dx
a + bx
(see 2.641 2)
sin kx
sin kx
k cos kx
dx = −
− 2
(a + bx)4
3b(a + bx)3
6b (a
+ bx)2
2
3 k
k sin kx
cos kx
−
dx
+ 2
6b (a + bx) 6b3 a + bx
(see 2.641 2)
k3
cos kx
cos kx
k sin kx
k cos kx
sin kx
+ 3
dx
dx = −
+ 2
+ 3
4
3
2
(a + bx)
3b(a + bx)
6b (a + bx)
6b (a + bx) 6b
a + bx
(see 2.641 1)
2
sin kx
k cos kx
sin kx
dx = −
−
5
4
(a + bx)
4b(a + bx)
12b2 (a + bx)3
k 2 sin kx
k 3 cos kx k4
sin kx
dx
+
+
3
2
4
4
24b (a + bx)
24b (a + bx) 24b
a + bx
(see 2.641 1)
cos kx
k sin kx
cos kx
dx = −
+
5
4
(a + bx)
4b(a + bx)
12b2 (a + bx)3
2
k4
k cos kx
k 3 sin kx
cos kx
+
dx
+
−
24b3 (a + bx)2
24b4 (a + bx) 24b4 a + bx
(see 2.641 2)
sin kx
sin kx
k cos kx
k 2 sin kx
k 3 cos kx
dx
=
−
−
+
+
(a + bx)6
5b(a + bx)5
20b2 (a +bx)4
60b3 (a + bx)3
120b4 (a + bx)2
k5
k 4 sin kx
cos kx
+
dx
−
120b5 (a + bx) 120b5 a + bx
(see 2.641 2)
cos kx
cos kx
k sin kx
k 2 cos kx
dx = −
+
+
6
5
2
4
(a + bx)
5b(a + bx)
20b (a + bx)
60b3 (a + bx)3
3
4
k5
k sin kx
k cos kx
sin kx
−
dx
−
−
4
2
5
120b (a + bx)
120b (a + bx) 120b5 a + bx
(see 2.641 1)
sin2m x
dx =
x
2m
m
m−1
ln x
(−1)m +
(−1)k
22m
22m−1
k=0
m
sin2m+1 x
(−1)m dx = 2m
(−1)k
x
2
k=0
3.
2.642
cos2m x
dx =
x
2m
m
2m + 1
k
m−1
ln x
1
+
2m
2m−1
2
2
k=0
m
1 cos2m+1 x
dx = 2m
x
2
k=0
2m + 1
k
2m
k
ci[(2m − 2k)x]
si[(2m − 2k + 1)x]
2m
k
ci[(2m − 2k)x]
ci[(2m − 2k + 1)x]
2.643
Trigonometric functions and powers
5.
6.
8.
2.643
3.4
2m
1
2m
m 2 x
m−1
(−1)m (−1)k+1
+ 2m−1
2
k=0
2m+1
sin
x
2m
k
cos(2m − 2k)x
+ (2m − 2k) si[(2m − 2k)x]
x
2m + 1
(−1)m (−1)k+1
2m
k
2
k=0
sin(2m − 2k + 1)x
×
− (2m − 2k + 1) ci[(2m − 2k + 1)x]
x
m
dx =
cos2m x
dx = −
x2
2m
1
2m
m 2 x
m−1
2m
1
− 2m−1
k
2
cos(2m − 2k)x
+ (2m − 2k) si[(2m − 2k)x]
x
k=0
⎧
m
cos2m+1 x
1 2m + 1 ⎨ cos(2m − 2k + 1)x
= − 2m
⎩
k
x2
2
x
k=0
⎫
⎬
+ (2m − 2k + 1) si[(2m − 2k + 1)x]
⎭
1.
2.
sin2m x
dx = −
x2
x2
7.
221
xp dx
xp−1 [p sin x + (q − 2)x cos x] q − 2
=−
+
q
sin x
q−1
(q − 1)(q − 2) sinq−1 x
xp dx
p(p − 1)
+
sinq−2 x (q − 1)(q − 2)
xp−2 dx
sinq−2 x
xp dx
xp−1 [p cos x − (q − 2)x sin x]
=−
q
cos x
(q − 1)(q − 2) cosq−1 x
p−2
p(p − 1)
dx
q−2
xp dx
x
+
+
q − 1 cosq−2 x (q − 1)(q − 2) cosq−2 x
2k−1
∞
−1
xn xn
k+1 2 2
dx =
+
B2k xn+2k
(−1)
sin x
n
(n + 2k)(2k)!
k=1
[|x| < π,
4.
xn
5.8
6.
n−1
n 2
1
−1
dx
= − n − [1 + (−1)n ] (−1) 2
Bn ln x −
sin x
nx
n!
xn dx
=
cos x
∞
k=0
∞
k=1
k= n
2
n > 0]
TU (333)(8b)
2n
2 2 −1
B2k x2k−n
(−1)k
(2k − n) · (2k)!
[n > 1, |x| > π]
,
+
π
|x| < , n > 0
2
|E2k |xn+2k+1
(n + 2k + 1)(2k)!
GU (333)(9b)
GU (333)(10b)
∞
1
|E2k |x2k−n+1
dx
n |En−1 |
=
[1
−
(−1)
ln
x
+
]
n
x cos x
2
(n − 1)!
(2k − n + 1) · (2k)!
k=0
k= n−1
2
+
π,
|x| <
2
GU (333)(11b)
222
Trigonometric Functions
7.
8.
2.644
∞
22k xn+2k−1
n
xn dx
n
n−1
=
−x
x
B2k
cot
x
+
+
n
(−1)k
2
n−1
(n + 2k − 1)(2k)!
sin x
k=1
[|x| < π, n > 1]
GU (333)(8c)
n
xn
n+1
cot x
2 n
n
dx
=− n +
Bn+1 ln x
− [1 − (−1)n ] (−1) 2
2
n+1
x
(n
+
1)x
(n
+ 1)!
sin x
∞
n (−1)k (2x)2k
− n+1
B2k
2
(2k − n − 1)(2k)!
k=1
k= n+1
2
9.
xn dx
= xn tan x + n
cos2 x
10.
xn
∞
k=1
(−1)k
2
2k
22k − 1 xn+2k−1
B2k
(n + 2k − 1) · (2k)!
+
n > 1,
1.
2.
3.
GU (333)(9c)
|x| <
π,
2
GU (333)(10c)
n+1
tan x
2n n n+1
dx
=
2
− [1 − (−1)n ] (−1) 2
− 1 Bn+1 ln x
2
n
cos x
x
(n + 1)!
∞
n (−1)k 22k − 1 (2x)2k
− n+1
B2k
x
(2k − n − 1)(2k)!
k=1
k= n+1
2
2.644
[|x| < π]
+
π,
|x| <
2
GU (333)(11c)
n−1
(2n − 2)(2n − 4) . . . (2n − 2k + 2)
sin x + (2n − 2k)x cos x
x dx
=
−
2n
(2n − 1)(2n − 3) . . . (2n − 2k + 3) (2n − 2k + 1)(2n − 2k) sin2n−2k+1 x
sin x
k=0
n−1
2
(n − 1)!
+
(ln sin x − x cot x)
(2n − 1)!!
n−1
(2n − 1)(2n − 3) . . . (2n − 2k + 1)
x dx
sin x + (2n − 2k − 1)x cos x
=−
2n+1
2n(2n − 2) . . . (2n − 2k + 2) (2n − 2k)(2n − 2k − 1) sin2n−2k x
sin
x
k=0
(2n − 1)!! x dx
+
2n n!
sin x
(see 2.644 5)
n−1
(2n − 2)(2n − 4) . . . (2n − 2k + 2)
x dx
(2n − 2k)x sin x − cos x
=
2n
cos x
(2n − 1)(2n − 3) . . . (2n − 2k + 3) (2n − 2k + 1)(2n − 2k) cos2n−2k+1 x
k=0
n−1
+
4.
2
(n − 1)!
(x tan x + ln cos x)
(2n − 1)!!
n−1
(2n − 1)(2n − 3) . . . (2n − 2k + 1)
x dx
(2n − 2k + 1)x sin x − cos x
=
2n+1
cos
x
2n(2n − 2) . . . (2n − 2k + 2) (2n − 2k)(2n − 2k − 1) cos2n−2k x
k=0
(2n − 1)!! x dx
+
2n n!
cos x
(see 2.644 6)
2.645
Trigonometric functions and powers
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
223
2k−1
∞
−1
x dx
k+1 2 2
=x+
B2k x2k+1
(−1)
sin x
(2k + 1)!
k=1
x dx
=
cos x
∞
k=0
|E2k |x2k+2
(2k + 2)(2k)!
x dx
= −x cot x + ln sin x
sin2 x
x dx
= x tan x + ln cos x
cos2 x
x dx
sin x + x cos x 1
x
dx
(see 2.644 5)
=
−
+
2 sin x
sin3 x
2 sin2 x
x sin x − cos x 1 x dx
x dx
=
+
(see 2.644 6)
cos3 x
2 cos2 x
2 cos x
2
x dx
x cos x
1
2
=−
−
− x cot x + ln (sin x)
3
sin4 x
3 sin3 x 6 sin2 x 3
x sin x
1
2
2
x dx
=
−
+ x tan x − ln (cos x)
cos4 x
3 cos3 x 6 cos2 x 3
3
3 x dx
x dx
x cos x
1
3x cos x
3
+
=−
−
−
−
8 sin x 8 sin x
sin5 x
4 sin4 x 12 sin3 x
8 sin2 x
(see 2.644 5)
x dx
cos x
x sin x
1
3x sin x
3
3
x dx
=
−
+
−
+
5
4
3
2
cos x
4 cos x 12 cos x 8 cos x 8 cos x 8
(see 2.644 6)
2.645
1.
m
x
dx
=
(−1)k
x
k
cosn x
p sin
3.
m
k=0
2.
2m
m
sin2m+1 x
dx =
(−1)k
n
k
cos x
m
xp dx
cosn−2k x
(see 2.643 2)
xp sin x
dx
(see 2.645 3)
cosn−2k x
k=0
xp
p
sin x dx
xp−1
=
−
dx
xp
cosn x
(n − 1) cosn−1 x n − 1 cosn−1 x
xp
[n > 1]
4.
5.
6.
(−1)k
m
k
(see 2.643 2)
GU (333)(12)
xp dx
sinn−2k x
k=0
p
m
cos2m+1 x
x cos x
k m
dx
=
dx
xp
(−1)
n
k
sin x
sinn−2k x
k=0
p−1
cos x
dx
xp
x
p
xp n = −
+
sin x
(n − 1) sinn−1 x n − 1 sinn−1 x
xp
cos2m x
dx =
sinn x
m
(see 2.643 1)
(see 2.645 6)
[n > 1]
(see 2.643 1)
GU (333)(13)
224
Trigonometric Functions
2.646
7.
8.
2.646
x
x cos x
x
+ ln tan
dx = −
sin x
2
sin2 x
x
x π
x sin x
dx =
− ln tan
+
cos2 x
cos x
2
4
p
1.
x tan x dx =
∞
(−1)
k=1
xp cot x dx =
2.
3.
∞
k=0
(−1)k
22k−1 − 1
B2k xp+2k
(p + 2k) · (2k)!
k+1 2
2k
22k B2k
xp+2k
(p + 2k)(2k)!
xp tan2 x dx = x tan x + ln cos x −
x2
2
x cot2 x dx = −x cot x + ln sin x −
x2
2
4.
2.647
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
+
p ≥ −1,
[p ≥ 1,
|x| <
π,
2
|x| < π]
GU (333)(12d)
GU (333)(13d)
xn
n
xn cos x dx
xn−1 dx
=
−
+
m
m−1
(m − 1)b (a + b sin x)m−1
(a + b sin x)
(m − 1)b (a + b sin x)
[m = 1]
xn sin x dx
xn
n
xn−1 dx
=
−
m
m−1
(m − 1)b (a + b cos x)m−1
(a + b cos x)
(m − 1)b (a + b cos x)
[m = 1]
π x
π x
x dx
= −x tan
−
+ 2 ln cos
−
1 + sin x
4
2
4
2
π x
π x
x dx
= x cot
−
−
+ 2 ln sin
1 − sin x
4
2
4
2
x
x
x dx
= x tan + 2 ln cos
1 + cos x
2
2
x
x
x dx
= −x cot + 2 ln cos
1 − cos x
2
2
x π
x cos x
x
2 dx = − 1 + sin x + tan 2 − 4
(1 + sin x)
x π
x
x cos x
2 dx = 1 − sin x + tan 2 + 4
(1 − sin x)
x
x sin x
x
2 dx = 1 + cos x − tan 2
(1 + cos x)
x
x sin x
x
2 dx = − 1 − cos x − cot 2
(1 − cos x)
MZ 247
MZ 247
PE (329)
PE (330)
PE (331)
PE (332)
MZ 247a
2.654
2.648
1.
2.
Trigonometric functions and powers
225
x
x + sin x
dx = x tan
1 + cos x
2
x
x − sin x
dx = −x cot
1 − cos x
2
x2 dx
GU (333)(16)
x sin x + cos x
GU (333)(17)
2 = b [(ax − b) sin x + (a + bx) cos x]
[(ax
−
b)
sin
x
+
(a
+
bx)
cos
x]
tan x
dx
2.651
GU (333)(18)
2 = a [a + (ax + b) tan x]
[a + (ax + b) tan x]
cos(x − t)
x dx
= cosec 2t x ln
− L(x + t) + L(x − t)
2.652
cos(x + t) cos(x − t)
cos(x + t)
!π
!,
+
!
!
t = nπ; |x| < ! − |t0 |! ,
2
where t0 is the
value of the argument t, which is reduced by multiples of the argument π to lie in the
LO III 288
interval − π2 , π2 .
2.653
√
√ sin x
√ dx = 2π S
1.
x
(cf. 8.251 21)
x
√
√ cos x
√ dx = 2π C
2.
x
(cf. 8.251 3)
x
2.649
2.654
1.
2.
3.
4.
5.
6.
7.
8.
√
Notation: Δ = 1 − k 2 sin2 x, k = 1 − k 2 :
xΔ
x sin x cos x
1
dx = − 2 + 2 E (x, k)
Δ
k
k
k2
x sin3 x cos x
2k 2 + 5
1 dx = − 4 F (x, k) +
E (x, k) − 4 3 3 − Δ2 x + k 2 sin x cos x Δ
4
Δ
9k
9k
9k
,
3
2
2
k
x sin x cos x
7k − 5
1 +
2
2
2
3
Δ
x
−
k
dx = − 4 F (x, k) +
E
(x,
k)
−
−
3k
sin
x
cos
x
Δ
Δ
9k
9k 4
9k 4
1
x cos x
x sin x dx
+ 2 arcsin (k sin x)
dx = − 2
Δ3
k Δ
kk
x sin x 1
x cos x dx
+ ln (k cos x + Δ)
=
Δ3
Δ
k
1
x sin x cos x dx
x
= 2 − 2 F (x, k)
3
Δ
k Δ k
3
x sin x cos x dx
1
2 − k 2 sin2 x
− 4 [E (x, k) + F (x, k)]
=x
3
Δ
k4 Δ
k
x sin x cos3 x dx
k2 sin2 x + k 2 − 2 k 2
1
+
=
x
F (x, k) + 4 E (x, k)
3
4
4
Δ
k Δ
k
k
226
Trigonometric Functions
2.655
2.655 Integrals containing sin x2 and cos x2
In integrals containing sin x2 and cos x2 , it is expedient to make the substitution x2 = u.
xp−1
p−1
p
2
2
cos x +
xp−2 cos x2 dx
1.
x sin x dx = −
2
2
xp−1
p−1
2.
xp cos x2 dx =
sin x2 −
xp−2 sin x2 dx
2
2
⎧
r
⎨
xn−4k+3 cos x2
xn−4k+1 sin x2
n
2
k
−
3.
x sin x dx = (n − 1)!!
(−1)
⎩
22k−1 (n − 4k + 3)!! 22k (n − 4k + 1)!!
k=1
⎫
⎬
(−1)r
xn−4r sin x2 dx
+ 2r
⎭
2 (n − 4r − 1)!!
+
: n ;,
GU (336)(4a)
r=
4
⎧
r
⎨
xn−4k+3 sin x2
xn−4k+1 cos x2
+ 2k
4.
xn cos x2 dx = (n − 1)!!
(−1)k−1 2k−1
⎩
2
(n − 4k + 3)!! 2 (n − 4k + 1)!!
k=1
⎫
⎬
r
(−1)
xn−4r cos x2 dx
+ 2r
⎭
2 (n − 4r − 1)!!
+
: n ;,
GU (336)(5a)
r=
4
cos2 x
5.
x sin x2 dx = −
2
sin2 x
6.
x cos x2 dx = −
2
x
1 π
C (x)
7.
x2 sin x2 dx = − cos x2 +
2
2 2
x
1 π
2
2
2
S (x)
8.
x cos x dx = sin x −
2
2 2
x2
1
9.
x3 sin x2 dx = − cos x2 + sin x2
2
2
x2
1
sin x2 + cos x2
10.
x3 cos x2 dx =
2
2
2.661
Trigonometric functions and exponentials
227
2.66 Combinations of trigonometric functions and exponentials
2.661
eax sinp x cosq x dx =
1
a2 + (p + q)2
eax sinp x cosq−1 x [a cos x + (p + q) sin x]
− pa eax sinp−1 x cosq−1 x dx + (q − 1)(p + q) eax sinp x cosq−2 x dx
TI (523)
=
a2
1
+ (p + q)2
eax sinp−1 x cosq x [a sin x − (p + q) cos x]
+ qa e
ax
sin
p−1
x cos
q−1
p−2
ax
q
x dx + (p − 1)(p + q) e sin
x cos x dx
TI (524)
=
a2
1
+ (p + q)2
eax sinp−1 x cosq−1 x a sin x cos x + q sin2 x − p cos2 x
p
p−2
ax
q−2
ax
q
x dx + p(p − 1) e sin
x cos x dx
+ q(q − 1) e sin x cos
TI (525)
1
= 2
a + (p + q)2
eax sinp−1 x cosq−1 x a sin x cos x + q sin2 x − p cos2 x
+q(q − 1) eax sinp−2 x cosq−2 x dx
− (q − p)(p + q − 1) eax sinp−2 x cosq x dx
TI (526)
=
1
a2 + (p + q)2
eax sinp−1 x cosq−1 x a sin x cos x + q sin2 x − p cos2 x
+p(p − 1) eax sinp−2 x cosq−2 x dx
p
ax
q−2
+ (q − p)(p + q − 1) e sin x cos
x dx
GU (334)(1a)
For p = m and q = n even integers, the integral eax sinm x cosn x dx can be reduced by means of
these formulas to the integral eax dx. However, when only m or only n is even, they can be reduced to
228
Trigonometric Functions
integrals of the form
e
ax
2.662 1.
eax sinn bx dx =
n
cos x dx or
1
a2 + n2 b2
eax cosn bx dx =
eax sinm x dx, respectively.
(a sin bx − nb cos bx) eax sinn−1 bx
n−2
2
ax
+ n(n − 1)b e sin
bx dx
2.
2.662
1
a2 + n2 b2
(a cos bx + nb sin bx) eax cosn−1 bx
2
ax
n−2
+ n(n − 1)b e cos
bx dx
eax sin2m bx dx
3.
=
m−1
k=0
(2m)!b2k eax sin2m−2k−1 bx
(2m − 2k)! [a2 + (2m)2 b2 ] [a2 + (2m − 2)2 b2 ] · · · [a2 + (2m − 2k)2 b2 ]
× [a sin bx − (2m − 2k)b cos bx] +
2m
m
=
m
eax
eax +
(−1)k
22m a 22m−1
k=1
[a2
(2m)2 b2 ] [a2
+
2m
m−k
a2
(2m)!b2m eax
+ (2m − 2)2 b2 ] · · · [a2 + 4b2 ] a
1
(a cos 2bkx + 2bk sin 2bkx)
+ 4b2 k 2
eax sin2m+1 bx dx
4.
m
(2m + 1)!b2k eax sin2m−2k bx [a sin bx − (2m − 2k + 1)b cos bx]
(2m − 2k + 1)! [a2 + (2m + 1)2 b2 ] [a2 + (2m − 1)2 b2 ] · · · [a2 + (2m − 2k + 1)2 b2 ]
k=0
m
2m + 1
eax (−1)k
= 2m
[a sin(2k + 1)bx − (2k + 1)b cos(2k + 1)bx]
2
2
a + (2k + 1)2 b2 m − k
=
k=0
eax cos2m bx dx =
5.8
m−1
k=0
+
=
(2m)!b2m eax
[a2 + (2m)2 b2 ] [a2 + (2m − 2)2 b2 ] · · · [a2 + 4b2 ] a
m
2m
eax
2m
1
eax +
[a cos 2kbx + 2kb sin 2kbx]
m 22m a 22m−1
m − k a2 + 4b2 k 2
k=1
6.
(2m)!b2k eax cos2m−2k−1 bx [a cos bx + (2m − 2k)b sin bx]
(2m − 2k)! [a2 + (2m)2 b2 ] [a2 + (2m − 2)2 b2 ] · · · [a2 + (2m − 2k)2 b2 ]
eax cos2m+1 bx dx
m
(2m + 1)!b2k eax cos2m−2k bx
(2m − 2k + 1)! [a2 + (2m − 1)2 b2 ] · · · [a2 + (2m − 2k + 1)2 b2 ]
k=0
m
1
eax 2m + 1
= 2m
[a cos(2k + 1)bx + (2k + 1)b sin(2k + 1)bx]
2
m − k a + (2k + 1)2 b2
2
=
k=0
2.663 eax (a sin bx − b cos bx)
1.
eax sin bx dx =
a2 + b 2
2.666
Trigonometric functions and exponentials
229
2.
3.
4.
2.664
eax sin bx (a sin bx − 2b cos bx)
2b2 eax
+
4b2 + a2
(4b2 + a2 ) a
a
eax
eax
− 2
cos 2bx + b sin 2bx
=
2a
a + 4b2 2
eax (a cos bx + b sin bx)
eax cos bx dx =
a2 + b 2
eax cos bx (a cos bx + 2b sin bx)
2b2 eax
eax cos2 bx dx =
+
2
2
4b + a
(4b2 + a2 ) a
a
eax
eax
+ 2
cos 2bx + b sin 2bx
=
2a
a + 4b2 2
eax sin2 bx dx =
1.
e
eax a sin(b + c)x − (b + c) cos(b + c)x
sin bx cos cx dx =
2
a2 + (b + c)2
a sin(b − c)x − (b − c) cos(b − c)x
+
a2 + (b − c)2
GU (334)(6b)
eax a cos cx + c sin cx a cos(2b + c)x + (2b + c) sin(2b + c)x
eax sin2 bx cos cx dx =
−
2
4
a2 + c 2
a2 + (2b + c)2
a cos(2b − c)x + (2b − c) sin(2b − c)x
−
a2 + (2b − c)2
eax a sin bx − b cos bx a sin(b + 2c)x − (b + 2c) cos(b + 2c)x
eax sin bx cos2 cx dx =
+
2
4
a2 + b 2
a2 + (b + 2c)2
a sin(b − 2c)x − (b − 2c) cos(b − 2c)x
+
a2 + (b − 2c)2
2.
3.
ax
GU (334)(6c)
GU (334)(6d)
2.665 eax dx
eax [a sin bx + (p − 2)b cos bx]
eax dx
a2 + (p − 2)2 b2
1.
=
−
+
p
p−1
2
sin bx
(p − 1)(p − 2)b
(p − 1)(p − 2)b2 sin
bx
sinp−2 bx
ax
eax [a cos bx − (p − 2)b sin bx]
a2 + (p − 2)2 b2
e dx
eax dx
=
−
+
2.
p
2
p−1
2
cos bx
(p − 1)(p − 2)b cos
bx
(p − 1)(p − 2)b
cosp−2 bx
TI (530)a
TI (529)a
By
successive
for p a natural number, we obtain integrals of the form
applications
2.665
ax
ofaxformulas
eax dx
eax dx
e dx
e dx
,
,
, which are not expressible in terms of a ﬁnite combination
,
sin bx
cos bx
cos2 bx
sin2 bx
of elementary functions.
2.666
eax
a
ax
p
p−1
ax
p−1
tan
1.
e tan x dx =
x−
x dx − eax tanp−2 x dx
TI (527)
e tan
p−1
p−1
a
eax cotp−1 x
+
2.
eax cotp x dx = −
TI (528)
eax cotp−1 x dx − eax cotp−2 x dx
p−1
p−1
eax tan x 1 eax dx
−
(see remark following 2.665)
3.
eax tan x dx =
a
a cos2 x
230
Trigonometric Functions
eax
(a tan x − 1) − a eax tan x dx
e tan x dx =
a
eax cot x 1 eax dx
ax
+
e cot x dx =
a
a sin2 x
eax
ax
2
(a cot x + 1) + a eax cot x dx
e cot x dx = −
a
2.667
4.
5.
6.
ax
2
(see 2.666 3)
(see remark following 2.665)
(see 2.666 5)
Integrals of type
R (x, eax , sin bx, cos cx) dx
Notation: sin t = − √
b
;
a2 + b 2
a
cos t = √
.
a2 + b 2
2.667 xp eax
p
p ax
(a sin bx − b cos bx) − 2
xp−1 eax (a sin bx − b cos bx) dx
1.
x e sin bx dx = 2
a + b2
a +
b2
p
xp eax
sin(bx + t) − √
xp−1 eax sin(bx + t) dx
= √
2
2
2
a +b
a + b2
xp eax
p
(a
cos
bx
+
b
sin
bx)
−
xp−1 eax (a cos bx + b sin bx) dx
2.
xp eax cos bx dx = 2
a + b2
a2 +
b2
p
xp eax
cos(bx + t) − √
xp−1 eax cos(bx + t) dx
=√
2
2
2
a +b
a + b2
n+1
(−1)k+1 n!xn−k+1
3.
xn eax sin bx dx = eax
sin(bx + kt)
2
2 k/2
k=1 (n − k + 1)! (a + b )
n+1
(−1)k+1 n!xn−k+1
n ax
ax
4.
x e cos bx dx = e
cos(bx + kt)
2
2 k/2
k=1 (n − k + 1)! (a + b )
eax
a2 − b 2
2ab
5.
xeax sin bx dx = 2
ax
−
sin
bx
−
bx
−
cos
bx
a + b2
a2 + b 2
a2 + b 2
eax
a2 − b 2
2ab
ax
6.
xe cos bx dx = 2
ax − 2
cos bx + bx − 2
sin bx
a + b2
a + b2
a + b2
⎧%
2
2
&
⎨
2
2
ax
2
a
2a
a
−
b
−
3b
e
7.
x2 eax sin bx dx = 2
x+
sin bx
ax2 −
2
a + b2 ⎩
a2 + b 2
(a2 + b2 )
⎫
%
2
&
⎬
2
2b
3a
−
b
4ab
− bx2 − 2
x
+
cos
bx
2
⎭
a + b2
(a2 + b2 )
TI 355
2.672
Trigonometric and hyperbolic functions
8.
231
⎧%
&
⎨
2 a2 − b 2
2a a2 − 3b2
e
2 ax
2
cos bx
x e cos bx dx = 2
x+
ax −
2
a + b2 ⎩
a2 + b 2
(a2 + b2 )
⎫
%
2
&
⎬
2
2b 3a − b
4ab
+ bx2 − 2
x+
sin bx
2
2
⎭
a +b
(a2 + b2 )
ax
GU (335), MZ 274-275
2.67 Combinations of trigonometric and hyperbolic functions
2.671 1.
sinh(ax + b) sin(cx + d) dx =
2.
3.
4.
a
cosh(ax + b) sin(cx + d)
a2 + c 2
c
− 2
sinh(ax + b) cos(cx + d)
a + c2
a
cosh(ax + b) cos(cx + d)
sinh(ax + b) cos(cx + d) dx = 2
a + c2
c
+ 2
sinh(ax + b) sin(cx + d)
a + c2
a
sinh(ax + b) sin(cx + d)
cosh(ax + b) sin(cx + d) dx = 2
a + c2
c
− 2
cosh(ax + b) cos(cx + d)
a + c2
a
sinh(ax + b) cos(cx + d)
cosh(ax + b) cos(cx + d) dx = 2
a + c2
c
+ 2
cosh(ax + b) sin(cx + d)
a + c2
GU (354)(1)
2.672
1.
2.
3.
4.
1
(cosh x sin x − sinh x cos x)
2
1
sinh x cos x dx = (cosh x cos x + sinh x sin x)
2
1
cosh x sin x dx = (sinh x sin x − cosh x cos x)
2
1
cosh x cos x dx = (sinh x cos x + cosh x sin x)
2
sinh x sin x dx =
232
2.673
Trigonometric Functions
2.673
sinh2m (ax + b) sin2n (cx + d) dx
1.
=
(−1)m
22m+2n
+
2m
m
(−1)n
22m+2n−2
2n
n
x+
m−1
n−1
j=0 k=0
2m
j
(−1)j+k
(2m −
2j)2 a2
2m (−1)k
m
(2n − 2k)c
k=0
2n n−1
(−1)m+n
22m+2n−1
2n
k
sin[(2n − 2k)(cx + d)]
k
+ (2n − 2k)2 c2
× {(2m − 2j)a sinh[(2m − 2j)(ax + b)] cos[(2n − 2k)(cx + d)]}
+(2 − 2k)c cosh[(2m − 2j)(ax + b)] sin[(2n − 2k)(cx + d)]
GU (354)(3a)
sinh2m (ax + b) sin2n−1 (cx + d) dx
2.
=
(−1)m+n
22m+2n−2
+
2m
m
n−1
k=0
n−1 m−1
n−1
(−1)
22m+2n−3
j=0 k=0
2n − 1
cos[(2n − 2k − 1)(cx + d)]
k
2n−1 2m
(−1)k
(2n − 2k − 1)c
(−1)j+k
(2m −
2j)2 a2
j
k
+ (2n − 2k − 1)2 c2
× {(2m − 2j)a sinh[(2m − 2j)(ax + b)] sin[(2n − 2k − 1)(cx + d)]}
−(2n − 2k − 1)c cosh[(2m − 2j)(ax + b)] cos[(2n − 2k − 1)(cx + d)]
GU (354)(3b)
3.
sinh2m−1 (ax + b) sin2n (cx + d) dx
2n =
n
22m+2n−2
+
2m−1
j
m−1
(−1)j
j=0
(2m − 2j − 1)a
(−1)n
22m+2n−3
m−1
n−1
j=0 k=0
cosh[(2m − 2j − 1)(ax + d)]
(−1)j+k
(2m − 2j −
2m−1
j
1)2 a2
2n k
+ (2n − 2k)2 c2
× {(2m − 2j − 1)a cosh[(2m − 2j − 1)(ax + b)] cos[(2n − 2k)(cx + d)]}
+(2n − 2k)c sinh[(2m − 2j − 1)(ax + b)] sin[(2n − 2k)(cx + d)]
GU (354)(3c)
2.673
Trigonometric and hyperbolic functions
233
sinh2m−1 (ax + b) sin2n−1 (cx + d) dx
4.
2n−1 m−1 n−1
(−1)j+k 2m−1
j
k
(−1)n−1 = 2m−2n−4
2 a2 + (2n − 2k − 1)2 c2
2
(2m
−
2j
−
1)
j=0
k=0
× {(2m − 2j − 1)a cosh[(2m − 2j − 1)(ax + b)] sin[(2n − 2k − 1)(cx + d)]}
−(2n − 2k − 1)c sinh[(2m − 2j − 1)(ax + b)] cos[(2n − 2k − 1)(cx + d)]
GU (354)(3d)
sinh2m (ax + b) cos2n (cx + d) dx
5.
2n m−1
2m
(−1)j j
2m
2n
n
sinh[(2m − 2j)(ax + b)]
x + 2m+2n−1
m
n
2
(2m − 2j)a
j=0
n−1
2n (−1)m 2m
m
k
+ 2m+2n−1
sin[(2n − 2k)(cx + d)]
2
(2n − 2k)c
k=0
2n m−1
(−1)j 2m
n−1
j
k
1
+ 2m+2n−2
2 a2 + (2n − 2k)2 c2
2
(2m
−
2j)
j=0
(−1)m
= 2m+2n
2
k=0
× {(2m − 2j)a sinh[(2m − 2j)(ax + b)] cos[(2n − 2k)(cx + d)]}
+(2 − 2k)c cosh[(2m − 2j)(ax + b)] sin[(2n − 2k)(cx + d)]
GU (354)(4a)
6.
sinh2m (ax + b) cos2n−1 (cx + d) dx
n−1
2n−1 (−1)m 2m
m
k
= 2m+2n−2
sin[(2n − 2k − 2)(cx + d)]
2
(2n − 2k − 1)c
k=0
2−1 m−1
(−1)j 2m
n−1
j
k
1
+ 2m+2n−3
2 a2 + (2n − 2k − 1)2 c2
2
(2m
−
2j)
j=0
k=0
× {(2m − 2j)a sinh[(2m − 2j)(ax + b)] cos[(2n − 2k − 1)(cx + d)]}
+(2n − 2k − 1)c cosh[(2m − 2j)(ax + b)] sin[(2n − 2k − 1)(cx + d)]
GU (354)(4a)
234
Trigonometric Functions
2.673
sinh2m−1 (ax + b) cos2n (cx + d) dx
7.
2n =
n
22m+2n−2
+
2m−1
j
m−1
(−1)j
j=0
(2m − 2j − 1)a
m−1
n−1
1
22m−2n−3
j=0 k=0
cosh[(2m − 2j − 1)(ax + d)]
(−1)j
2m
j
2n k
(2m − 2j − 1)2 a2 + (2n − 2k)2 c2
× {(2m − 2j − 1)a cosh[(2m − 2j − 1)(ax + b)] cos[(2n − 2k)(cx + d)]}
+(2n − 2k)c sinh[(2m − 2j − 1)(ax + b)] sin[(2n − 2k)(cx + d)]
GU (354)(4b)
sinh2m−1 (ax + b) cos2n−1 (cx + d) dx
8.
=
m−1
n−1
1
22m+2n−4
j=0 k=0
(−1)j
2m−1
j
2n−1 k
(2m − 2j − 1)2 a2 + (2n − 2k − 1)2 c2
× {(2m − 2j − 1)a cosh[(2m − 2j − 1)(ax + b)] cos[(2n − 2k − 1)(cx + d)]}
+(2n − 2k − 1)c sinh[(2m − 2j − 1)(ax + b)] sin[(2n − 2k − 1)(cx + d)]
GU (354)(4b)
9.
cosh2m (ax + b) sin2n (cx + d) dx
2m 2n m
n
22m+2n
=
m−1
(−1)k 2n
(−1)n 2m
m
k
sin[(2n − 2k)(cx + d)]
x + 2m+2n−1
2
(2n − 2k)c
2n +
+
n
22m+2n−1
(−1)n
22m+2n−2
k=0
m−1
2m
j
j=0
(2m − 2j)a
m−1
n−1
j=0 k=0
sinh[(2m − 2j)(ax + b)]
(−1)k
2m
j
2n k
(2m − 2j)2 a2 + (2n − 2k)2 c2
× {(2m − 2j)a sinh[(2m − 2j)(ax + b)] cos[(2n − 2k)(cx + d)]}
+(2n − 2k)c cosh[(2m − 2j)(ax + b)] sin[(2n − 2k)(cx + d)]
GU (354)(5a)
2.673
Trigonometric and hyperbolic functions
235
cosh2m−1 (ax + b) sin2n (cx + d) dx
10.
2n =
n
22m+2n−2
+
m−1
2m−1
j
j=0
(2m − 2j − 1)a
m−1
n−1
(−1)n
22m+2n−3
j=0 k=0
sinh[(2m − 2j − 1)(ax + b)]
(−1)k
2m−1
j
2n k
(2m − 2j − 1)2 a2 + (2n − 2k)2 c2
× {(2m − 2j − 1)a sinh[(2m − 2j − 1)(ax + b)] cos[(2n − 2k)(cx + d)]}
+(2n − 2k)c cosh[(2m − 2j − 1)(ax + b)] sin[(2n − 2k)(cx + d)]
GU (354)(5a)
cosh2m (ax + b) sin2n−1 (cx + d) dx
11.
n−1
(−1)k+1 2n−1
(−1)n−1 2m
m
k
cos[(2n − 2k − 1)(cx + d)]
=
22m+2n−2
(2n − 2k − 1)c
k=0
2n−1 m−1 n−1
(−1)k 2m
j
k
(−1)n−1 + 2m+2n−3
2 a2 + (2n − 2k − 1)2 c2
2
(2m
−
2j)
j=0
k=0
× {(2m − 2j)a sinh[(2m − 2j)(ax + b)] sin[(2n − 2k − 1)(cx + d)]}
−(2n − 2k − 1)c cosh[(2m − 2j)(ax + b)] cos[(2n − 2k − 1)(cx + d)]
GU (354)(5b)
12.
cosh2m−1 (ax + b) sin2n−1 (cx + d) dx
2n−1 m−1 n−1
(−1)k 2m−1
j
k
(−1)n−1 = 2m+2n−4
2 a2 + (2n − 2k − 1)2 c2
2
(2m
−
2j
−
1)
j=0
k=0
× {(2m − 2j − 1)a sinh[(2m − 2j − 1)(ax + b)] sin[(2n − 2k − 1)(cx + d)]}
−(2n − 2k − 1)c cosh[(2m − 2j − 1)(ax + b)] cos[(2n − 2k − 1)(cx + d)]
GU (354)(5b)
236
Trigonometric Functions
2.673
cosh2m (ax + b) cos2n (cx + d) dx
2m 2n 13.
m
n
22m+2n
=
2m x+
2n +
+
n
22m+2n−1
2
n−1
m
22m+2n−1
k=0
m−1
2m
j
j=0
(2m − 2j)a
22m+2n−2
j=0 k=0
sin[(2n − 2k)(cx + d)]
sinh[(2m − 2j)(ax + b)]
2m
j
m−1
n−1
1
k
(2n − 2k)c
2n k
(2m − 2j)2 a2 + (2n − 2k)2 c2
× {(2m − 2j)a sinh[(2m − 2j)(ax + b)] cos[(2n − 2k)(cx + d)]}
+(2n − 2k)c cosh[(2m − 2j)(ax + b)] sin[(2n − 2k)(cx + d)]
GU (354)(6)
cosh2m−1 (ax + b) cos2n (cx + d) dx
14.
2n =
n
22m+2n−2
+
m−1
2m−1
j
j=0
(2m − 2j − 1)a
2m−1
j
m−1
n−1
1
22m+2n−3
j=0 k=0
sinh[(2m − 2j − 1)(ax + b)]
2n k
(2m − 2j − 1)2 a2 + (2n − 2k)2 c2
× {(2m − 2j − 1)a sinh[(2m − 2j − 1)(ax + b)] cos[(2n − 2k)(cx + d)]}
+(2n − 2k)c cosh[(2m − 2j − 1)(ax + b)] sin[(2n − 2k)(cx + d)]
GU (354)(6)
15.
cosh2m (ax + b) cos2n−1 (cx + d) dx
2m =
m
22m+2n−2
+
n−1
k=0
1
22m+2n−3
2n−1 k
(2n − 2k − 1)c
m−1
n−1
j=0 k=0
sin[(2n − 2k − 1)(cx + d)]
2m
j
2n−1 k
(2m − 2j)2 a2 + (2n − 2k − 1)2 c2
× {(2m − 2j)a sinh[(2m − 2j)(ax + b)] cos [(2n − 2k − 1)(cx + d)]}
+(2n − 2k − 1)c cosh[(2m − 2j)(ax + b)] sin[(2n − 2k − 1)(cx + d)]
GU (354)(6)
2.711
The logarithm
237
cosh2m−1 (ax + b) cos2n−1 (cx + d) dx
16.
=
m−1
n−1
1
22m+2n−4
j=0 k=0
2m−1
j
2n−1 k
(2m − 2j − 1)2 a2 + (2n − 2k − 1)2 c2
× {(2m − 2j − 1)a sinh[(2m − 2j − 1)(ax + b)] cos[(2n − 2k − 1)(cx + d)]}
+(2n − 2k − 1)c cosh[(2m − 2j − 1)(ax + b)] sin[(2n − 2k − 1)(cx + d)]
GU (354)(6)
2.674
eax sinh bx sin cx dx =
1.
eax sinh bx cos cx dx =
2.
eax cosh bx sin cx dx =
3.
eax cosh bx cos cx dx =
4.
e(a+b)x
[(a + b) sin cx − c cos cx]
2 [(a + b)2 + c2 ]
e(a−b)x
−
[(a − b) sin cx − c cos cx]
2 [(a − b)2 + c2 ]
e(a+b)x
[(a + b) cos cx + c sin cx]
2 [(a + b)2 + c2 ]
(a−b)x
e
−
[(a − b) cos cx + c sin cx]
2 [(a − b)2 + c2 ]
e(a+b)x
[(a + b) sin cx − c cos cx]
2 [(a + b)2 + c2 ]
e(a−b)x
+
[(a − b) sin cx − c cos cx]
2 [(a − b)2 + c2 ]
e(a+b)x
[(a + b) cos cx + c sin cx]
2 [(a + b)2 + c2 ]
e(a−b)x
+
[(a − b) cos cx + c sin cx]
2 [(a − b)2 + c2 ]
MZ 379
2.7 Logarithms and Inverse-Hyperbolic Functions
2.71 The logarithm
2.711
ln x dx = x ln x − m
m
m
lnm−1 x dx
x (−1)k (m + 1)m(m − 1) · · · (m − k + 1) lnm−k x
m+1
m
=
k=0
(m > 0)
TI (603)
238
Logarithms and Inverse-Hyperbolic Functions
2.721
2.72–2.73 Combinations of logarithms and algebraic functions
2.721
xn lnm x dx =
1.
m
xn+1 lnm x
−
n+1
n+1
xn lnm−1 x dx
(see 2.722)
For n = −1
m
ln x dx
lnm+1 x
2.
=
x
m+1
For n = −1 and m = −1
dx
= ln (ln x)
3.
x ln x
m
xn+1 lnm−k x
(−1)k (m + 1)m(m − 1) · · · (m − k + 1)
2.722
xn lnm x dx =
m+1
(n + 1)k+1
TI (604)
k=0
2.723
1.
2.
3.
2.724
ln x
1
x ln x dx = x
−
n + 1 (n + 1)2
2
ln x
2 ln x
2
−
xn ln2 x dx = xn+1
+
n + 1 (n + 1)2
(n + 1)3
3
ln x
3 ln2 x
6 ln x
6
−
xn ln3 x dx = xn+1
+
−
n + 1 (n + 1)2
(n + 1)3
(n + 1)4
n
1.
2.
2.725
1.
2.
n+1
xn dx
xn+1
n+1
m =−
m−1 + m − 1
(ln x)
(m − 1) (ln x)
TI 375
TI 375
xn dx
m−1
(ln x)
For m = 1
n
x dx
= li xn+1
ln x
1
(a + bx)m+1 dx
(a + bx)m+1 ln x −
(m + 1)b
x
m m m−k k k+1
b x
1
m
m+1
m+1
k a
(a + bx)
ln x −
(a + bx) ln x dx =
−a
(m + 1)b
(k + 1)2
(a + bx)m ln x dx =
k=0
For m = −1, see 2.727 2.
2.726
(a + bx)2
a2
1
−
1.
(a + bx) ln x dx =
ln x − ax + bx2
2b
2b
4
b2 x3
1 abx2
2.
(a + bx)2 ln x dx =
(a + bx)3 − a3 ln x − a2 x +
+
3b
2
9
TI 374
2.731
Combinations of logarithms and algebraic functions
(a + bx)3 ln x dx =
3.
2.727
8
1.
2.8
3.
4.
5.
2.728
1
dx
ln x dx
ln x
=
+
−
(a + bx)m
b(m − 1)
(a + bx)m−1
x(a + bx)m−1
For m = 1
1
1 ln(a + bx) dx
ln x dx
= ln x ln(a + bx) −
(see 2.728 2)
a + bx
b
b
x
1
x
ln x dx
ln x
+
ln
=−
2
(a + bx)
b(a + bx) ab a + bx
1
x
ln x dx
ln x
1
+
ln
=−
+
(a + bx)3
2b(a + bx)2
2ab(a + bx) 2a2 b a + bx
√
√
(a + bx)1/2 − a1/2
ln x dx
2
√
=
[a > 0]
(ln x − 2) a + bx − 2 a ln
x1/2 a + bx b √
√
a + bx
2
(ln x − 2) a + bx + 2 −a arctan
[a < 0]
=
b
−a
xm ln(a + bx) dx =
2.9
2.
3.
4.
TI 376
m+1 dx
1
x
xm+1 ln(a + bx) − b
m+1
a + bx
bx
bx
ln(a + bx)
= ln a ln x +
Φ − , 2, 1
x
a
a
[a > 0]
m+1
1
(−a)m+1
1 (−1)k xm−k+2 ak−1
m+1
x ln(a + bx) dx =
−
ln(a
+
bx)
+
x
m+1
bm+1
m+1
(m − k + 2)bk−1
k=1
ax
1 2 a2
1 x2
−
x ln(a + bx) dx =
x − 2 ln(a + bx) −
2
b
2 2
b
3
3
ax2
a2 x
1 3 a
1 x
2
−
+ 2
x ln(a + bx) dx =
x + 3 ln(a + bx) −
3
b
3 3
2b
b
ax3
a2 x2
1 4 a4
a3 x
1 x4
x3 ln(a + bx) dx =
−
+
−
x − 4 ln(a + bx) −
4
b
4 4
3b
2b2
b3
⎧
⎨
1
x
x2n ln x2 + a2 dx =
x2n+1 ln x2 + a2 + (−1)n 2a2n+1 arctan
2n + 1 ⎩
a
⎫
n
(−1)n−k 2n−2k 2k+1 ⎬
a
x
−2
⎭
2k + 1
m
1.
2.731
1 3
1
1
(a + bx)4 − a4 ln x − a3 x + a2 bx2 + ab2 x3 + b3 x4
4b
4
3
16
1.
2.729
239
k=0
240
Logarithms and Inverse-Hyperbolic Functions
x2n+1 ln x2 + a
2.7327
2
dx =
2.732
⎧
⎨
1
x2n+2 + (−1)n a2n+2 ln x2 + a2
2n + 2 ⎩
⎫
n+1
⎬
(−1)n−k
a2n−2k+2 x2k
+
⎭
k
k=1
2.733
1.
2.
3.
4.
5.
2.734
x
ln x2 + a2 dx = x ln x2 + a2 − 2x + 2a arctan
a
2
1
x2 + a2 ln x2 + a2 − x2
x ln x + a2 dx =
2
2 3
2
1 3 2
x
2
2
2
2
3
x ln x + a dx =
x ln x + a − x + 2a x − 2a arctan
3
3
a
x4
1 4
+ a2 x2
x − a4 ln x2 + a2 −
x3 ln x2 + a2 dx =
4
2
2 5 2 2 3
2
1 5 2
x
4
2
2
4
5
x ln x + a dx =
x ln x + a − x + a x − 2a x + 2a arctan
5
5
3
a
!
!
x2n ln !x2 − a2 ! dx
DW
DW
DW
DW
DW
!
!
n
! 2
!
!x + a!
1
2!
2n+1
2n−2k 2k+1
!
!
!
a
−2
ln x − a + a
ln !
x
x
x − a!
2k + 1
k=0
1
! 2
!
! n+1
2n+2
! 2
1
2n+1
2
2n+2
2
2n−2k+2
2k
2.735
x
a
x
ln !x − a ! −
ln !x − a ! dx =
−a
x
2n + 2
k
1
=
2n + 1
2n+1
k=1
2.736
1.
2.
3.
4.
5.
!
!
!
!x + a!
!
!
!
!
ln !x2 − a2 ! dx = x ln !x2 − a2 ! − 2x + a ln !!
x − a!
!
!
!
!
1 2
x − a2 ln !x2 − a2 ! − x2
x ln !x2 − a2 ! dx =
2
!
!
! 2
! 2
!x + a!
!
! 2 3
1
2
2!
3
2!
2
3
!
!
!
!
x ln x − a dx =
x ln x − a − x − 2a x + a ln !
3
3
x − a!
! 2
!
! x4
! 2
1 4
3
2!
4
2!
2 2
!
!
x − a ln x − a −
−a x
x ln x − a dx =
4
2
!
!
! 2
! 2
!x + a!
!
! 2 5 2 2 3
1
4
2!
5
2!
4
5
!
!
!
!
x ln x − a dx =
x ln x − a − x − a x − 2a x + a ln !
5
5
3
x − a!
DW
DW
DW
DW
DW
2.74 Inverse hyperbolic functions
2.741 x x
1.
arcsinh dx = x arcsinh − x2 + a2
a
a
DW
2.813
Arcsines and arccosines
2.
arccosh
3.
4.
2.742
x x
dx = x arccosh − x2 − a2
a
a
x 2
= x arccosh + x − a2
a
241
+
,
x
arccosh > 0
a
+
,
x
arccosh < 0
a
DW
DW
x a x
dx = x arctanh + ln a2 − x2
a
a 2
x a x
arccoth dx = x arccoth + ln x2 − a2
a
a 2
arctanh
1.
x arcsinh
x
dx =
a
x2
a2
+
2
4
x arccosh
x
dx =
a
x2
a2
−
2
4
x2
a2
−
2
4
2.
=
DW
DW
x x 2
−
x + a2
a
4
x x 2
x − a2
arccosh −
a
4
x x 2
x − a2
arccosh +
a
4
arcsinh
DW
+
,
x
arccosh > 0
a
+
,
x
arccosh < 0
a
DW
2.8 Inverse Trigonometric Functions
2.81 Arcsines and arccosines
2.811
arcsin
x
a
n
dx = x
n/2
k=0
+ a2 − x2
(n+1)/2
(−1)k−1
k=1
2.812
n
x
· (2k)! arcsin
2k
a
(−1)k
arccos
x
a
n
dx = x
n/2
(−1)k
k=0
+ a2 − x2
n
2k − 1
n
x
· (2k)! arccos
2k
a
(n+1)/2
k=1
(−1)k
n−2k
n
2k − 1
· (2k − 1)! arcsin
x
a
n−2k+1
n−2k
· (2k − 1)! arccos
2.813 x
x
+ a2 − x2
arcsin dx = sign(a) x arcsin
1.11
a
|a|
2
x 2
x
x
arcsin
− 2x
2.9
dx = x arcsin
+ 2 a2 − x2 arcsin
a
|a|
|a|
⎡
3
x 3
x
x
3.
arcsin
dx = sign(a) ⎣x arcsin
+ 3 a2 − x2 arcsin
a
|a|
|a|
⎤
x
− 6 a2 − x2 ⎦
− 6x arcsin
|a|
2
x
a
n−2k+1
242
Inverse Trigonometric Functions
2.814
2.814 x
x 1.
arccos dx = x arccos − a2 − x2
a
a
2
x
x 2
x
2.
arccos
dx = x arccos
− 2 a2 − x2 arccos − 2x
a
a
a
3
3
x
x 2
x
x
arccos
3.
dx = x arccos
− 3 a2 − x2 arccos
− 6x arccos + 6 a2 − x2
a
a
a
a
2.82 The arcsecant, the arccosecant, the arctangent, and the arccotangent
2.821 x
a
x
1.
arccosec dx = arcsin dx = x arcsin + a ln x + x2 − a2
a
x
2
a
= x arcsin − a ln x + x2 − a2
x
2.
arcsec
DW
a
a
dx = x arccos − a ln x +
x
x
a
= x arccos − a ln x + x2 − a2
x
x
dx =
a
+
a
π,
0 < arcsin <
x
2 ,
+ π
a
− < arcsin < 0
2
x
arccos
x2 − a2
+
a
π,
0 < arccos <
x
2 ,
+ π
a
− < arccos < 0
2
x
DW
2.822 x a x
1.8
arctan dx = x arctan − ln a2 + x2
a
a 2
x a x
2.
arccot dx = x arccot − ln a2 + x2
a
a 2
1 2
x ax
x
9
x + a2 arctan −
3.
x arctan dx =
a
2
a
2
2
ax πx
1 2
x
x
+
−
x + a2 arctan
4.9
x arccot dx =
a
2
4
2
a
ax2
1
x
x 1
5.9
x2 arctan dx = x3 arctan + a3 ln x2 + a2 −
a
3
a 6
6
1
ax2
πx3
x
x 1
+
6.9
x2 arccot dx = − x3 arctan − a3 ln x2 + a2 +
a
3
a 6
6
6
2.83 Combinations of arcsine or arccosine and algebraic functions
DW
DW
n+1
dx
xn+1
x
1
x
x
√
arcsin −
2.831
x arcsin dx =
(see 2.263 1, 2.264, 2.27)
2
a
n
+
1
a
n
+
1
− x2
an+1
n+1
dx
x
x
1
x
x
√
arccos +
2.832
xn arccos dx =
(see 2.263 1, 2.264, 2.27)
2
a
n+1
a n+1
a − x2
arccos x
arcsin x
dx and
dx) cannot be expressed as a
1.
For n = −1, these integrals (that is,
x
x
ﬁnite combination of elementary functions.
n
2.838
Arcsine or arccosine and algebraic functions
2.
2.8339
1.
2.
3.
4.
5.
6.
2.834
π 1
arccos x
dx = − ln −
x
2 x
243
arcsin x
dx
x
x2
a2
x 2
x
2
−
+
a −x
arcsin
2
4
|a| 4
1 2
πx2
x 2
x
x
2
2
− sign(a)
2x − a arcsin
+
a −x
x arccos dx =
a
4
4
|a| 4
3
x
x
1 2
x
arcsin
+
x + 2a2
a2 − x2
x2 arcsin dx = sign(a)
a
3
|a| 9
3
3
x
πx
x
1 2
x
2
2
2
2
− sign(a)
arcsin
+
x + 2a
a −x
x arccos dx =
a
6
3
|a| 9
4
x
3a4
1 x
x
−
+ x 2x2 + 3a2
a2 − x2
x3 arcsin dx = sign(a)
arcsin
a
4
32
|a| 32
%
&
4
4
4
8x
−
3a
πx
x
1
x
x3 arccos dx =
− sign(a)
arcsin
+ x 2x2 + 3a2
a2 − x2
a
8
32
|a| 32
x
x arcsin dx = sign(a)
a
√
a2 − x2
x
√
1
x 1 a + a2 − x2
x
1
arccos dx = − arccos − ln
2.
2
x
a
x
a a
x
.
2
arcsin x
2
(a − b)(1 − x)
arcsin x
2.835
− √
a > b2
dx = −
arctan
2
2
2
(a + bx)
b(a + bx) b a − b
(a + b)(1 + x)
2
(a + b)(1 + x) + (b − a)(1 − x)
1
arcsin x
− √
=−
a < b2
ln b(a + bx) b b2 − a2
(a + b)(1 + x) − (b − a)(1 − x)
√
1
c + 1x
arcsin x
x arcsin x
√
√
+
dx
=
−
[c > −1]
arctan
2.8368
2
2)
2
2
2c
(1
+
cx
2c
c
+
1
1
(1 + cx )
√ −x
1 − x2 + x −(c + 1)
1
arcsin x
+ ln √
[c < −1]
=−
2
2c (1 + cx ) 4c −(c + 1)
1 − x2 − x −(c + 1)
1.
1
x 1 a+
1
x
arcsin dx = − arcsin − ln
2
x
a
x
a a
2.837 x arcsin x
√
dx = x − 1 − x2 arcsin x
1.
2
1−x
x arcsin x
x
x2
1
2
√
2.
−
dx =
1 − x2 arcsin x + (arcsin x)
4
2
4
1 − x2
3
2x 1 2
x arcsin x
x3
√
+
−
x +2
3.
dx =
1 − x2 arcsin x
9
3
3
1 − x2
2.838 arcsin x
x arcsin x 1 <
1.
dx = √
+ ln 1 − x2
2
2
3
1−x
(1 − x2 )
244
Inverse Trigonometric Functions
2.
2.841
arcsin x
1 1−x
x arcsin x
<
dx = √
+ ln
2
2
1+x
3
1−x
(1 − x2 )
2.84 Combinations of the arcsecant and arccosecant with powers of x
2.841
1.
x arcsec
x2 arcsec
2.
a
1 2
a
arccos dx =
x arccos − a x2 − a2
x
2
x
a
1 2
2
2
x arccos + a x − a
=
2
x
x
dx =
a
x
dx =
a
=
3.
DW
a a 2
a
− x x − a2 −
ln x + x2 − a2
x 2
2 +
a
π,
0 < arccos <
x
2
a a 2
a3
3
2
2
2
ln x + x − z
x arccos + x x − a +
x 2
2
+π
,
a
< arccos < π
2
x
arccos
1
3
+
a
π,
0 < arccos <
x
2,
+π
a
< arccos < π
2
x
a
1
dx =
x
3
x3 arccos
3
DW
a
1 2
x
a
x arcsin + a x2 − a2
x arccosec dx = arcsin dx =
a
x
2
x
a
1 2
2
2
x arcsin − a x − a
=
2
x
+
a
π,
0 < arcsin <
x
2 ,
+ π
a
− < arcsin < 0
2
x
DW
2.85 Combinations of the arctangent and arccotangent with algebraic functions
2.851
2.852
3.
2.853
1.
2.
xn+1
x
a
x
dx =
arctan −
a
n+1
a n+1
xn arccot
xn+1
x
a
x
dx =
arccot +
a
n+1
a n+1
1.
2.
xn arctan
xn+1 dx
a2 + x2
xn+1 dx
a2 + x2
For n = −1
arctan x
dx cannot be expressed as a ﬁnite combination of elementary functions.
x
arccot x
π
arctan x
dx = ln x −
dx
x
2
x
1 2
x ax
x
dx =
x + a2 arctan −
a
2
a
2
1 2
x ax
x
x + a2 arccot +
x arccot dx =
a
2
a
2
x arctan
2.859
Arctangent and arccotangent with algebraic functions
245
ax2
x3
x a3 2
x
dx =
arctan +
ln x + a2 −
a
3
a
6
6
3
3
x
x a
ax2
πx3
x
ln x2 + a2 +
+
x2 arccot dx = − arctan −
4.9
a
3
a
6
6
6
1
x
1
a2 + x2
x
1
ln
arctan dx = − arctan −
2.854
2
a
x
a 2a
x2
x
arctan x
1
β − αx
α + βx
√
arctan
x
2.855
dx
=
−
ln
(α + βx)2
α2 + β 2
α + βx
1 + x2
2.856
1 ln 1 + x2 dx
x arctan x
1
2
1.
−
arctan
x
ln
1
+
x
dx
=
1 + x2
2
2
1 + x2
2
1
x arctan x
1 2
2.
dx = x arctan x − ln 1 + x2 − (arctan x)
2
1+x
2
2
3
1
x arctan x
x arctan x
1
2
x
+
1
+
x
arctan
x
−
3.
dx
=
−
dx
2
1+x
2
2
1 + x2
3.
x2 arctan
9
TI (689)
TI (405)
(see 2.8511)
x
1
x arctan x
1
2 2
− x arctan x + (arctan x)
dx = − x2 + ln 1 + x2 +
2
1+x
6
3
3
2
&
% n
(2n − 2k)!!(2n − 1)!!
x
arctan x dx
1 (2n − 1)!!
arctan x arctan x
2.857
+
n+1 =
(2n)!!(2n − 2k + 1)!! (1 + x2 )n−k+1
2 (2)!!
(1 + x2 )
k=1
n
1
(2n − 1)!!(2n − 2k)!!
1
+
2
(2n)!!(2n − 2k + 1)!!(n − k + 1) (1 + x2 )n−k+1
k=1
√
√
2
x arctan x
x
√
2.858
dx = − 1 − x2 arctan x + 2 arctan √
− arcsin x
1 − x2
1
−
x2
arctan x
a + bx2
x arctan x
1
<
2.859
[a < b]
dx = √
− √
arctan
2
b − a√
3
a b−a
a a + bx
√
(a + bx2 )
x arctan x
a + bx2 − a − b
1
√
= √
+ √
[a > b]
ln √
2a a − b
a a + bx2
a + bx2 + a − b
4.
4
3
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3–4 Deﬁnite Integrals of Elementary
Functions
3.0 Introduction
3.01 Theorems of a general nature
3.011 Suppose that f (x) is integrable† over the largest of the intervals (p, q), (p, r), (r, q). Then (depending on the relative positions of the points p, q, and r) it is also integrable over the other two intervals,
and we have
q
r
f (x) dx =
p
q
f (x) dx +
p
f (x) dx.
FI II 126
r
3.012 The ﬁrst mean-value theorem. Suppose (1) that f (x) is continuous and that g(x) is integrable
over the interval (p, q), (2) that m ≤ f (x) ≤ M , and (3) that g(x) does not change sign anywhere in the
interval (p, q). Then, there exists at least one point ξ (with p ≤ ξ ≤ q) such that
q
q
f (x)g(x) dx = f (ξ) g(x) dx.
FI II 132
p
p
3.013 The second mean-value theorem. If f (x) is monotonic and non-negative throughout the interval
(p, q), where p <q, and if g(x) is integrable over that interval, then there exists at least one point ξ (with
p ≤ ξ ≤ q) such that
1.
p
q
ξ
f (x)g(x) dx = f (p) g(x) dx
p
Under the conditions of Theorem 3.013 1, if f (x) is nondecreasing, then
q
q
f (x)g(x) dx = f (q) g(x) dx
[p ≤ ξ ≤ q] .
2.
p
ξ
If f (x) is monotonic in the interval (p, q), where p <q, and if g(x) is integrable over that interval, then
∗ We omit the deﬁnition of deﬁnite and multiple integrals since they are widely known and can easily be found in any
textbook on the subject. Here we give only certain theorems of a general nature which provide estimates, or which reduce
the given integral to a simpler one.
†A
q
function f (x) is said to be integrable over the interval (p, q), if the integral
f (x) dx exists. Here, we usually mean
p
the existence of the integral in the sense of Riemann. When it is a matter of the existence of the integral in the sense of
Stieltjes or Lebesgue, etc., we shall speak of integrability in the sense of Stieltjes or Lebesgue.
247
248
Introduction
q
3.
ξ
q
f (x)g(x) dx = f (p) g(x) dx + f (q) g(x) dx
p
or
p
q
4.
3.020
ξ
q
f (x)g(x) dx = A g(x) dx + B g(x) dx
p
[p ≤ ξ ≤ q] ,
ξ
p
[p ≤ ξ ≤ q] ,
ξ
where A and B are any two numbers satisfying the conditions
A ≥ f (p + 0)
A ≤ f (p + 0)
B ≤ f (q − 0)
B ≥ f (q − 0)
and
and
[if f decreases],
[if f increases].
In particular,
q
ξ
q
5.
f (x)g(x) dx = f (p + 0) g(x) dx + f (q − 0) g(x) dx
p
p
FI II 138
ξ
3.02 Change of variable in a deﬁnite integral
β
3.020
ψ
f (x) dx =
α
f [g(t)]g (t) dt;
x = g(t).
ϕ
This formula is valid under the following conditions:
1.
f (x) is continuous on some interval A ≤ x ≤ B containing the original limits of integration α
and β.
2.
The equalities α = g(ϕ) and β = g(ψ) hold.
3.
g(t) and its derivative g (t) are continuous on the interval ϕ ≤ t ≤ ψ.
4.
As t varies from ϕ to ψ, the function g(t) always varies in the same direction from g(ϕ) = α to
g(ψ) = β.∗
3.021
β
The integral
f (x) dx can be transformed into another integral with given limits ϕ and ψ by
α
means of the linear substitution
x=
β
1.
f (x) dx =
α
β−α
ψ−ϕ
ψ
f
ϕ
αψ − βϕ
β−α
t+
:
ψ−ϕ
ψ−ϕ
β−α
αψ − βϕ
t+
ψ−ϕ
ψ−ϕ
dt
In particular, for ϕ = 0 and ψ = 1,
∗ If this last condition is not satisﬁed, the interval ϕ ≤ t ≤ ψ should be partitioned into subintervals throughout each of
which the condition is satisﬁed:
β
ϕ1
ϕ2
ψ
f (x) dx =
f [g(t)]g (t) dt +
f [g(t)]g (t) dt + · · · +
f [g(t)]g (t) dt.
α
ϕ
ϕ1
ϕn−1
3.033
General formulas
β
2.
1
f (x) dx = (β − α) f ((β − α)t + α) dt
α
0
For ϕ = 0 and ψ = ∞,
β
f (x) dx = (β − α)
3.
α
3.022
1.
249
∞
f
0
α + βt
1+t
dt
(1 + t)2
The following formulas also hold:
β
β
f (x) dx =
f (α + β − x) dx
α
β
2.
α
β
f (β − x) dx
f (x) dx =
0
0
α
3.
α
f (x) dx =
−α
f (−x) dx
−α
3.03 General formulas
3.031
1.
Suppose that a function f (x) is integrable over the interval (−p, p) and satisﬁes the relation
f (−x) = f (x) on that interval. (A function satisfying the latter condition is called an even
function.) Then,
p
p
f (x) dx = 2 f (x) dx.
FI II 159
−p
2.
0
Suppose that f (x) is a function that is integrable on the interval (−p, p) and satisﬁes the relation
f (−x) = −f (x) on that interval. (A function satisfying the latter condition is called an odd
function). Then,
p
f (x) dx = 0.
FI II 159
−p
3.032 1.
0
π
2
π
2
f (sin x) dx =
f (cos x) dx,
0
where f (x) is a function that is integrable on the interval (0, 1).
2π
π 2.
f (p cos x + q sin x) dx = 2
f
p2 + q 2 cos x dx,
0
0
where f (x) is integrable on the interval − p2 + q 2 , p2 + q 2 .
π2
π2
3.
f (sin 2x) cos x dx =
f cos2 x cos x dx,
0
FI II 159
FI II 160
0
where f (x) is integrable on the interval (0, 1).
3.033
1.
If f (x + π) = f (x) and f (−x) = f (x), then
FI II 161
250
Introduction
∞
f (x)
0
sin x
dx =
x
3.034
π
2
f (x) dx
LO V 277(3)
0
If f (x + π) = −f (x) and f (−x) = f (x), then
π2
∞
sin x
dx =
f (x)
f (x) cos x dx
x
0
0
2.
LO V 279(4)
In formulas
∞3.033, it is assumed that the integrals in the left members of the formulas exist.
q
f (px) − f (qx)
3.034
dx = [f (0) − f (+∞)] ln ,
x
p
0
if f (x) is continuous for x ≥0 and if there exists a ﬁnite limit f (+∞) = lim f (x).
x→+∞
3.035
1.
2.
f α + exi + f α + e−xi
2π
dx =
f (α + p)
[|p| < 1]
2
1
+
2p
cos
x
+
p
1
−
p2
0
π
1 − p cos x
f α + exi + f α + e−xi dx = π {f (α + p) + f (α)}
2
1
−
2p
cos
x
+
p
0
3.
0
FI II 633
π
π
−xi
−f α+e
f α+e
1 − 2p cos x + p2
[|p| < 1]
xi
sin x dx =
LA 230(16)
BE 169
π
{f (α + p) − f (α)}
π
[|p| < 1]
BE 169
In formulas 3.035, it is assumed that the function f is analytic in the closed unit circle with its center
at the point α.
3.036
π
π
2
sin2 x
1.11
f
f sin2 x dx
p <1
dx
=
2
1 + 2p cos x + p
0
0 π
2
sin2 x
f
=
dx
p ≥1
2
p
0
2.
π
F (n) (cos x) sin2n x dx = (2n − 1)!!
0
3.037
LA 228(6)
π
F (cos x) cos nx dx
B 174
0
If f is analytic in the circle of radius r and if
f [r (cos x + i sin x)] = f1 (r, x) + if2 (r, x),
then
1.
2.
3.
∞
f1 (r, x)
π −p f re
dx =
2
2
2p
0 p +x
∞
x dx
π −p − f (0)
f re
f2 (r, x) 2
=
2
p +x
2
0
∞
f2 (r, x)
π
dx = [f (r) − f (0)]
x
2
0
LA 230(19)
LA 230(20)
LA 230(21)
3.045
Improper integrals
251
∞
π f2 (r, x)
dx = 2 f (r) − f re−p
2 + x2 )
x
(p
2p
0
∞
∞
x dx
√
3.038
F qx + p 1 + x2 =
F (p cosh x + q sinh x) sinh x dx
2
−∞ 1 + x
−∞
∞
F sign p · p2 − q 2 cosh x sinh2 x dx
= 2q
4.
LA 230(22)
0
[If F is a function with a continuous derivative in the interval (−∞, ∞), all these integrals converge.]
3.04 Improper integrals
3.041 Suppose that a function f (x) is deﬁned on an interval (p, +∞) and that it is integrable over an
arbitrary ﬁnite subinterval of the form (p, P ). Then, by deﬁnition
P
+∞
f (x) dx = lim
f (x) dx,
P →+∞ p
p
+∞
f (x) dx exists or that it converges.
if this limit exists. If it does exist, we say that the integral
p
Otherwise, we say that the integral diverges.
3.042 Suppose that a function f (x) is bounded and integrable in an arbitrary interval (p, q − η) (for
0 < η < q − p) but is unbounded in every interval (q − η, q) to the left of the point q. The point q is then
called a singular point. Then, by deﬁnition,
q−η
q
f (x) dx = lim
f (x) dx,
η→0 p
p
q
f (x) dx exists or that it converges.
if this limit exists. In this case, we say that the integral
p
3.043 If not only the integral of f (x) but also the integral of |f (x)| exists, we say that the integral of
f (x) converges absolutely.
+∞
3.044 The integral
f (x) dx converges absolutely if there exists a number α > 1 such that the limit
p
lim {xα |f (x)|}
x→+∞
exists. On the other hand, if
lim {x|f (x)|} = L > 0,
x→+∞
+∞
|f (x)| dx diverges.
the integral
p
3.045
q
Suppose that the upper limit q of the integral
f (x) dx is a singular point. Then, this integral
p
converges absolutely if there exists a number α <1 such that the limit
lim [(q − x)α |f (x)|]
x→q
exists. On the other hand, if
lim [(q − x)|f (x)|] = L > 0,
x→q
q
f (x) dx diverges.
the integral
p
252
Introduction
3.046
3.046 Suppose that the functions f (x) and g(x) are deﬁned on the interval (p, +∞), that f (x) is
integrable over every ﬁnite interval of the form (p, P ), that the integral
P
f (x) dx
p
is a bounded function of P , that g(x) is monotonic, and that g(x) →0 as x → +∞. Then, the integral
+∞
f (x)g(x) dx
p
converges.
FI II 577
3.05 The principal values of improper integrals
3.051 Suppose that a function f (x) has a singular point r somewhere inside the interval (p, q), that
f (x) is deﬁned at r, and that f (x) is integrable over every portion of this interval that does not contain
the point r. Then, by deﬁnition
q
q
r−η
f (x) dx = lim
f (x) dx +
f (x) dx .
η→0
η →0
p
r+η p
Here, the limit must exist for independent modes of approach of η and η to zero. If this limit does not
exist but the limit
q
r−η
f (x) dx +
f (x) dx
lim
η→0
p
r+η
"q
does exist, we say that this latter limit is the principal value of the improper integral p f (x) dx, and we
q
say that the integral
f (x) dx exists in the sense of principal values.
FI II 603
p
3.052 Suppose that the function f (x) is continuous over the interval (p, q) and vanishes at only one
point r inside this interval. Suppose that the ﬁrst derivative f (x) exists in a neighborhood of the point
r. Suppose that f (r) = 0 and that the second derivative f (r) exists at the point r itself. Then,
q
dx
FI II 605
p f (x)
diverges, but exists in the sense of principal values.
3.053 A divergent integral of a positive function cannot exist in the sense of principal values.
3.054 Suppose that the function f (x) has no singular points in the interval (−∞, +∞). Then, by
deﬁnition
Q
+∞
f (x) dx = lim
f (x) dx.
P →−∞
Q→+∞
−∞
P
Here, the limit must exist for independent approach of P and Q to ±∞. If this limit does not exist but
the limit
+P
lim
f (x) dx
P →+∞ −P
does exist, this last limit is called the principal value of the improper integral
+∞
f (x) dx.
FI II 607
−∞
3.055 The principal value of an improper integral of an even function exists only when this integral
FI II 607
converges (in the ordinary sense).
3.112
Rational functions
253
3.1–3.2 Power and Algebraic Functions
3.11 Rational functions
∞
1.
−∞
r2
p + qx
π
(p − qr cos λ)
(principal value)
dx =
2
+ 2rx cos λ + x
r sin λ
(see also 3.194 8 and 3.252 1 and 2)
3.11211
Integrals of the form
BI (22)(14)
∞
gn (x) dx
, where
h
−∞ n (x)hn (−x)
gn (x) = b0 x2n−2 + b1 x2n−4 + · · · bn−1 ,
hn (x) = a0 xn + a1 xn−1 + · · · an
[All roots of hn (x) lie in the upper half-plane.]
∞
πi Mn
gn (x) dx
=
,
1.
h
(x)h
(−x)
a
n
0 Δn
−∞ n
where
!
!
!a1 a3 a5
0 !!
!
!a0 a2 a4
0 !!
!
! 0 a1 a3
0 !!,
Δn = !
! ..
!
.
..
!.
!
!
!
!0 0 0
an !
2.
∞
g1 (x) dx
−∞ h1 (x)h1 (−x)
3.8
4.11
∞
∞
b1
a2
a1
b2
a4
a3
···
..
0
0
.
!
bn−1 !!
0 !!
0 !!.
!
!
!
an !
πib0
a0 a1
g3 (x) dx
= πi
−∞ h3 (x)h3 (−x)
∞
g4 (x) dx
−∞ h4 (x)h4 (−x)
∞
g5 (x) dx
6.
!
! b0
!
! a0
!
!
Mn = ! 0
! ..
!.
!
!0
JE
−b0 + aa0 2b1
g2 (x) dx
= πi
a0 a1
−∞ h2 (x)h2 (−x)
5.
=
JE
−∞ h5 (x)h5 (−x)
a0 a1 b 2
a3
a0 (a0 a3 − a1 a2 )
−a2 b0 + a0 b1 −
a0 b 3
(a0 a3 − a1 a2 )
a4
a0 (a0 a23 + a21 a4 − a1 a2 a3 )
JE
b0 (−a1 a4 + a2 a3 ) − a0 a3 b1 + a0 a1 b2 +
= πi
= πi
JE
M5
,
a0 Δ5
where
M5 = b0 −a0 a4 a5 + a1 a24 + a22 a5 − a2 a3 a4 + a0 b1 (−a2 a5 + a3 a4 )
a0 b 4 −a0 a1 a5 + a0 a23 + a21 a4 − a1 a2 a3 ,
+ a0 b2 (a0 a5 − a1 a4 ) + a0 b3 (−a0 a3 + a1 a2 ) +
a5
Δ5 = a20 a25 − 2a0 a1 a4 a5 − a0 a2 a3 a5 + a0 a23 a4 + a21 a24 + a1 a22 a5 − a1 a2 a3 a4
JE
254
Power and Algebraic Functions
3.121
3.12 Products of rational functions and expressions that can be reduced to square
roots of ﬁrst- and second-degree polynomials
3.121
1
1.
0
2.
3.
∞
sin kλ
1
dx
√
=
2
cosec
λ
1 − 2x cos λ + x2 x
2k − 1
BI (10)(17)
k=1
1
dx
1
π
[0 < p < q]
=
q
−
px
q(q − p)
x(1 − x)
0
1
π
11∓r
1+r
dx
1∓x
=± ∓
arctan
2
1
−
2rx
+
r
1
±
x
4r
r
1
±
r
1−r
0
BI (10)(9)
LI (14)(5, 16)
3.13–3.17 Expressions that can be reduced to square roots of third- and fourthdegree polynomials and their products with rational functions
Notation: In 3.131–3.137 we set: α = arcsin
c−u
,
b−u
.
(a − c)(b − u)
,
(b − c)(a − u)
.
(a − c)(u − b)
a−u
,
λ = arcsin
,
κ = arcsin
(a − b)(u − c)
a−b
u−a
a−c
a−b
μ = arcsin
,
ν = arcsin
,
p=
,
u−b
u−c
a−c
γ = arcsin
3.131
1.
2.
3.
4.
5.
6.
7.
u−c
,
b−c
a−c
, β = arcsin
a−u
δ = arcsin
q=
b−c
.
a−c
u
dx
2
=√
F (α, p)
a−c
(a − x)(b − x)(c − x)
−∞
c
dx
2
=√
F (β, p)
a−c
(a − x)(b − x)(c − x)
u
u
dx
2
=√
F (γ, q)
a−c
(a − x)(b − x)(x − c)
c
b
dx
2
=√
F (δ, q)
a−c
(a − x)(b − x)(x − c)
u
u
dx
2
=√
F (κ, p)
a−c
(a − x)(x − b)(x − c)
b
a
dx
2
=√
F (λ, p)
a−c
(a − x)(x − b)(x − c)
u
u
dx
2
=√
F (μ, q)
a
−c
(x − a)(x − b)(x − c)
a
[a > b > c ≥ u]
BY (231.00)
[a > b > c > u]
BY (232.00)
[a > b ≥ u > c]
BY (233.00)
[a > b > u ≥ c]
BY (234.00)
[a ≥ u > b > c]
BY (235.00)
[a > u ≥ b > c]
BY (236.00)
[u > a > b > c]
BY (237.00)
3.133
Square roots of polynomials
∞
8.
u
3.132
c
1.
u
dx
2
=√
F (ν, q)
a
−c
(x − a)(x − b)(x − c)
x dx
2
=√
[c F (β, p) + (a − c) E (β, p)] − 2
a−c
(a − x)(b − x)(c − x)
c
b
u
BY (232.19)
√
2a
=√
F (γ, q) − 2 a − c E (γ, q)
a−c
(a − x)(b − x)(x − c)
x dx
BY (233.17)
2
=√
(b − a) Π δ, q 2 , q + a F (δ, q)
a−c
(a − x)(b − x)(x − c)
x dx
b
x dx
(a − x)(x − b)(x − c)
2
=√
a−c
BY (234.16)
(b − c) Π κ, p2 , p + c F (κ, p)
[a ≥ u > b > c]
a
5.
u
(a − u)(c − u)
b−u
[a > b > u ≥ c]
u
4.
BY (238.00)
[a > b ≥ u > c]
3.
[u ≥ a > b > c]
[a > b > c > u]
u
2.
255
BY (235.16)
√
x dx
2c
=√
F (λ, p) + 2 a − c E (λ, p)
a−c
(a − x)(x − b)(x − c)
[a > u ≥ b > c]
u
6.
a
BY (236.16)
2
= √
a(a − b) Π(μ, 1, q) + b2 F (μ, q)
b a−c
(x − a)(x − b)(x − c)
x dx
[u > a > b > c]
3.133 1.
BY (237.16)
u
dx
2
√
=
[F (α, p) − E (α, p)]
3
(a − b) a − c
(a − x) (b − x)(c − x)
−∞
[a > b > c ≥ u]
c
2.
u
dx
(a −
x)3 (b
− x)(c − x)
=
2
2
√
[F (β, p) − E (β, p)] +
a−c
(a − b) a − c
BY (231.08)
c−u
(a − u)(b − u)
[a > b > c > u]
BY (232.13)
u
dx
(b − u)(u − c)
2
2
√
=
E (γ, q) −
3
(a
−
b)(a
−
c)
a−u
(a
−
b)
a
−
c
(a − x) (b − x)(x − c)
c
BY (233.09)
[a > b ≥ u > c]
b
dx
2
√
=
BY (234.05)
E (δ, q) [a > b > u ≥ c]
3
(a − b) a − c
(a − x) (b − x)(x − c)
u
.
u
dx
u−b
2
2
√
=
[F (κ, p) − E (κ, p)] +
3
a − b (a − u)(u − c)
(a − b) a − c
(a − x) (x − b)(x − c)
b
3.
4.
5.
[a > u > b > c]
BY (235.04)
256
Power and Algebraic Functions
∞
6.
u
8.
.
dx
2
2
√
=
E (ν, q) +
3
a
−
b
(b − a) a − c
(x − a) (x − b)(x − c)
u−b
(u − a)(u − c)
[u > a > b > c]
√
2
dx
2 a−c
√
E (α, p) −
=
F (α, p)
3
(a − b) a − c
(a − x)(b − x) (c − x) (a − b)(b
−∞
− c)
c−u
2
−
b − c (a − u)(b − u)
[a > b > c ≥ u]
√
c
2
dx
2 a−c
√
E (β, p) −
=
F (β, p)
(a − b)(b − c)
(a − b) a − c
(a − x)(b − x)3 (c − x)
u
BY (238.05)
[a > b > c > u]
√
u
dx
2
2 a−c
√
E (γ, q)
=
F (γ, q) −
(a − b)(b − c)
(a − x)(b − x)3 (x − c) (b − c) a − c c
2
(a − u)(u − c)
+
(a − b)(b − c)
b−u
[a > b > u > c]
√
a
dx
2
2 a−c
√
E (λ, p)
=
F (λ, p) −
(a − b)(b − c)
(a − x)(x − b)3 (x − c) (a − b) a − c u
2
(a − u)(u − c)
+
(a − b)(b − c)
u−b
[a > u > b > c]
√
u
2
dx
2 a−c
√
E (μ, q) −
=
F (μ, q)
3
(a
−
b)(b
−
c)
(b
−
c)
a−c
(x − a)(x − b) (x − c)
a
BY (232.14)
[u > a > b > c]
√
2
dx
2 a−c
√
E (ν, q) −
=
F (ν, q)
3
(a
−
b)(b
−
c)
(b
−
c)
a−c
(x − a)(x − b) (x − c)
u
u−a
2
−
a − b (u − b)(u − c)
[u ≥ a > b > c]
.
u
dx
b−u
2
2
√
=
E (α, p) +
3
b
−
c
(a
−
u)(c − u)
(c − b) a − c
(a − x)(b − x)(c − x)
−∞
BY (237.12)
7.
3.133
u
9.
10.
11.
12.
13.
b
14.
u
15.
b
BY (233.10)
BY (236.09)
∞
[a > b > c > u]
BY (231.09)
u
dx
(a − x)(b − x)(x −
c)3
dx
(a − x)(x − b)(x −
c)3
=
=
2
2
√
[F (δ, q) − E (δ, q)] +
b−c
(b − c) a − c
.
BY (238.04)
BY (231.10)
b−u
(a − u)(u − c)
[a > b > u > c]
BY (234.04)
[a ≥ u > b > c]
BY (235.01)
2
√
E (κ, p)
(b − c) a − c
3.134
Square roots of polynomials
a
16.
u
dx
2
2
√
=
E (λ, p) −
(b − c)(a − c)
(b − c) a − c
(a − x)(x − b)(x − c)3
a
dx
2
2
√
[F (μ, q) − E (μ, q)] +
a−c
(b − c) a − c
=
(x − a)(x − b)(x − c)3
[u > a > b > c]
∞
18.
u
=
3(a −
+
c
2.
u
u
3.
dx
(a − x)5 (b − x)(c − x)
b
4.
u
=
dx
(a −
x)5 (b
(a −
x)5 (b
c
=
− x)(x − c)
dx
− x)(x − c)
=
5.
b
u
b)2
u−a
(u − b)(u − c)
BY (237.13)
BY (238.03)
2
[(3a − b − 2c) F (α, p) − 2(2a − b − c) E (α, p)]
3
(a − c)
.
2
3(a − c)(a − b)
(c − u)(b − u)
(a − u)3
[a > b > c ≥ u]
BY (231.08)
2
[(3a − b − 2c) F (β, p) − 2(2a − b − c) E (β, p)]
3(a− b)2 (a − c)3
2 4a2 − 3ab − 2ac + bc − u(3a − 2b − c)
c−u
+
3(a − b)(a − c)2
(a − u)3 (b − u)
BY (232.13)
[a > b > c > u]
2
[2(2a − b − c) E (γ, q) − (a − b) F (γ, q)]
3(a −
(a − c)3
.
2 5a2 − 3ab − 3ac + bc − 2u(2a − b − c)
(b − u)(u − c)
−
3(a − b)2 (a − c)2
(a − u)3
[a > b ≥ u > c]
BY (233.09)
b)3
3(a −
−
BY (236.10)
u
dx
5
(a − x) (b − x)(c − x)
−∞
dx
2
√
[F (ν, q) − E (ν, q)]
=
3
(b − c) a − c
(x − a)(x − b)(x − c)
[u ≥ a > b > c]
3.134 1.
(a − u)(u − b)
u−c
[a > u ≥ b > c]
u
17.
257
b)2
2
[2(2a − b − c) E (δ, q) − (a − b) F (δ, q)]
3
(a − c)
.
2
3(a − b)(a − c)
(b − u)(u − c)
(a − u)3
[a > b > u ≥ c]
BY (234.05)
dx
(a −
x)5 (x
− b)(x − c)
=
2
[(3a − b − 2c) F (κ, p) − 2(2a − b − c) E (κ, p)]
(a − c)3
3(a −
.
2 4a2 − 2ab − 3ac + bc − u(3a − b − 2c)
u−b
+
3(a − b)2 (a − c)
(a − u)3 (u − c)
BY (235.04)
[a > u > b > c]
b)2
258
Power and Algebraic Functions
∞
6.
u
7.
3.134
dx
2
=
[2(2a − b − c) E (ν, q) − (a − b) F (ν, q)]
5
2
(x − a) (x − b)(x − c) 3(a − b) (a − c)3
2
.
2 4a − 2ab − 3ac + bc + u(b + 2c − 3a)
u−b
+
3(a − b)2 (a − c)
(u − a)3 (u − c)
BY (238.05)
[u > a > b > c]
u
dx
2
√
=
2 (b − c)2 a − c
5
3(a
−
b)
(a − x)(b − x) (c − x)
−∞
× [2(a − c)(a + c − 2b) E (α, p) + (b − c)(3b − a − 2c) F (α, p)]
2 3ab − ac + 2bc − 4b2 − u(2a − 3b + c)
c−u
−
3(a − b)(b − c)2
(a − u)(b − u)3
[a > b > c ≥ u]
BY (231.09)
c
8.
u
dx
(a − x)(b −
u
c
a
u
a
3(a −
b)2 (b
2
√
− c)2 a − c
dx
(a − x)(b −
x)5 (x
− c)
=
3(a −
b)2 (b
2
√
− c)2 a − c
× [(a − b)(2a − 3b + c) F (γ, q) + 2(a − c)(2b − a − c) E (γ, q)]
.
2 3ab + 3bc − ac − 5b2 − 2u(a − 2b + c)
(a − u)(u − c)
+
3(a − b)2 (b − c)2
(b − u)3
[a > b > u > c]
BY (233.10)
10.
11.
− x)
=
× [(b − c)(3b − a − 2c) F (β, p) + 2(a − c)(a − 2b + c) E (β, p)]
.
(a − u)(c − u)
2
+
3(a − b)(b − c)
(b − u)3
[a > b > c > u]
BY (232.14)
9.
x)5 (c
dx
2
√
=
2 (b − c)2 a − c
5
3(a
−
b)
(a − x)(x − b) (x − c)
× [(b − c)(3b − 2c − a) F (λ, p) + 2(a − c)(a + c − 2b) E (λ, p)]
.
2 3ab + 3bc − ac − 5b2 + 2u(2b − a − c)
(a − u)(u − c)
+
2
2
3(a − b) (b − c)
(u − b)3
[a > u > b > c]
BY (236.09)
u
dx
(x − a)(x − b)5 (x − c)
=
2
√
3(a − b)2 (b − c)2 a − c
× [(a − b)(2a + c − 3b) F (μ, q) + 2(a − c)(2b − a − c) E (μ, q)]
.
(u − a)(u − c)
2
+
3(a − b)(b − c)
(u − b)3
[u > a > b > c]
BY (237.12)
3.134
Square roots of polynomials
∞
12.
u
13.
14.
15.
16.
17.
18.
259
dx
2
√
=
2 (b − c)2 a − c
5
3(a
−
b)
(x − a)(x − b) (x − c)
× [(a − b)(2a + c − 3b) F (ν, q) + 2(a − c)(2b − c − a) E (ν, q)]
2 3bc + 2ab − ac − 4b2 + u(3b − a − 2c)
u−a
−
3(a − b)2 (b − c)
(u − b)3 (u − c)
BY (238.04)
[u ≥ a > b > c]
u
dx
2
=
[2(a + b − 2c) E (α, p) − (b − c) F (α, p)]
5
2
(a − x)(b − x)(c − x)
3(b − c) (a − c)3
−∞
.
2 ab − 3ac − 2bc + 4c2 + u(2a + b − 3c)
b−u
+
2
3(a − c)(b − c)
(a − u)(c − u)3
[a > b > c > u]
By (231.10)
b
dx
2
=
[(2a + b − 3c) F (δ, q) − 2(a + b − 2c) E (δ, q)]
5
2
(a − x)(b − x)(x − c)
3(b − c) (a − c)3
u
.
2 ab − 3ac − 2bc + 4c2 + u(2a + b − 3c)
b−u
+
2
3(b − c) (a − c)
(a − u)(u − c)3
[a > b > u > c]
BY (234.04)
u
dx
2
=
[2(a + b − 2c) E (κ, p) − (b − c) F (κ, p)]
5
2 (a − c)3
(a
−
x)(x
−
b)(x
−
c)
3(b
−
c)
b
.
2
(a − u)(u − b)
+
3(a − c)(b − c)
(u − c)3
[a ≥ u > b > c]
BY (235.20)
a
dx
2
=
[2(a + b − 2c) E (λ, p) − (b − c) F (λ, p)]
5
2
(a
−
x)(x
−
b)(x
−
c)
3(b
−
c)
(a − c)3
u
.
2 ab − 3ac − 3bc + 5c2 + 2u(a + b − 2c)
(a − u)(u − b)
−
2
2
3(b − c) (a − c)
(u − c)3
[a > u ≥ b > c]
BY (236.10)
u
dx
2
=
[(2a + b − 3c) F (μ, q) − 2(a + b − 2c) E (μ, q)]
5
2
(x − a)(x − b)(x − c)
3(b − c) (a − c)3
a
2 4c2 − ab − 2ac − bc + u(3a + 2b − 5c)
u−a
+
2
3(b − c)(a − c)
(u − b)(u − c)3
[u > a > b > c]
BY (237.13)
∞
dx
2
=
[(2a + b − 3c) F (ν, q) − 2(a + b − 2c) E (ν, q)]
5
2
3
(x − a)(x − b)(x − c)
3(b − c) (a − c)
u
.
(u − a)(u − b)
2
+
3(a − c)(b − c)
(u − c)3
[u ≥ a > b > c]
BY (238.03)
260
3.135
1.6
2.
3.
4.
5.
6.
7.
Power and Algebraic Functions
u
3.135
dx
2
√
=
[(b − c) F (α, p) − (2a − b − c) E (α, p)]
3 (c − x)3
(a
−
b)(b
−
c)2 a − c
(a
−
x)(b
−
x)
−∞
2(b + c − 2u)
+
2
(b − c) (a − u)(b − u)(c − u)
BY (231.13)
[a > b > c > u]
a
dx
2
√
=
[(b − c) F (λ, p) − 2(2a − b − c) E (λ, p)]
3
3
(a
−
b)(b
−
c)2 a − c (a − x)(x − b) (x − c)
u
a−u
2(a − b − c + u)
+
(a − b)(b − c)(a − c) (u − b)(u − c)
BY (236.15)
[a > u > b > c]
u
dx
2
√
=
[(2a − b − c) E (μ, q) − 2(a − b) F (μ, q)]
3
3
(a − b)(b − c)2 a−c
(x − a)(x − b) (x − c)
a
u−a
2
+
(a − c)(b − c) (u − b)(u − c)
BY (236.14)
[u > a > b > c]
∞
dx
2
√
=
[(2a − b − c) E (ν, q) − 2(a − b) F (ν, q)]
3
3
(a − b)(b − c)2 a−c
(x − a)(x − b) (x − c)
u
u−a
2
−
(a − b)(b − c) (u − b)(u − c)
BY (238.13)
[u ≥ a > b > c]
u
dx
2
=
[(2b − a − c) E (α, p) − (b − c) F (α, p)]
3
3
(a − x) (b − x)(c − x)
(a − b)(b − c) .
(a − c)3
−∞
b−u
2
+
(b − c)(a − c) (a − u)(c − u)
BY(231.12)
[a > b > c > u]
b
dx
2
=
[(a − b) F (δ, q) + (2b − a − c) E (δ, q)]
3
3
(a − x) (b − x)(x − c)
(b − c)(a − b) .
(a − c)3
u
b−u
2
+
(b − c)(a − c) (a − u)(u − c)
BY (234.03)
[a > b > u > c]
u
dx
2
=
[(b − c) F (κ, p) − (2b − a − c) E (κ, p)]
3
3
(a − x) (x − b)(x − c)
(a − b)(b − c) .
(a − c)3
b
u−b
2
+
(a − b)(a − c) (a − u)(u − c)
BY (235.15)
[a > u > b > c]
3.136
Square roots of polynomials
∞
8.
u
dx
2
=
[(a + c − 2b) E (ν, q) − (a − b) F (ν, q)]
3
3
(x − a) (x − b)(x − c)
(a − b)(b − c) .
(a − c)3
+
9.
10.
11.
12.
3.136
1.
u
u
261
2
(a − b)(a − c)
u−b
(u − a)(u − c)
[u > a > b > c]
BY (238.14)
dx
2
√
=
[(a + b − 2c) E (α, p) − 2(b − c) F (α, p)]
3
3
(b
−
c)(a
−
b)2 a−c
(a − x) (b − x) (c − x)
−∞
c−u
2
−
(a − b)(b − c) (a − u)(b − u)
BY (231.11)
[a > b > c ≥ u]
c
dx
2
√
=
[(a + b − 2c) E (β, p) − 2(b − c) F (β, p)]
2 (b − c) a − c
3
3
(a
−
b)
(a − x) (b − x) (c − x)
u
c−u
2
+
(a − b)(a − c) (a − u)(b − u)
BY (232.15)
[a > b > c > u]
u
dx
2
√
=
[(a − b) F (γ, q) − (a + b − 2c) E (γ, q)]
2 (b − c) a − c
3
3
(a
−
b)
(a − x) (b − x) (x − c)
c
2
2
2 a + b − ac − bc − u(a + b − 2c)
u−c
+
(a − b)2 (b − c)(a − c)
(a − u)(b − u)
BY (233.11)
[a > b > u > c]
∞
dx
2
√
=
[(a − b) F (ν, q) − (a + b − 2c) E (ν, q)]
2
3
3
(x − a) (x − b) (x − c) (a − b) (b − c) a − c
u
2u − a − b
+
2
(a − b) (u − a)(u − b)(u − c)
BY (238.15)
[u > a > b > c]
dx
3
(a − x) (b − x)3 (c − x)3
−∞
=
2
(a −
− c)2 (a − c)3
× (b − c)(a + b − 2c) F (α, p) − 2 c2 + a2 + b2 − ab − ac − bc E (α, p)
b)2 (b
+
2[c(a − c) + b(a − b) − u(2a − c − b)]
(a − b)(a − c)(b − c)2 (a − u)(b − u)(c − u)
[a > b > c > u]
BY (231.14)
262
Power and Algebraic Functions
∞
2.
u
dx
3
(x − a) (x − b)3 (x − c)3
=
1.6
(a −
− c)2 (a − c)3
× (a − b)(2a − b − c) F (ν, q) − 2 a2 + b2 + c2 − ab − ac − bc E (ν, q)
2[u(a + b − 2c) − a(a − c) − b(b − c)]
(a − b)2 (a − c)(b − c) (u − a)(u − b)(u − c)
[u > a > b > c]
BY (238.16)
dx
2
a−r
√
, p − F (α, p)
=
Π α,
a−c
(a − r) a − c
−∞ (r − x) (a − x)(b − x)(c − x)
u
c
[a > b > c ≥ u]
2.
u
u
3.
c
b
u
dx
2
√
=
(r − x) (a − x)(b − x)(x − c) (r− a)(r − b) a − c
r−a
, q + (r − b) F (δ, q)
× (b − a) Π δ, q 2
r−b
[a > b > u ≥ c, r = b]
BY (233.02)
BY (234.18)
dx
2
a−b
√
,p
=
Π λ,
a
−r
(a
−
r)
a
−
c
(x − r) (a − x)(x − b)(x − c)
u
a
r = c]
a
8
7.
dx
2
b−c
√
,q
=
Π γ,
r−c
(r − c) a − c
(r − x) (a − x)(b − x)(x − c)
dx
2
√
=
(c
−
r)(b
−
r) a − c
(x − r) (a − x)(x − b)(x − c)
2c− r
, p + (b − r) F (κ, p)
× (c − b) Π κ, p
b−r
[a ≥ u > b > c, r = b]
BY (235.17)
b
BY (232.17)
u
5.
BY (231.15)
dx
2(c − b)
√
=
(r − x) (a − x)(b − x)(c − x) (r − b)(r − c) a − c
2
r−b
√
,p +
× Π β,
F (β, p)
r−c
(r − b) a − c
[a > b > c > u, r = 0]
[a > b ≥ u > c,
4.
6.
2
b)2 (b
+
3.137
3.137
[a > u ≥ b > c,
u
r = a]
dx
2
√
=
(x − r) (x − a)(x − b)(x − c) (b− r)(a − r) a − c
b−r
, q + (a − p) F (μ, q)
× (b − a) Π μ,
a−b
[u > a > b > c, r = a]
BY (236.02)
BY (237.17)
3.139
Square roots of polynomials
∞
8.
u
2
r−c
√
, q − F (ν, q)
=
Π ν,
a−c
(r − c) a − c
(x − r) (x − a)(x − b)(x − c)
dx
[u ≥ a > b > c]
3.138
u
1.
0
2.
3.
4.
5.
dx
x(1 − x) (1 − k 2 x)
√
= 2 F arcsin u, k
[0 < u < 1]
√
dx
[0 < u < 1]
2
= 2 F arccos u, k
u
x(1 − x) k + k 2 x
√
1
dx
1−u
<
,k
[0 < u < 1]
=
2
F
arcsin
k
2
u
x(1 − x) x − k
u
√
dx
<
[0 < u < 1]
= 2 F arctan u, k
2
0
x(1 + x) 1 + k x
u
√
dx
< = F 2 arctan u, k
2
0
x 1 + x2 + 2 k − k 2 x
1
<
[0 < u < 1]
1
6.
u
u
7.
a
a
8.
u
263
dx
< =F
x k 2 (1 + x2 ) + 2 (1 + k 2 ) x
dx
(x − α) [(x −
m)2
+
n2 ]
1
=√ F
p
dx
1
=√ F
2
2
p
(α − x) [(x − m) + n ]
BY (238.06)
PE (532), JA
PE(533)
PE (534)
PE (535)
JA
√
π
− 2 arctan u, k
2
2 arctan
2 arccot
[0 < u < 1]
u−α
p+m−α
,
p
2p
JA
[α < u] ,
α−u
p−m+α
,
p
2p
[u < α] ,
where p = (m − α)2 + n2 .
√
1− 3−u
√
,
3.139 Notation α = arccos
1+ 3−u
√
3+1−u
,
γ = arccos √
3−1+u
u
dx
1
√
F (α, sin 75◦ )
= √
1.
4
3
3
−∞ 1 − x
1
dx
1
√
F (β, sin 75◦ )
2.
= √
4
3
3
1
−
x
u
√
3−1+u
β = arccos √
,
3+1−u
√
u−1− 3
√ .
δ = arccos
u−1+ 3
H 66 (285)
H 65 (284)
264
3.
4.
5.
6.
7.
8.
9.
10.
11.
Power and Algebraic Functions
u
∞
3.139
dx
1
√
F (γ, sin 15◦ )
H 65 (283)
= √
4
3
3
x −1
1
∞
dx
1
√
F (δ, sin 15◦ )
H 65 (282)
= √
4
3
3
x −1
u
3
1
dx
1
1
√ √
√
=
MO 9
Γ
3
2π 3 3 2
1 − x3
0
√
3
1
x dx
2
1 3
√
= √
MO 9
Γ
π 34
3
1 − x3
0
1
1 √
4
1 − x3 dx =
27 F (β, sin 75◦ ) − 2u 1 − u3
BY (244.01)
5
u
√
1
√
1
1
x dx
2 1 − u3
4
√
= 3− 4 − 3 4 F (β, sin 75◦ ) + 2 3 E (β, sin 75◦ ) − √
BY (244.05)
3+1−u
1 − x3
u
√
1 m
2(m − 2) 1 xm−3 dx
x dx
2um−2 1 − u3
√
√
+
=
BY (244.07)
2m − 1
2m − 1 u 1 − x3
1 − x3
u
√
u
√
1
x dx
2 u3 − 1
4
− 14
◦
◦
4
√
BY (240.05)
= 3 + 3 F (γ, sin 15 ) − 2 3 E (γ, sin 15 ) + √
3−1+u
x3 − 1
1
√
u
dx
1 + u + u2
1
2
◦
◦
√
√
√
√
[F
(α,
sin
75
= √
)
−
2
E
(α,
sin
75
)]
+
4
3
27
3 1+ 3−u 1−u
−∞ (1 − x) 1 − x
[u = 1]
12.
u
BY (246.06)
√
dx
1 + u + u2
2
1
◦
◦
√ √
√
[F (δ, sin 15 ) − 2 E (δ, sin 15 )] + √ = √
4
27
3 u−1+ 3 u−1
(x − 1) x3 − 1
[u = 1]
√
u
(1 − x) dx
2− 3
[F (α, sin 75◦ ) − E (α, sin 75◦ )]
= √
√
2 √
4
27
−∞ 1 +
3−x
1 − x3
√
1
(1 − x) dx
2− 3
√
[F (β, sin 75◦ ) − E (β, sin 75◦ )]
= 4
√
2 √
27
u 1+
3−x
1 − x3
√
√
√
u
2 3−2
2− 3
(x − 1) dx
u3 − 1
√
√
− 4
E (γ, sin 15◦ )
=
√
2 √
u2 − 2u − 2
3
27
1
1+ 3−x
x3 − 1
√ √
√
∞
2 2− 3
2− 3
(x − 1) dx
u3 − 1
√
√
−
E (δ, sin 15◦ )
=
√
2 √
4
2 − 2u − 2
u
3
27
3
u
1+ 3−x
x −1
&
√ % √ √
u
(1 − x) dx
2 + 3 2 4 3 1 − u3
− E (α, sin 75◦ )
= √
√
2 √
4
u2 − 2u − 2
27
−∞ 1 −
3−x
1 − x3
√
u
(x − 1) dx
2+ 3
√
[F (γ, sin 15◦ ) − E (γ, sin 15◦ )]
= 4
√
2 √
3
27
1
1− 3−x
x −1
BY (242.03)
13.
14.
15.
16.
17.
18.
BY (246.07)
BY (244.04)
BY (240.08)
BY (242.07)
BY (246.08)
BY (240.04)
3.141
Square roots of polynomials
√
(x − 1) dx
2+ 3
[F (δ, sin 15◦ ) − E (δ, sin 15◦ )]
= √
√
2 √
4
3
27
u
1− 3−x
x −1
2
u
x + x + 1 dx
1
E (α, sin 75◦ )
= √
√
2 √
4
3
−∞ 1 +
3−x
1 − x3
2
1
x + x + 1 dx
1
E (β, sin 75◦ )
= √
√ 2 √
4
3
3
u x−1+
3
1−x
2
u
x + x + 1 dx
1
E (γ, sin 15◦ )
= √
√
2 √
4
3
3
1
3+x−1
x −1
2
∞
x + x + 1 dx
1
E (δ, sin 15◦ )
= √
√ 2 √
4
3
3
u
x−1+ 3
x −1
√
u
(x − 1) dx
2+ 3
4
◦
√
E (γ, sin 15 ) − √
F (γ, sin 15◦ )
= √
4
4
2
3
27 √
27
1 (x + x + 1) x − 1
√
2 − 3 2(u − 1) 3 + 1 − u
√
√
− √
3
3 − 1 + u u3 − 1
19.
20.
21.
22.
23.
24.
265
∞
BY (242.05)
BY (246.01)
BY (244.02)
BY (240.01)
BY (242.01)
BY (240.09)
√
2
u
1 + 3 − x dx
1
,√
+
Π α, p2 , sin 75◦
= √
√
√
2
4
3
−∞
1 + 3 − x − 4 3p2 (1 − x) 1 − x3
25.
1
26.
u
u
27.
1
u
3.141
√
2
1 + 3 − x dx
1
,√
+
Π β, p2 , sin 75◦
= √
√
√
2
4
3
1 + 3 − x − 4 3p2 (1 − x) 1 − x3
BY (244.03)
√
2
1 − 3 − x dx
1
,√
+
Π γ, p2 , sin 15◦
= √
√
√
2
4
3
1 − 3 − x − 4 3p2 (x − 1) x3 − 1
BY (240.02)
√
2
1 − 3 − x dx
1
,√
+
Π δ, p2 , sin 15◦
= √
√
√
2
4
3
1 − 3 − x − 4 3p2 (x − 1) x3 − 1
BY (242.02)
∞
28.
BY (246.02)
Notation: In 3.141 and 3.142 we set:
a−c
c−u
u−c
,
β = arcsin
,
γ = arcsin
α = arcsin
a−u
b−u
b−c
.
.
(a − c)(b − u)
(a − c)(u − b)
a−u
,
κ = arcsin
,
λ = arcsin
,
δ = arcsin
(b − c)(a − u)
(a − b)(u − c)
a−b
u−a
a−c
a−b
b−c
μ = arcsin
,
ν = arcsin
,
p=
,
q=
.
u−b
u−c
a−c
a−c
266
Power and Algebraic Functions
√
a−x
(a − u)(c − u)
dx = 2 a − c [F (β, p) − E (β, p)] + 2
(b − x)(c − x)
b−u
c
1.
u
u
c
√
a−x
dx = 2 a − c E (γ, q)
(b − x)(x − c)
u
√
a−x
dx = 2 a − c E (δ, q) − 2
(b − x)(x − c)
2.
b
3.
u
4.
b
a
u
u
6.
a
7.
u
u.
8.
c
u
BY (233.01)
(b − u)(u − c)
a−u
[a > b > u ≥ c]
√
a−x
(a − u)(u − b)
dx = 2 a − c [F (κ, p) − E (κ, p)] + 2
(x − b)(x − c)
u−c
BY (234.06)
[a ≥ u > b > c]
BY (235.07)
√
x−a
dx = −2 a − c E (μ, q) + 2
(x − b)(x − c)
[a > u ≥ b > c]
BY (236.04)
(u − a)(u − c)
u−b
[a > b > c > u]
BY (237.03)
BY (232.07)
√
b−x
2(a − b)
dx = 2 a − c E (γ, q) − √
F (γ, q)
(a − x)(x − c)
a−c
BY (233.04)
[a > b > u ≥ c]
√
x−b
(a − u)(u − b)
2(b − c)
dx = 2 a − c E (κ, p) − √
F (κ, p) − 2
(a − x)(x − c)
u−c
a−c
BY (234.07)
10.
b
a.
11.
u
u.
a
[a > b ≥ u > c]
[a > b ≥ u > c]
√
b−x
(b − u)(u − c)
2(a − b)
dx = 2 a − c E (δ, q) − √
F (δ, q) − 2
(a − x)(x − c)
a−u
a−c
u.
12.
BY (232.06)
[u > a > b > c]
√
2(b − c)
b−x
(a − u)(c − u)
dx = √
F (β, p) − 2 a − c E (β, p) + 2
(a − x)(c − x)
b−u
a−c
c.
9.
[a > b > c > u]
√
a−x
dx = 2 a − c [F (λ, p) − E (λ, p)]
(x − b)(x − c)
5.
b.
3.141
[a ≥ u > b > c]
BY (235.06)
√
x−b
2(b − c)
dx = 2 a − c E (λ, p) − √
F (λ, p)
(a − x)(x − c)
a−c
[a > u ≥ b > c]
√
2(a − b)
x−b
(u − a)(u − c)
dx = √
F (μ, q) − 2 a − c E (μ, q) + 2
(x − a)(x − c)
u−b
a−c
[u > a > b > c]
BY (236.03)
BY (237.04)
3.141
Square roots of polynomials
c
√
c−x
dx = −2 a − c E (β, p) + 2
(a − x)(b − x)
13.
u
u
14.
c
b
15.
u
16.
b
a
17.
u
u
18.
a
c
19.
u
u
c
b
21.11
22.
(a − u)(c − u)
b−u
[a > b > c > u]
BY (232.08)
√
x−c
dx = 2 a − c [F (γ, q) − E (γ, q)]
(a − x)(b − x)
[a > b ≥ u > c]
√
x−c
(b − u)(u − c)
dx = 2 a − c [F (δ, q) − E (δ, q)] + 2
(a − x)(b − x)
a−u
u
20.
267
√
x−c
dx = 2 a − c E (κ, p) − 2
(a − x)(x − b)
√
x−c
dx = 2 a − c E (λ, p)
(a − x)(x − b)
[a > b > u ≥ c]
BY (233.03)
BY (234.08)
(a − u)(u − b)
u−c
[a ≥ u > b > c]
BY (235.07)
[a > u ≥ b > c]
BY (236.01)
√
x−c
(u − a)(u − c)
dx = 2 a − c [F (μ, q) − E (μ, q)] + 2
(x − a)(x − b)
u−b
[u > a > b > c]
BY (237.05)
2√
(b − x)(c − x)
dx =
a − c [(2a − b − c) E (β, p) − (b − c) F (β, p)]
a−x
3
(a − u)(c − u)
2
+ (2b − 2a + c − u)
3
b−u
[a > b > c > u]
BY (232.11)
2√
(x − c)(b − x)
dx =
a − c [(2a − b − c) E (γ, q) − 2(a − b) F (γ, q)]
a−x
3
2
−
(a − u)(b − u)(u − c)
3
[a > b ≥ u > c]
BY (233.06)
2√
(x − c)(b − x)
dx =
a − c [2(b − a) F (δ, q) + (2a − b − c) E (δ, q)]
a−x
3
u
(b − u)(u − c)
2
+ (b + c − a − u)
3
a−u
[a > b > u ≥ c]
u
2√
(x − b)(x − c)
dx =
a − c [(2a − b − c) E (κ, p) − (b − c) F (κ, p)]
a−x
3
b
(a − u)(u − b)
2
+ (b + 2c − 2a − u)
3
u−c
[a ≥ u > b > c]
BY (234.11)
BY (235.10)
268
Power and Algebraic Functions
a
23.
11
u
u
24.
a
c
25.
u
u
26.
c
b
27.
28.
29.
30.11
3.141
2√
(x − b)(x − c)
dx =
a − c [(2a − b − c) E (λ, p) − (b − c) F (λ, p)]
a−x
3
2
+
(a − u)(u − b)(u − c)
3
[a > u ≥ b > c]
BY (236.07)
2√
(x − b)(x − c)
dx =
a − c [2(a − b) F (μ, q) + (b + c − 2a) E (μ, q)]
x−a
3
(u − a)(u − b)
2
+ (u + 2a − 2b − c)
3
u−c
[u > a > b > c]
BY (237.08)
2√
(a − x)(c − x)
dx =
a − c [(2b − a − c) E (β, p) − (b − c) F (β, p)]
b−x
3
(a − u)(c − u)
2
+ (a + c − b − u)
3
b−u
[a > b > c > u]
BY (232.10)
2√
(a − x)(x − c)
dx =
a − c [(2b − a − c) E (γ, q) + (a − b) F (γ, q)]
b−x
3
2
−
(a − u)(b − u)(u − c)
3
[a > b ≥ u > c]
BY (233.05)
2√
(a − x)(x − c)
dx =
a − c [(a − b) F (δ, q) + (2b − a − c) E (δ, q)]
b−x
3
u
(b − u)(u − c)
2
+ (2a + c − 2b − u)
3
a−u
[a > b > u ≥ c]
u
2√
(a − x)(x − c)
dx =
a − c [(b − c) F (κ, p) + (a + c − 2b) E (κ, p)]
x−b
3
b
(a − u)(u − b)
2
+ (2b − a − 2c + u)
3
u−c
[a ≥ u > b > c]
a
2√
(a − x)(x − c)
dx =
a − c [(a + c − 2b) E (λ, p) + (b − c) F (λ, p)]
x−b
3
u
2
−
(a − u)(u − b)(u − c)
3
[a > u ≥ b > c]
u
2√
(x − a)(x − c)
dx =
a − c [(a + c − 2b) E (μ, q) − (a − b) F (μ, q)]
x
−
b
3
a
(u − a)(u − c)
2
+ (u + b − a − c)
3
u−b
[u > a > b > c]
BY (234.10)
BY (235.11)
BY (236.06)
BY (237.06)
3.142
Square roots of polynomials
c
31.
u
u
32.
c
b
33.
34.
35.
36.
3.142
2√
(a − x)(b − x)
dx =
a − c [2(b − c) F (β, p) + (2c − a − b) E (β, p)]
c−x
3
(a − u)(c − u)
2
+ (a + 2b − 2c − u)
3
b−u
[a > b > c > u]
BY (232.09)
2√
(a − x)(b − x)
dx =
a − c [(a + b − 2c) E (γ, q) − (a − b) F (γ, q)]
x−c
3
2
+
(a − u)(b − u)(u − c)
3
[a > b ≥ u > c]
BY (233.07)
2√
(a − x)(b − x)
dx =
a − c [(a + b − 2c) E (δ, q) − (a − b) F (δ, q)]
x
−
c
3
u
(b − u)(u − c)
2
+ (2c − 2a − b + u)
3
a−u
[a > b > u ≥ c]
u
2√
(a − x)(x − b)
dx =
a − c [(a + b − 2c) E (κ, p) − 2(b − c) F (κ, p)]
x
−
c
3
b
(a − u)(u − b)
2
+ (u + c − a − b)
3
u−c
[a ≥ u > b > c]
a
2√
(a − x)(x − b)
dx =
a − c [(a + b − 2c) E (λ, p) − 2(b − c) F (λ, p)]
x−c
3
u
2
−
(a − u)(u − b)(u − c)
3
[a > u ≥ b > c]
u
2√
(x − a)(x − b)
dx =
a − c [(a + b − 2c) E (μ, q) − (a − b) F (μ, q)]
x−c
3
a
(u − a)(u − c)
2
+ (u + 2c − a − 2b)
3
u−b
[u > a > b > c]
u
1.
−∞
b
2.
u
+2
3.
b
BY (234.09)
BY (235.09)
BY (236.05)
BY (237.07)
.
√
a−x
b−u
2(a − c)
2
2 a−c
E (α, p) +
dx = √
F (α, p) −
(b − x)(c − x)3
b−c
b−c
(a − u)(c − u)
a−c
[a > b > c > u]
√
a−x
2 a−c
a−b
√
E (δ, q)
F (δ, q) −
dx = 2
3
(b − x)(x − c)
b−c
(b − c) .a − c
u
269
a−c
b−c
BY (231.05)
b−u
(a − u)(u − c)
[a > b > u > c]
√
2
a−x
2 a−c
E (κ, p) − √
dx =
F (κ, p)
(x − b)(x − c)3
b−c
a−c
[a ≥ u > b > c]
BY (234.13)
BY (235.12)
270
Power and Algebraic Functions
√
2
a−x
(a − u)(u − b)
2 a−c
2
E (λ, p) − √
dx =
F (λ, p) −
(x − b)(x − c)3
b−c
b−c
u−c
a−c
a
4.
u
u
5.
a
BY (236.12)
[a > u ≥ b > c]
√
x−a
u−a
2(a − b)
2 a−c
√
E (μ, q) −
dx =
F (μ, q) − 2
3
(x − b)(x − c)
b−c
(u − b)(u − c)
(b − c) a − c
[u > a > b > c]
√
x−a
2(a − b)
2 a−c
√
E (ν, q) −
dx =
F (ν, q)
(x − b)(x − c)3
b−c
(b − c) a − c
BY (237.10)
[u ≥ a > b > c]
√
2 a−c
a−b
a−x
c−u
dx =
E (α, p) − 2
3
(b − x) (c − x)
b−c
b − c (a − u)(b − u)
BY (238.09)
[a > b > c ≥ u]
√
a−x
2 a−c
dx =
E (β, p)
[a > b > c > u]
3
(b − x) (c − x)
b−c
u
√
u
2 a−c
2
a−x
(a − u)(u − c)
dx =
[F (γ, q) − E (γ, q)] +
3 (x − c)
(b
−
x)
b
−
c
b
−
c
b−u
c
BY (231.03)
∞
6.
u
u
7.
−∞
c
8.
9.
3.142
a
10.
u
u
11.
a
u
[a > b > u > c]
√
2 a−c
2
a−x
(a − u)(u − c)
dx =
E (λ, p) +
(x − b)3 (x − c)
c−b
b−c
u−b
BY (233.15)
[a > u > b > c]
BY (236.11)
√
2 a−c
x−a
dx
=
[F (μ, q) − E (μ, q)]
(x − b)3 (x − c)
b−c
∞
12.
u
−∞
c.
u
u.
15.
c
b.
u
2
b−x
dx = √
E (α, p)
(a − x)3 (c − x)
a−c
2
b−x
2(a − b)
dx = √
E (β, p) −
(a − x)3 (c − x)
a−c
a−c
14.
16.
[u > a > b > c]
√
2 a−c
x−a
u−a
dx =
[F (ν, q) − E (ν, q)] + 2
3
(x − b) (x − c)
b−c
(u − b)(u − c)
.
13.
BY (232.01)
[u ≥ a > b > c]
BY (238.10)
[a > b > c ≥ u]
BY (231.01)
c−u
(a − u)(b − u)
[a > b > c > u]
2
b−x
(b − u)(u − c)
2
dx = √
[F (γ, q) − E (γ, q)] +
(a − x)3 (x − c)
a−c
a−u
a−c
2
b−x
dx = √
[F (δ, q) − E (δ, q)]
(a − x)3 (x − c)
a−c
BY (237.09)
BY (232.05)
[a > b ≥ u > c]
BY (233.13)
[a > b > u ≥ c]
BY (234.15)
3.142
Square roots of polynomials
u.
17.
18.
.
u
19.
−∞
b.
20.
u
21.
b
22.
u
u.
23.
a
25.
∞
.
[a > b > c > u]
b−u
(a − u)(u − c)
[a > b > u > c]
BY (234.14)
x−b
2
dx = √
[F (κ, p) − E (κ, p)]
(a − x)(x − c)3
a−c
[a > u ≥ b > c]
x−b
2
b−c
dx = √
E (μ, q) − 2
(x − a)(x − c)3
a−c
a−c
u−a
(u − b)(u − c)
BY (237.11)
x−b
2
dx = √
E (ν, q)
[u ≥ a > b > c]
3
(x
−
a)(x
−
c)
a
−c
u
√
u 2 a−c
2(b − c)
c−x
√
dx
=
E (α, p) −
F (α, p)
3 (b − x)
(a
−
x)
a
−
b
(a
−
b) a − c
−∞
√
u
u
27.
c
b
28.
u
[a > b > c ≥ u]
2 a−c
2(b − c)
c−x
√
dx =
E (β, p) −
F (β, p) − 2
3
(a − x) (b − x)
a−b
(a − b) a − c
26.
BY (235.03)
BY (236.14)
[u > a > b > c]
.
c
BY (238.07)
BY (231.04)
[a ≥ u > b > c]
x−b
(a − u)(u − b)
2
2
√
dx
=
[F
(λ,
p)
−
E
(λ,
p)]
+
(a − x)(x − c)3
a−c
u−c
a−c
a.
BY (235.08)
[u > a > b > c]
.
b−x
b−u
2
dx = √
[F (α, p) − E (α, p)] + 2
(a − x)(c − x)3
(a
−
u)(c − u)
a−c
b−x
2
dx = − √
E (δ, q) + 2
(a − x)(x − c)3
a−c
u.
24.
.
u−b
(a
−
u)(u − c)
b
[a > u > b > c]
.
∞.
2
x−b
u−b
dx = √
[F (ν, q) − E (ν, q)] + 2
3
(x − a) (x − c)
(u − a)(u − c)
a−c
u
2
x−b
dx = − √
E (κ, p) + 2
(a − x)3 (x − c)
a−c
271
BY (238.01)
BY (231.07)
c−u
(a − u)(b − u)
[a > b > c > u]
BY (232.03)
2 a−c
2
x−c
(b − u)(u − c)
2
dx =
E (γ, q) − √
F (γ, q) −
3
(a − x) (b − x)
a−b
a−b
a−u
a−c
√
[a > b ≥ u > c]
BY (233.14)
x−c
2 a−c
2
dx =
E (δ, q) − √
F (δ, q)
(a − x)3 (b − x)
a−b
a−c
[a > b > u ≥ c]
BY (234.20)
√
272
Power and Algebraic Functions
u
29.
b
√
x−c
2(b − c)
2 a−c
√
dx =
E (κ, p)
F (κ, p) −
(a − x)3 (x − b)
a−b
(a − b) .
a−c
+2
a−c
a−b
u−b
(a − u)(u − c)
[a > u > b > c]
∞
30.
u
√
2
2(a − c)
x−c
2 a−c
√
dx
=
E (ν, q) +
F
(ν,
q)
−
(x − a)3 (x − b)
a−b
a−b
a−c
u
31.
−∞
c
32.
u
u
33.
c
a
34.
u
36.
3.143
1.
6
2.
3.144
1.
.
BY (235.13)
u−b
(u − a)(u − c)
[u > a > b > c]
√
c−x
c−u
2 a−c
[F (α, p) − E (α, p)] + 2
dx =
3
(a − x)(b − x)
a−b
(a − u)(b − u)
BY (238.08)
[a > b > c ≥ u]
BY (231.06)
[a > b > c > u]
2
x−c
(a − u)(u − c)
2 a−c
E (γ, q) +
dx = −
3
(a − x)(b − x)
a−b
a−b
b−u
BY (232.04)
[a > b > u > c]
√
2
x−c
(a − u)(u − c)
2 a−c
[F (λ, p) − E (λ, p)] +
dx =
(a − x)(x − b)3
a−b
a−b
u−b
BY (233.16)
√
c−x
2 a−c
[F (β, p) − E (β, p)]
dx =
(a − x)(b − x)3
a−b
√
[a > u > b > c]
√
x−c
2 a−c
E (μ, q)
[u > a > b > c]
dx =
(x − a)(x − b)3
a−b
a
√
∞
b−c
x−c
u−a
2 a−c
E (ν, q) − 2
dx =
3
(x − a)(x − b)
a−b
a − b (u − b)(u − c)
u
BY (236.13)
[u ≥ a > b > c]
BY (238.11)
u
35.
3.143
√ √
2 (1 − u) √
4
arctan
,2 2
2−1
(1 + u)
u
√ ∞
dx
2
u2 − 1
1
√
,
= F arccos 2
4
2
u
+
1
2
1
+
x
u
1
dx
1
√
= F
2
1 + x4
1+
1
Notation: α = arcsin √
.
2
u −u+1
√ ∞
dx
3
= F α,
2
2
x(x − 1) (x − x + 1)
u
[u ≥ 1]
BY (237.01)
H 66 (286)
H 66 (287)
BY (261.50)
3.144
Square roots of polynomials
∞
2.
u
∞
3.
u
∞
4.
u
5.
6.
2(2u − 1)
dx
273
=
− 4E
x3 (x − 1)3 (x2 − x + 1)
u(u − 1) (u2 − u + 1)
√ 3
α,
2
BY (261.54)
[u > 1]
&
√ √ 3
3
2u − 1
α,
− E α,
+ 2
2
2 u(u − 1) (u2 − u − 1)
% (2x − 1)2 dx
=4 F
x3 (x − 1)3 (x2 − x + 1)
[u > 1]
√ &
√ 3
3
α,
− E α,
2
2
% dx
4
<
=
F
3
3
x(x − 1) (x2 − x + 1)
[u ≥ 1]
√ 3
α,
2
∞
(2x − 1)2 dx
<
[u > 1]
= 4E
3
u
x(x − 1) (x2 − x + 1)
√ √ ∞.
x(x − 1)
3
3
4
1
α,
− F α,
3 dx = 3 E
2
2
3
2
(x − x + 1)
u
∞
7.
u
∞
8.
u
u
∞
10.
u
.
dx
(2x − 1)2
∞
9.
dx
(2x − 1)2
∞
11.
u
(2x −
1)2
BY (261.56)
BY (261.52)
BY (261.51)
BY (261.53)
[u > 1]
√ &
√ 1
3
3
u(u − 1)
α,
− E α,
+
2
2
2(2u − 1) u2 − u + 1
% 1
x(x − 1)
=
F
x2 − x + 1
3
x2 − x + 1
=E
x(x − 1)
[u > 1]
√
3
3
u(u − 1)
α,
−
2
2(2u − 1) u2 − u + 1
dx
4
= E
2
3
x(x − 1) (x − x + 1)
dx
40
E
=
5
5
2
3
x (x − 1) (x − x + 1)
dx
44
<
F
=
27
5
x(x − 1) (x2 − x + 1)
BY (261.57)
[u > 1]
BY (261.58)
√ √ 1
2
3
3
u(u − 1)
α,
− F α,
−
2
3
2
2u − 1 u2 − u + 1
[u > 1]
BY (261.55)
√ 2
√ 2(2u − 1) 9u − 9u − 1
3
3
4
α,
− F α,
− 2
3
2
3 u3 (u − 1)3 (u2 − u + 1)
[u > 1]
BY (261.54)
√ √ 2(2u − 1) u(u − 1)
3
3
56
<
α,
−
E α,
+
2
27
2
3
9 (u2 − u + 1)
[u > 1]
BY (261.52)
274
Power and Algebraic Functions
∞
12.
u
3.145
u
1.
α
u
2.
√ √ 3
3
16
1
E α,
F α,
=
−
4
2
27
2
27
2
(2x − 1) x(x − 1) (x − x + 1)
8 5u2 − 5u + 2
u(u − 1)
−
3
9(2u − 1)
u2 − u + 1
[u > 1]
dx
dx
1
=√ F
2
2
pq
(x − α)(x − β) [(x − m) + n ]
.
q(u − α) 1
2 arctan
,
p(u − β) 2
[β < α < u]
.
BY (261.55)
(p + q)2 + (α − β)2
pq
dx
(α − x)(x − β) [(x − m)2 + n2 ]
.
q(α − u) 1 −(p − q)2 + (α − β)2
2 arccot
,
p(u − β) 2
pq
[β < u < α]
.
.
β
dx
q(β − u) 1 (p + q)2 + (α − β)2
1
,
= √ F 2 arctan
pq
p(α − u) 2
pq
(x − α)(x − β) [(x − m)2 + n2 ]
u
[u < β < α]
β
1
=√ F
pq
3.
3.145
.
where (m − α)2 + n2 = p2 , and (m − β)2 + n2 = q 2 .∗
4.
Set
2
2
2
2
(m1 − m) + (n1 + n) = p2 ,
(m1 − m) + (n1 − n) = p21 ,
.
2
(p + p1 ) − 4n2
cot α =
2;
4n2 − (p − p1 )
then
u
dx
2
,= p+p F
+
1
m−n tan α
2
[(x − m)2 + n2 ] (xm1 ) + n21
α + arctan
√
u − m 2 pp1
,
n
p + p1
[m − n tan α < u < m + n cot α]
3.146
1
1.
0
2.
0
1
1
dx
π 1√
√
= +
2K
4
1+x
8
4
1 − x4
x2
dx
π
√
=
4
4
1+x
8
1−x
√ 2
2
BI (13)(6)
BI (13)(7)
∗ Formulas 3.145 are not valid for α + β = 2m. In this case, we make the substitution x − m = z, which leads to one of
the formulas in 3.152.
3.147
Square roots of polynomials
1
3.
0
1√
x4
dx
π
√
=− +
2K
4
1+x
8
4
1 − x4
√ 2
2
BI (13)(8)
.
3.147
Notation: In 3.147–3.151 we set: α = arcsin
.
β = arcsin
.
δ = arcsin
275
(a − c)(d − u)
,
(a − d)(c − u)
.
(a − c)(u − d)
,
(c − d)(a − u)
γ = arcsin
(b − d)(u − c)
,
(b − c)(u − d)
κ = arcsin
.
(b − d)(c − u)
,
(c − d)(b − u)
(a − c)(b − u)
,
(b − c)(a − u)
.
(a − c)(u − b)
(b − d)(a − u)
,
μ = arcsin
,
λ = arcsin
(a − b)(u − c)
(a − b)(u − d)
.
.
.
(b − d)(u − a)
(b − c)(a − d)
(a − b)(c − d)
,
q=
,
r=
.
ν = arcsin
(a − d)(u − b)
(a − c)(b − d)
(a − c)(b − d)
.
d
1.
u
dx
2
=
F (α, q)
(a − x)(b − x)(c − x)(d − x)
(a − c)(b − d)
[a > b > c > d > u]
u
2.
d
u
dx
(a − x)(b − x)(c − x)(x − d)
c
u
F (β, r)
=
2
(a − c)(b − d)
dx
(a − x)(b − x)(x − c)(x − d)
=
2
(a − c)(b − d)
dx
(a − x)(b − x)(x − c)(x − d)
=
2
(a − c)(b − d)
u
b
dx
(a − x)(x − b)(x − c)(x − d)
=
2
(a − c)(b − d)
a
u
BY (253.00)
F (δ, q)
BY (254.00)
F (κ, q)
BY (255.00)
F (λ, r)
[a ≥ u > b > c > d]
7.11
BY (254.00)
F (γ, r)
[a > b > u ≥ c > d]
6.
(a − c)(b − d)
[a > b ≥ u > c > d]
b
5.
2
[a > b > c > u ≥ d]
u
4.
(a − x)(b − x)(c − x)(x − d)
=
[a > b > c ≥ u > d]
c
3.
dx
BY (251.00)
BY (256.00)
dx
2
=
F (μ, r)
(a − x)(x − b)(x − c)(x − d)
(a − c)(b − d)
[a > u ≥ b > c > d]
BY (257.00)
276
Power and Algebraic Functions
u
8.
a
dx
(x − a)(x − b)(x − c)(x − d)
=
2
(a − c)(b − d)
3.148
F (ν, q)
[u > a > b > c > d]
3.148
1.
d
8
u
u
2.
x dx
=
(a − x)(b − x)(c − x)(d − x)
(a − c)(b − d)
d
c
3.
u
b
5.
u
6.
u
8
(a − x)(b − x)(c − x)(x − d)
a
7.
u
x dx
(a − x)(b − x)(x − c)(x − d)
x dx
(a − x)(b − x)(x − c)(x − d)
b
(a − x)(b − x)(c − x)(x − d)
x dx
c
x dx
=
=
2
(a − c)(b − d)
2
(a − c)(b − d)
a−d
, q + c F (α, q)
(d − c) Π α,
a−c
[a > b > c > d > u]
BY (251.03)
[a > b > c ≥ u > d]
BY (252.11)
d−c
, r + a F (β, r)
(d − a) Π β,
a−c
(c − b) Π γ,
c−d
, r + b F (γ, r)
b−d
[a > b > c > u ≥ d]
u
4.
2
x dx
(a − x)(x − b)(x − c)(x − d)
=
=
2
(a − c)(b − d)
2
(a − c)(b − d)
=
2
(a − c)(b − d)
x dx
2
=
(a − x)(x − b)(x − c)(x − d)
(a − c)(b − d)
(c − d) Π δ,
8.
a
x dx
(x − a)(x − b)(x − c)(x − d)
=
2
(a − c)(b − d)
[a > b ≥ u > c > d]
BY (254.10)
[a > b > u ≥ c > d]
BY (255.17)
[a ≥ u > b > c > d]
BY (256.11)
b−c
, q + a F (κ, q)
(b − a) Π κ,
a−c
a−b
, r + c F (λ, r)
(b − c) Π λ,
a−c
(a − d) Π μ,
b−a
, r + d F (μ, r)
b−d
(a − b) Π ν,
d
1.
u
dx
x (a − x)(b − x)(c − x)(d − x)
2
= cd (a − c)(b − d)
BY (257.11)
a−d
, q + b F (ν, q)
b−d
[u > a > b > c > d]
3.149
BY (253.11)
b−c
, q + d F (δ, q)
b−d
[a > u ≥ b > c > d]
u
BY (258.00)
BY (258.11)
c(a − d)
, q + d F (α, q)
(c − d) Π α,
d(a − c)
[a > b > c > d > u]
BY (251.04)
3.149
Square roots of polynomials
u
2.
d
dx
x (a − x)(b − x)(c − x)(x − d)
=
c
3.
u
2
ad (a − c)(b − d)
dx
x (a − x)(b − x)(c − x)(x − d)
=
u
4.
2
bc (a − c)(b − d)
277
a(d − c)
, r + d F (β, r)
d(a − c)
[a > b > c ≥ u > d]
BY (252.12)
(a − d) Π β,
b(c − d)
, r + c F (γ, r)
c(b − d)
[a > b > c > u ≥ d]
BY (253.12)
(b − c) Π γ,
dx
x (a − x)(b − x)(x − c)(x − d)
d(b − c)
= , q + c F (δ, q)
(d − c) Π δ,
c(b − d)
cd (a − c)(b − d)
[a > b ≥ u > c > d]
BY (254.11)
c
2
b
5.
u
dx
x (a − x)(b − x)(x − c)(x − d)
=
u
6.
2
a(b − c)
, q + b F (κ, q)
× (a − b) Π κ,
b(a − c)
ab (a − c)(b − d)
[a > b > u ≥ c > d]
BY (255.18)
dx
x (a − x)(x − b)(x − c)(x − d)
c(a − b)
= , r + b F (λ, r)
× (c − b) Π λ,
b(a − c)
bc (a − c)(b − d)
[a ≥ u > b > c > d]
BY (256.12)
b
2
a
7.
u
8.
a
u
dx
x (a − x)(x − b)(x − c)(x − d)
2
d(b − a)
=
, r + a F (μ, r)
× (d − a) Π μ,
a(b − d)
ad (a − c)(b − d)
[a > u ≥ b > c > d]
BY (257.12)
dx
x (x − a)(x − b)(x − c)(x − d)
=
2
ab (a − c)(b − d)
b(a − d)
, q + a F (ν, q)
a(b − d)
[u > a > b > c > d]
BY (258.12)
(b − a) Π ν,
278
Power and Algebraic Functions
3.151 1.
d
u
dx
(p − x) (a − x)(b − x)(c − x)(d − x)
=
u
2.
d
dx
(p − x) (a − x)(b − x)(c − x)(x − d)
=
c
3.
u
dx
2
=
(p − x) (a − x)(b − x)(c − x)(x − d) (p− b)(p − c) (a − c)(b − d)
(c − d)(p − b)
× (c − b) Π γ,
, r + (p − c) F (γ, r)
(b − d)(p − c)
[a > b > c > u ≥ d, p = c] BY (253.39)
b
dx
(p − x) (a − x)(b − x)(x − c)(x − d)
5.
u
=
b
2
(p− a)(p − d) (a − c)(b − d)
(d − c)(p − a)
× (d − a) Π β,
, r + (p − d) F (β, r)
(a − c)(p − d)
[a > b > c ≥ u > d, p = d] BY (252.39)
dx
2
=
(p − x) (a − x)(b − x)(x − c)(x − d) (p− c)(p − d) (a − c)(b − d)
(b − c)(p − d)
× (c − d) Π δ,
, q + (p − c) F (δ, q)
(b − d)(p − c)
[a > b ≥ u > c > d, p = c] BY (254.39)
c
6.
2
(p− c)(p − d) (a − c)(b − d)
(a − d)(p − c)
× (d − c) Π α,
, q + (p − d) F (α, q)
(a − c)(p − d)
[a > b > c > d > u, p = d] BY (251.39)
u
4.
3.151
u
2
(p− a)(p − b) (a − c)(b − d)
(b − c)(p − a)
× (b − a) Π κ,
, q + (p − b) F (κ, q)
(a − c)(p − b)
[a > b > u ≥ c > d, p = b] BY (255.38)
dx
(x − p) (a − x)(x − b)(x − c)(x − d)
=
2
(b− p)(p − c) (a − c)(b − d)
(a − b)(p − c)
× (b − c) Π λ,
, r + (p − b) F (λ, r)
(a − c)(p − b)
[a ≥ u > b > c > d, p = b] BY (256.39)
3.152
Square roots of polynomials
a
7.
u
dx
(p − x) (a − x)(x − b)(x − c)(x − d)
=
u
8.
a
2
(p− a)(p − d) (a − c)(b − d)
(b − a)(p − d)
× (a − d) Π μ,
, r + (p − a) F (μ, r)
(b − d)(p − a)
[a > u ≥ b > c > d, p = a] BY (257.39)
dx
(p − x) (x − a)(x − b)(x − c)(x − d)
=
3.152
Notation: In 3.152–3.163 we set:
279
2
(p− a)(p − b) (a − c)(b − d)
(a − d)(p − b)
× (a − b) Π ν,
, q + (p − a) F (ν, q)
(b − d)(p − a)
[u > a > b > c > d, p = a] BY (258.39)
u
α = arctan ,
b
β = arccot
u
a
b
a2 + b 2
u
ε = arccos ,
ξ = arcsin
δ = arccos ,
,
b
u
a2 + u 2
a b2 − u2
a u2 − b 2
ζ = arcsin
,
κ
=
arcsin
,
b a2 − u 2
u a2 − b 2
√
u 2 − a2
a2 − b 2
a
,
q
=
,
μ = arcsin
,
ν
=
arcsin
u2 − b 2
u
a
b
a
b
r= √
,
s= √
,
t= .
2
2
2
2
a
a +b
a +b
a2 + b 2
,
a2 + u 2
u
η = arcsin ,
b
2
a − u2
λ = arcsin
,
a2 − b 2
γ = arcsin
u
1.
∞
2.
u
u
3.
[a > b > 0]
H 62(258), BY (221.00)
dx
1
= F (β, q)
2
2
2
2
a
(x + a ) (x + b )
[a > b > 0]
H 63 (259), BY (222.00)
0
b
4.
u
u
5.
∞
6.
u
(x2
+
a2 ) (x2
+
b2 )
dx
(x2
+
a2 ) (b2
+
a2 ) (b2
−
1
=√
F (γ, r)
2
a + b2
[b ≥ u > 0]
H 63 (260)
−
x2 )
1
=√
F (δ, r)
2
a + b2
[b > u ≥ 0]
H 63 (261), BY (213.00)
1
=√
F (ε, s)
2
a + b2
[u > b > 0]
H 63 (262), BY (211.00)
[u > b > 0]
H 63 (263), BY (212.00)
dx
(x2
+
a2 ) (x2
=
x2 )
dx
(x2
b
dx
1
F (α, q)
a
0
u
b
−
b2 )
dx
1
=√
F (ξ, s)
2
2
2
2
2
a + b2
(x + a ) (x − b )
280
Power and Algebraic Functions
u
7.
=
1
F (η, t)
a
[a > b ≥ u > 0]
H 63 (264), BY (219.00)
=
1
F (ζ, t)
a
[a > b > u ≥ 0]
H 63 (265), BY (220.00)
=
1
F (κ, q)
a
[a ≥ u > b > 0]
H 63 (266), BY (217.00)
dx
1
= F (λ, q)
2
2
2
2
a
(a − x ) (x − b )
[a > u ≥ b > 0]
H 63 (257), BY (218.00)
1
F (μ, t)
a
[u > a > b > 0]
H 63 (268), BY (216.00)
dx
1
= F (ν, t)
a
(x2 − a2 ) (x2 − b2 )
[u ≥ a > b > 0]
H 64(269), BY (215.00)
0
b
8.
u
u
9.
a
10.
u
u
11.
∞
u
3.153
u
1.
0
u
2.
0
u
u
u
5.
b
6.
u
−
x2 ) (x2
−
b2 )
dx
(x2
−
a2 ) (x2
−
b2 )
x2 dx
(x2
+
a2 ) (x2
=
+
b2 )
x2 dx
(a2 + x2 ) (b2 − x2 )
=u
=
a2 + u 2
− a E (α, q)
b2 + u2
a2
+
b2
[u > 0,
a > b]
a2
E (γ, r) − √
F (γ, r) − u
a2 + b 2
x dx
+
x2 ) (b2
−
x2 )
=
b
a
8.
u
b 2 − u2
a2 + u 2
BY (214.05)
a
a2 + b2 E (δ, r) − √
F (δ, r)
2
a + b2
BY (213.06)
b2
1 2
=√
F (ε, s) − a2 + b2 E (ε, s) +
(u + a2 ) (u2 − b2 )
u
a2 + b 2
(a2 + x2 ) (x2 − b2 )
x2 dx
x2 dx
(a2 − x2 ) (b2 − x2 )
x2 dx
(a2
BY (221.09)
2
−
x2 ) (b2
−
x2 )
= a {F (η, t) − E (η, t)}
= a {F (ζ, t) − E (ζ, t)} + u
[u > b > 0]
BY (211.09)
[a > b ≥ u > 0]
BY (219.05)
b2 − u2
a2 − u 2
[a > b > u ≥ 0]
u
7.
dx
(a2
(a2
0
−
x2 )
2
b
−
x2 ) (b2
[b > u ≥ 0]
4.
−
x2 )
[b ≥ u > 0]
b
3.
−
x2 ) (b2
dx
a
12.
(a2
(a2
b
dx
3.153
x2 dx
(a2 − x2 ) (x2 − b2 )
= a E (κ, q) −
x2 dx
= a E (λ, q)
(a2 − x2 ) (x2 − b2 )
BY (220.06)
1 2
(a − u2 ) (u2 − b2 )
u
[a ≥ u > b > 0]
BY (217.05)
[a > u ≥ b > 0]
BY (218.06)
3.154
Square roots of polynomials
9.
u
6
a
1
10.
0
3.154
u
1.
0
x2 dx
(1 +
u
b
3.
u
4.
u
5.
b
6.
u
u
7.
b
1
k2
1 + k2
−E
2
π , 1 − k2
4
a
8.
u
BI (14)(9)
x2 ) (b2
+
−
x2 )
4
x dx
(a2
+
x2 ) (x2
−
b2 )
x4 dx
(a2
−
x2 ) (b2
−
x2 )
x4 dx
(a2
u > 0]
2
1
2a − b2 a2 F (γ, r) − 2 a4 − b4 E (γ, r)
= √
(a2 + x2 ) (b2 − x2 ) 3 a2 + b2
b2 − u2
u 2
2
2
− 2b − a + u
3
a2 + u 2
[a ≥ u > 0]
(a2
0
=
BY (216.06)
a2 + u 2
b2 + u2
BY (221.09)
x4 dx
x4 dx
b
+
k 2 x2 )
[u > a > b > 0]
u 2
a 2
2 a + b2 E (α, q) − b2 F (α, q) +
u − 2a2 − b2
=
3
3
(x2 + a2 ) (x2 + b2 )
u
x2 ) (1
u 2 − a2
u2 − b2
x4 dx
0
(x2 − a2 ) (x2 − b2 )
= a {F (μ, t) − E (μ, t)} + u
[a > b,
2.
x2 dx
281
−
x2 ) (b2
−
x2 )
2
1
2a − b2 a2 F (δ, r) − 2 a4 − b4 E (δ, r)
= √
2
2
3 a +b
u 2
+
(a + u2 ) (b2 − u2 )
3
[b > u ≥ 0]
BY (213.06)
2
1
2b − a2 b2 F (ε, s) + 2 a4 − b4 E (ε, s)
= √
2
2
3 a +b
2b2 − 2a2 + u2 2
+
(u + a2 ) (u2 − b2 )
3u
[u > b > 0]
BY (211.09)
=
=
4
x dx
(a2
−
x2 ) (x2
−
b2 )
BY (214.05)
=
u
a 2
2a + b2 F (η, t) − 2 a2 + b2 E (η, t) +
(a2 − u2 ) (b2 − u2 )
3
3
[a > b ≥ u > 0]
BY (219.05)
a 2
2a + b2 F (ζ, t) − 2 a2 + b2 E (ζ, t)
3
b2 − u2
u 2
2
2
+ u + a + 2b
3
a2 − u 2
[a > b > u ≥ 0]
BY (220.06)
a 2
2 a + b2 E (κ, q) − b2 F (κ, q)
3
u2 + 2a2 + 2b2 2
(a − u2 ) (u2 − b2 )
−
3u
[a ≥ u > b > 0]
BY (217.05)
u
x dx
a 2
2 a + b2 E (λ, q) − b2 F (λ, q) +
=
(a2 − u2 ) (u2 − b2 )
2
2
2
2
3
3
(a − x ) (x − b )
4
[a > u ≥ b > 0]
BY (218.06)
282
Power and Algebraic Functions
u
9.
a
3.155
1.
a
u
2.
3.
4.
5.9
6.
7.
8.
x4 dx
(x2
−
a2 ) (x2
−
b2 )
=
a 2
2a + b2 F (μ, t) − 2 a2 + b2 E (μ, t)
3
u 2 − a2
u 2
2
2
+ u + 2a + b
3
u2 − b 2
[u > a > b > 0]
(a2 − x2 ) (x2 − b2 ) dx =
3.155
BY (216.06)
u
a 2
a + b2 E (λ, q) − 2b2 F (λ, q) −
(a2 − u2 ) (u2 − b2 )
3
3
[a > u ≥ b > 0]
u
a 2
a + b2 E (μ, t) − a2 − b2 F (μ, t)
(x2 − a2 ) (x2 − b2 ) dx =
3
a
u 2 − a2
u 2
2
2
+ u − a − 2b
3
u2 − b 2
[u > a > b > 0]
u
a 2
2b F (α, q) − a2 + b2 E (α, q)
(x2 + a2 ) (x2 + b2 ) dx =
3
0
a2 + u 2
u 2
2
2
+ u + a + 2b
3
b2 + u2
[a > b, u > 0]
u
1
(a2 + x2 ) (b2 − x2 ) dx =
a2 + b2 a2 F (γ, r) − a2 − b2 E (γ, r)
3
0
b2 − u2
u 2
2
2
+ u + 2a − b
3
a2 + u 2
[a ≥ u > 0]
b
1 2
(a2 + x2 ) (b2 − x2 ) dx =
a + b2 a2 F (δ, r) + b2 − a2 E (δ, r)
3
u
u 2
(a + u2 ) (b2 − u2 )
+
3
[b > u ≥ 0]
u
1
(a2 + x2 ) (x2 − b2 ) dx =
a2 + b2 b2 − a2 E (ε, s) − b2 F (ε, s)
3
b
u 2 + a2 − b 2 2
(a + u2 ) (u2 − b2 )
+
3u
[u > b > 0]
u
a 2
a + b2 E (η, t) − a2 − b2 F (η, t)
(a2 − x2 ) (b2 − x2 ) dx =
3
0
u 2
(a − u2 ) (b2 − u2 )
+
3
[a > b ≥ u > 0]
b
a 2
a + b2 E (ζ, t) − a2 − b2 F (ζ, t)
(a2 − x2 ) (b2 − x2 ) dx =
3
u
b2 − u2
u
+ u2 − 2a2 − b2
3
a2 − u 2
[a > b > u ≥ 0]
BY (218.11)
BY (216.10)
BY (221.08)
BY (214.12)
BY (213.13)
BY (211.08)
BY (219.11)
BY (220.05)
3.156
3.156
∞
6
u
x2
x2
b
dx
1
√
=
a2 b 2 a2 + b 2
(x2 + a2 ) (x2 − b2 )
u > 0]
a2 + b
2
4.
x2
u
dx
(x2
+
a2 ) (x2
−
b2 )
=
1
√
a2 + b
a2 b 2 a2 + b 2
u2 − b 2
1
− 2
b u a2 + u 2
2
u
6.
x2
b
a
7.
u
x2
dx
1
= 2 E (κ, q)
2
2
2
2
ab
(a − x ) (x − b )
dx
(a2
−
x2 ) (x2
−
b2 )
=
BY (213.09)
BY (211.11)
E (ξ, s) − b2 F (ξ, s)
[u ≥ b > 0]
b
dx
b2 − u2
1
1
= 2 {F (ζ, t) − E (ζ, t)} + 2
2
2
2
2
2
ab
b u a2 − u 2
(a − x ) (b − x )
u x
BY (222.04)
E (ε, s) − b2 F (ε, s)
[u > b > 0]
∞
BY (217.09)
b2 + u2
1
− 2 E (β, q)
2
2
a +u
ab
2
dx
1
√
a F (δ, r) − a2 + b2 E (δ, r)
=
2
2
2
2
2
2
2
2
(x + a ) (b − x ) a b a + b
1 2
+ 2 2
(a + u2 ) (b2 − u2 )
a b u
[b > u > 0]
u
3.
1
= 2
2
2
2
2
2
ub
x (x + a ) (x + b )
b
u
dx
[a ≥ b,
2.
5.
283
u
a 2
a + b2 E (κ, q) − 2b2 F (κ, q)
(a2 − x2 ) (x2 − b2 ) dx =
3
b
u 2 − a2 − b 2 2
(a − u2 ) (u2 − b2 )
+
3u
[a ≥ u > b > 0]
9.
1.
Square roots of polynomials
BY (212.06)
[a > b > u > 0]
BY (220.09)
[a ≥ u > b > 0]
BY (217.01)
1
1 2
E (λ, q) − 2 2
(a − u2 ) (u2 − b2 )
2
ab
a b u
[a > u ≥ b > 0]
u
dx
u 2 − a2
1
1
= 2 {F (μ, t) − E (μ, t)} + 2
2
ab
a u u2 − b 2
(x2 − a2 ) (x2 − b2 )
a x
BY (218.12)
[u > a > b > 0]
BY (216.09)
[u ≥ a > b > 0]
BY (215.07)
8.
∞
9.
u
x2
dx
(x2
−
a2 ) (x2
−
b2 )
=
1
{F (ν, t) − E (ν, t)}
ab2
284
3.157
Power and Algebraic Functions
u
1.
0
2.
3.
b
[b ≥ u > 0,
11.
(p −
x2 )
dx
1
b
√
,r
=
Π δ, 2
2
2
2
2
2
2
2
b −p
(p − b ) a + b
(a + x ) (b − x )
b > u ≥ 0, p = b2
[u ≥ b > 0]
u
0
10.
p = 0]
BY (214.13)a
2
BY (213.02)
u
7.
9.
2
dx
1
p
2
√
F
(ε,
s)
=
,
s
+
p
−
b
b2 Π ε,
2
p − b2
(a2 + x2 ) (x2 − b2 ) p (p − b2 ) a2 + b2 b (p − x )
u > b > 0, p = b2
BY (211.14)
∞
dx
1
a2 + p
√
=
, s − F (ξ, s)
Π ξ, 2
2
a + b2
(a2 + p) a2 + b2
(a2 + x2 ) (x2 − b2 )
u (x − p)
8.
BY (221.13)
dx
1
a +p
=−
, q − F (β, q)
BY (222.11)
Π β,
2
a (a2 + p)
a2
(x2 + a2 ) (x2 + b2 )
u (p − x )
u
b 2 p + a2
dx
1
2
√
, r + p F (γ, r)
=
a Π γ,
2
p (a2 + b2 )
p (p + a2 ) a2 + b2
(a2 + x2 ) (b2 − x2 )
0 (p − x )
∞
u
6.
b2
p + b2
Π α,
, q + F (α, q)
p
p
[p = 0]
4.
5.
dx
1
=
a (p + b2 )
(p − x2 ) (x2 + a2 ) (x2 + b2 )
3.157
dx
b2
1
Π η, , t
=
ap
p
(p − x2 ) (a2 − x2 ) (b2 − x2 )
b
[a > b ≥ u > 0;
BY (212.12)
p = b]
BY (219.02)
dx
1
=
2 ) (p − b2 )
2
2
2
2
a
(p
−
a
(a − x ) (b − x )
u (p −
2
2
2
b
p
−
a
× b − a2 Π ζ, 2
, t + p − b2 F (ζ, t)
a (p − b2 )
a > b > u ≥ 0; p = b2
BY (220.13)
u
p a2 − b 2
dx
1
2
2
b Π κ, 2
, q + p − b F (κ, q)
=
2
ap (p − b2 )
a (p − b2 )
(a2 − x2 ) (x2 − b2 )
b (p − x )
a ≥ u > b > 0; p = b2
BY (217.12)
a
a2 − b 2
dx
1
Π λ, 2
,q
=
2
2
a (a − p)
a −p
(a2 − x2 ) (x2 − b2 )
u (x − p)
a > u ≥ b > 0; p = a2
BY (218.02)
u
dx
2
2
(x − a2 ) (x2 − b2 )
a (p − x )
2
p − b2
1
2
2
a
Π
μ,
F
(μ,
t)
−
b
,
t
+
p
−
a
=
a (p − a2 ) (p − b2 ) p − a2
u > a > b > 0; p = a2 , p = b2
BY (216.12)
x2 )
3.158
Square roots of polynomials
∞
12.
(x2 − p)
u
dx
(x2 − a2 ) (x2 − b2 )
=
285
p
1 Π ν, 2 , t − F (ν, t)
ap
a
[u ≥ a > b > 0;
3.158 1.
u
0
0
5.
b
6.
u
<
dx
=
3
(a2 + x2 ) (b2 − x2 )
a2
1
√
E (γ, r)
a2 + b 2
u > 0]
BY (221.06)
u ≥ 0]
BY (222.03)
[b ≥ u > 0]
dx
1
u
<
= √
E (δ, r) − 2 2
2
2
2
a
(a
+ b2 )
3
a a +b
(a2 + x2 ) (b2 − x2 )
u
 b > 0]
∞
8.
u
<
0
dx
(a2 + x2 ) (b2 −
=
3
x2 )
10.
u
u2 − b 2
u 2 + a2
BY (211.05)
BY (212.03)
1
u
√
{F (γ, r) − E (γ, r)} + b 2 a2 + b 2
b2 (a2 + u2 ) (b2 − u2 )
[b > u > 0]
∞
BY (213.08)
dx
1
<
= √
{F (ξ, s) − E (ξ, s)}
2
3
a a2 + b 2
(a2 + x2 ) (x2 − b2 )
[u ≥ b > 0]
u
9.
(x2 + b2 )
u
1
{F (α, q) − E (α, q)} + a (a2 − b2 )
a2 (u2 + a2 ) (u2 + b2 )
[b > u ≥ 0]
7.
=
BY (222.05)
dx
1
<
{F (β, q) − E (β, q)}
=
2
a
(a
− b2 )
3
(a2 + x2 ) (x2 + b2 )
0
3
a2 )
[a > b,
u
BY (221.05)
u ≥ 0]
[a > b;
u
dx
(x2 +
4.
u > 0]
2
dx
1
u
<
a E (β, q) − b2 F (β, q) − = 2 2
2
2
2
ab (a − b )
3
b (a + u2 ) (b2 + u2 )
(x2 + a2 ) (x2 + b2 )
<
∞
BY (215.12)
2
1
a E (α, q) − b2 F (α, q)
2
2
(a − b )
[a > b,
u
3.
3
ab2
[a > b;
∞
u
=
(x2 + a2 ) (x2 + b2 )
2.
dx
 0]
BY (212.04)
286
Power and Algebraic Functions
u
11.
<
0
(a2
b
12.
 b ≥ u > 0]
BY (219.07)
[a > b > u ≥ 0]
BY (220.10)
a
F (κ, q) − E (κ, q) +
u
u2 − b 2
a2 − u 2
[a > u > b > 0]
a u2 − b 2
E (ν, t) −
u u 2 − a2
BY (217.10)
1
1
F (η, t) − 2 2
ab2
b (a − b2 )
dx
(x2 − a2 ) (x2 −
=
3
b2 )
[u > a > b > 0]
a E (η, t) − u
[a > b > u > 0]
b2 F (λ, q) − a2 E (λ, q) + au
a2 − u 2
b2 − u2
BY (215.04)
a2 − u 2
u2 − b 2
x2 dx
(x2 + a2 ) (x2 +
=
3
b2 )
∞
u
BY (218.04)
[u > a > b > 0]
1
b 2 u 2 − a2
− 2 F (ν, t)
a E (ν, t) −
2
2
u u −b
ab
BY (216.11)
[u ≥ a > b > 0]
BY (215.06)
a
{F (α, q) − E (α, q)}
a2 − b 2
[a > b,
2.
BY (219.06)
1
a
E (μ, t) − 2 F (μ, t)
b2 (a2 − b2 )
ab
dx
1
<
= 2 2
b (a − b2 )
3
(x2 − a2 ) (x2 − b2 )
<
0
3
x2 )
a E (η, t) − u
dx
1
<
= 2 2
ab (a − b2 )
3
(a2 − x2 ) (x2 − b2 )
a
3.159
−
x2 )
[a > u > b > 0]
u
17.
(b2
1
= 2 2
a (a − b2 )
dx
1
<
=
2
a (b − a2 )
3
(x2 − a2 ) (x2 − b2 )
0
−
3
x2 )
dx
b
dx
3.159
u > 0]
BY (221.12)
x2 dx
a
u
<
= 2
{F (β, q) − E (β, q)} + 2
2 + u2 ) (b2 + u2 )
a
−
b
3
(a
(x2 + a2 ) (x2 + b2 )
[a > b,
u ≥ 0]
BY (222.10)
3.159
Square roots of polynomials
u
3.
0
u
7.
b
<
∞
u
9.
x2 dx
1
u
=√
{F (δ, r) − E (δ, r)} + 2
2 + b2
a
+
b2
3
a
(a2 + x2 ) (b2 − x2 )
0
u
u
11.
b
u
u
BY (213.07)
x2 dx
(a2 + x2 ) (b2 −
3
x2 )
[u > b > 0]
BY (211.13)
[u ≥ b > 0]
BY (212.01)
1
u
−√
=
E (γ, r)
2
2
2
2
2
a + b2
(a + u ) (b − u )
BY (214.07)
x dx
1
u
<
=√
{F (ξ, s) − E (ξ, s)} + 2
2
2
2
3
a +b
(a + u ) (u2 − b2 )
(a2 + x2 ) (x2 − b2 )
<
x2 dx
=
3
(a2 − x2 ) (b2 − x2 )
a2
1
− b2
a E (η, t) − u
[u > b > 0]
b2 − u2
1
− F (η, t)
2
2
a −u
a
[a > b ≥ u > 0]
12.
b
b2 − u2
a2 + u 2
2
0
13.
BY (214.04)
[b > u > 0]
∞
10.
2
x2 dx
1
<
=√
E (ξ, s)
2 + b2
3
a
(a2 + x2 ) (x2 − b2 )
<
BY (222.07)
1
=√
{F (γ, r) − E (γ, r)}
2
a + b2
(b2 − x2 )
[b > u ≥ 0]
x dx
u2 − b 2
1
a
<
=√
E (ε, s) −
2
2
2
2
u (a + b ) u2 + a2
3
a +b
(a2 + x2 ) (x2 − b2 )
u
3
x2 )
2
8.
u ≥ 0]
[b ≥ u > 0]
b
BY (221.11)
x dx
 0]
2
x2 dx
1
<
a E (β, q) − b2 F (β, q)
=
2
2
a (a − b )
3
(x2 + a2 ) (x2 + b2 )
0
2
1
u
a E (α, q) − b2 F (α, q) − 2
2
2
−b )
(a + u ) (b2 + u2 )
[a > b,
u
5.
3
a (a2
[a > b,
∞
u
=
(x2 + a2 ) (x2 + b2 )
4.
x2 dx
<
287
<
x2 dx
(a2 −
<
3
x2 )
(b2 − x2 )
x2 dx
(a2 −
3
x2 )
=
(x2 − b2 )
=
BY (212.10)
BY (219.04)
a
1
E (ζ, t) − F (ζ, t)
a2 − b 2
a
1
a (a2 − b2 )
[a > b > u ≥ 0]
3
2 − b2
u
a
b2 F (κ, q) − a2 E (κ, q) +
u a2 − u 2
[a > u > b > 0]
BY (220.08)
BY (217.06)
288
Power and Algebraic Functions
∞
14.
u
u
15.
x2 dx
<
3
(x2 − a2 ) (x2 − b2 )
<
0
a
16.
u
u
17.
=
3
(a2 − x2 ) (b2 − x2 )
<
∞
18.
u
2
x dx
=
3
(x2 − a2 ) (x2 − b2 )
x2 dx
<
3
(x2 − a2 ) (x2 − b2 )
a2
1
a u2 − b 2
− E (ν, t) + F (ν, t)
u u 2 − a2
a
1
2
a − b2
x2 dx
1
<
= 2
a
−
b2
3
(a2 − x2 ) (x2 − b2 )
a
x2 dx
a
= 2
a − b2
3.161
u
[u > a > b > 0]
a2 − u 2
− a E (η, t)
b2 − u2
[a > b > u > 0]
a2 − u 2
a F (λ, q) − a E (λ, q) + u
u2 − b 2
a
E (μ, t)
− b2
1
= 2
a − b2
b2
a E (ν, t) −
u
∞
1.
u
BY (218.07)
[u > a > b > 0]
BY (216.01)
u 2 − a2
u2 − b 2
2.
3.
4.
BY (215.11)
a2 b2 − u2 2a2 + b2
1 2
2
2
= 3 4 2 a + b E (β, q) − b F (β, q) +
3a b
3a2 b4 u3
x4 (x2 + a2 ) (x2 + b2 )
dx
[a > b,
BY (219.12)
[a > u > b > 0]
[u ≥ a > b > 0]
3.161
BY (215.09)
u > 0]
BY (222.04)
2 2
dx
1
√
a 2a − b2 F (δ, r) − 2 a4 − b4 E (δ, r)
=
4
4
2
2
2
2
2
2
(x + a ) (b − x ) 3a b a + b
u
a2 b2 + 2u2 a2 − b2 2
+
(b − u2 ) (a2 + u2 )
3a4 b4 u3
[b > u > 0]
BY (213.09)
√
u
a2 + b 2
dx
2b2 − a2
2 a2 − b 2
√
=
F (ε, s) +
E (ε, s)
4
3
a4 b 4
(x2 + a2 ) (x2 − b2 ) 3a4 b2 a2 + b2
b x
1
+ 2 2 3 (u2 + a2 ) (u2 − b2 )
3a b u
[u > b > 0]
BY (211.11)
∞
4
dx
1
√
2 a − b4 E (ξ, s) + b2 2b2 − a2 F (ξ, s)
=
4
4
2
2
4
2
2
2
2
(x + a ) (x − b ) 3a b a +b
u x
a2 b2 + u2 2a2 − b2
u2 − b 2
−
2
4
3
3a b u
u 2 + a2
[u ≥ b > 0]
BY (212.06)
b
x4
3.162
Square roots of polynomials
b
5.
u
6.
7.
⎧
⎨ dx
1
2a2 + b2 F (ζ, t) − 2 a2 + b2 E (ζ, t)
= 3 4
x4 (a2 − x2 ) (b2 − x2 ) 3a b ⎩
⎫
2
2
2
2 2
2
2
2a + b u + a b a b − u ⎬
+
u3
a2 − u 2 ⎭
[a > b > u > 0]
dx
1 = 3 4 2 a2 + b2 E (κ, q) − b2 F (κ, q)
2
2
2
2
3a
b
(a − x ) (x − b )
b
1
+ 2 2 3 (a2 − u2 ) (u2 − b2 )
3a b u
[a ≥ u > b > 0]
⎧
a
dx
1 ⎨ = 3 4 2 a2 + b2 E (λ, q) − b2 F (λ, q)
4
(a2 − x2 ) (x2 − b2 ) 3a b ⎩
u x
⎫
⎬
2 a2 + b 2 u 2 + a 2 b 2 2
2 ) (u2 − b2 )
−
(a
−
u
⎭
au3
[a > u ≥ b > 0]
u
8.
x4
a
⎧
1 ⎨ 2
2a + b2 F (μ, t) − 2 a2 + b2 E (μ, t)
= 3 4
3a b ⎩
⎫
2
a + 2b2 u2 + a2 b2 b2 u2 − a2 ⎬
+
au3
u2 − b 2 ⎭
⎧
∞
dx
1 ⎨ 2
2a + b2 F (ν, t) − 2 a2 + b2 E (ν, t)
= 3 4
4
2
2
2
2
⎩
(x − a ) (x − b ) 3a b
u x
⎫
⎬
ab2 2
+ 3 (u − a2 ) (u2 − b2 )
⎭
u
1.
0
u
<
BY (217.14)
BY (218.12)
dx
2
(x − a2 ) (x2 − b2 )
[u ≥ a > b > 0]
3.162
BY (220.09)
u
x4
9.
289
dx
5
(x2 + a2 ) (x2 + b2 )
=
[u > a > b > 0]
BY (216.09)
BY (215.07)
2
3a − b2 F (α, q) − 2 2a2 − b2 E (α, q)
−
u a2 4a2 − 3b2 + u2 3a2 − 2b2
<
+
3
3a4 (a2 − b2 ) (u2 + a2 ) (u2 + b2 )
[a > b, u > 0]
BY (221.06)
1
3a3
(a2
2
b2 )
290
Power and Algebraic Functions
2.
3.
4.
5.
6.
7.
8.
∞
3.162
2
dx
1
<
3a − b2 F (β, q) − 2 2a2 − b2 E (β, q)
=
2
3
2
2
5
u
(x2 + a2 ) (x2 + b2 ) 3a (a − b ) .
u2 + b 2
u
+ 2 2
2
3a (a − b ) (a2 + u2 )3
[a > b, u ≥ 0]
BY (222.03)
2
u
2
2
2
a 2a − 4b
dx
3b − a
<
=
2 F (α, q) +
2 E (α, q)
2
2
2
5
3ab (a − b ) .
3b4 (a2 − b2 )
0
(x2 + a2 ) (x2 + b2 )
u 2 + a2
u
+ 2 2
2
3b (a − b ) (u2 + b2 )3
[a > b, u > 0]
BY (221.05)
∞
2 2
dx
1
<
a − 2b2 E (β, q) + b2 3b2 − a2 F (β, q)
=
2 2a
4
2
2
5
3ab (a − b )
u
(x2 + a2 ) (x2 + b2 )
u b2 3a2 − 4b2 + u2 2a2 − 3b2
<
−
3
3b4 (a2 − b2 ) (u2 + a2 ) (u2 + b2 )
[a > b, u ≥ 0]
BY (222.05)
u
2
dx
1
<
<
2 b + 2a2 E (γ, r) − a2 F (γ, r)
=
5
3
0
(a2 + x2 ) (b2 − x2 ) 3a4 (a2 + b2 )
.
b2 − u2
u
+ 2 2
2
3a (a + b ) (a2 + u2 )3
[b ≥ u > 0]
BY (214.15)
b
2
dx
1
<
<
4a + 2b2 E (δ, r) − a2 F (δ, r)
=
5
3
u
(a2 + x2 ) (b2 − x2 ) 3a4 (a2 + b2 )
.
2 2
u a 5a + 3b2 + u2 4a2 + 2b2
b2 − u2
−
2
3
4
2
2
3a (a + b )
(a2 + u2 )
[b > u > 0]
BY (213.08)
u
dx
1
<
<
3a2 + 2b2 F (ε, s) − 4a2 + 2b2 E (ε, s)
=
5
3
b
(a2 + x3 ) (x2 − b2 ) 3a4 (a2 + b2 )
.
2
3a + b2 u2 + 2 2a2 + b2 a2
u2 − b 2
+
2
3
3a2 (a2 + b2 ) u
(u2 + a2 )
[u > b > 0]
BY (211.05)
∞
2
dx
1
<
<
3a + 2b2 F (ξ, s) − 4a2 + 2b2 E (ξ, s)
=
5
3
u
(a2 + x2 ) (x2 − b2 ) 3a4 (a2 + b2 )
.
u2 − b 2
u
+ 2 2
3a (a + b2 ) (a2 + u2 )3
[u > b > 0]
BY (212.03)
3.162
Square roots of polynomials
9.
u
<
0
10.
11.
12.
13.
14.
15.
dx
(a2 + x2 ) (b2 −
=
5
x2 )
<
1
3
b2 )
291
2a2 + 3b2 F (γ, r) − 2a2 + 4b2 E (γ, r)
3b4 (a2 +
u 3a3 + 4b2 b2 − 2a2 + 3b2 u2
<
+
3
3b4 (a2 + b2 ) (a2 + u2 ) (b2 − u2 )
[b > u > 0]
BY (214.10)
∞
2
dx
1
<
<
2a + 4b2 E (ξ, s) − b2 F (ξ, s)
=
5
3
u
(a2 + x2 ) (x2 − b2 )
3b4 (a2 + b2 )
2
u 3a + 4b2 b2 − 2a2 + 3b2 u2
<
+
3
3b4 (a2 + b2 ) (a2 + u2 ) (u2 − b2 )
[u > b > 0]
u
2a 2b2 − a2
dx
2a2 − 3b2
<
F (η, t) +
=
2 E (η, t)
3ab4 (a2 − b2 )
5
3b4 (a2 −
b2 ) 0
(a2 − x2 ) (b2 − x2 )
2
u 3a − 5b2 b2 − 2 a2 − 2b2 u2
a2 − u 2
+
2
b2 − u2
3b4 (a2 − b2 ) (b2 − u2 )
[a > b > a > 0]
a
2a a2 − 2b2
dx
3b2 − a2
<
=
2 F (λ, q) +
2 E (λ, q)
5
3ab2 (a2 − b2 ) 3b4 (a2− b2
)
u
(a2 − x2 ) (x2 − b2 )
u 2 2b2 − a2 u2 + 3a2 − 5b2 b2
a2 − u 2
+
2
u2 − b 2
3b4 (a2 − b2 ) (u2 − b2 )
[a > u > b > 0]
u
2a 2b2 − a2
dx
2a2 − 3b2
<
F (μ, t) +
=
2 E (μ, t)
3ab4 (a2 − b2 )
5
3b4 (a2 − b2 )
a
(x2 − a2 ) (x2 − b2 )
u 2 − a2
u
+ 2 2
2
2
2
3b (a − b ) (u − b ) u2 − b2
[u > a > b > 0]
2
∞
4b − 2a2 a
dx
2a2 − 3b2
<
F (ν, t)
=
E (ν, t) +
2
3ab4 (a2 − b2 )
5
3b4 (a2 − b2)
u
(x2 − a2 ) (x2 − b2 )
3b2 − a2 u2 − 4b2 − 2a2 b2 u2 − a2
−
2
u2 − b 2
3b2 u (a2 − b2 ) (u2 − b2 )
[u ≥ a > b > 0]
u
dx
1
<
4a2 − 2b2 E (η, t) − a2 − b2 F (η, t)
=
2
3
2 − b2 )
5
0
(a2 − x2 ) (b2 − x2 ) 3a (a
2
u 5a − 3b2 a2 − 4a2 − 2b2 u2
b2 − u2
−
a (a2 − u2 )
a2 − u 2
[a > b ≥ u > 0]
BY (212.04)
BY (219.06)
BY (218.04)
BY (216.11)
BY (215.06)
BY (219.07)
292
Power and Algebraic Functions
b
16.
 b > u ≥ 0]
3a3
dx
<
(x2 +
2.
u
u
3.
2
3a − b2 F (κ, q) − 4a2 − 2b2 E (κ, q)
− 2 2a2 − b2 a2 + b2 − 3a2 u2 u2 − b2
,
+
2
a2 − u 2
3a2 u (a2 − b2 ) (a2 − u2 )
[a > u > b > 0]
BY (217.10)
3
a2 )
=
(x2 +
3
b2 )
∞
4.
u
1
ab2
dx
(a2
a2
−
(a2
2
b2 )
−
b2 )
a2 + b2 E (α, q) − 2b2 F (α, q)
u
(a2 + u2 ) (b2 + u2 )
[a > b, u > 0]
3
3
(x2 + a2 ) (b3 − x2 )
1
<
=
a2 b 2
+
BY (220.10)
2
b2 )
(a2
2
dx
1
<
a + b2 E (β, q) − 2b2 F (β, q)
=
2
2
2
2
3
3
ab (a − b )
(x2 + a2 ) (x2 + b2 )
u
−
2
2
2
b (a − b ) (a2 + u2 ) (b2 + u2 )
[a > b, u ≥ 0]
<
0
3a3
1
3a3
−
∞
E (ζ, r) −
2
dx
1
<
4a − 2b2 E (ν, t) − a2 − b2 F (ν, t)
=
2
3
2
2
5
(x2 − a2 ) (x2 − b2 ) 3a (a − b )
4a2 − 2b2 a2 + b2 − 3a2 u2 u2 − b2
+
2
u 2 − a2
3a2 u (a2 − b2 ) (u2 − a2 )
[u > a > b > 0]
BY (215.04)
0
2
b2 )
(a2
3.163
b2
(a2 + b2 )
(a2
+
b2 )
3
BY (221.07)
BY (222.12)
2
a F (γ, r) − a2 − b2 E (γ, r)
u
(a2 + u2 ) (b2 − u2 )
[b > u > 0]
dx
b 2 − a2
1
<
<
=
E (ξ, s) − <
F (ξ, s)
3
3
3
3
(x2 + a2 ) (x2 − b2 )
a2 b2 (a2 + b2 )
a2 (a2 + b2 )
u
+
2
2
2
b (a + b ) (u2 + a2 ) (u2 − b2 )
[u > b > 0]
BY (214.15)
BY (212.05)
3.165
Square roots of polynomials
u
5.
<
0
∞
6.
u
dx
=
3
3
(a2 − x2 ) (b2 − x2 )
293
a2 + b 2
1
F
(η,
t)
−
2 E (η, t)
ab2 (a2 − b2 )
ab2(a2− b2 )
4
a + b 4 − a2 + b 2 u 2 u
+
2
a2 b2 (a2 − b2 )
(a2 − u2 ) (b2 − u2 )
[a > b > u > 0]
2
BY (279.08)
2
a +b
dx
1
<
F (ν, t) −
= 2 2
2 E (ν, t)
2
2
ab (a − b )
3
3
ab (a2 − b2 )
(x2 − a2 ) (x2 − b2 )
1
+
u (a2 − b2 ) (u2 − a2 ) (u2 − b2 )
[u > a > b > 0]
BY (215.10)
.
2
1
u2 − ρρ
(ρ − ρ)
,
r=
.
3.164 Notation: α = arccos 2
−
u + ρρ
2
ρρ
∞
dx
1
= √ F (α, r)
BY (225.00)
1.
ρρ
(x2 + ρ2 ) (x2 + ρ2 )
u
∞
2u (u2 + ρ2 ) (u2 + ρ2 )
1
x2 dx
=
−
E (α, r)
2.
2
2
2√
2
2
2
2
2
4
2
2
(x + ρ ) (x + ρ )
(ρ + ρ) (u − ρ ρ )
(ρ + ρ) ρρ
u (x − ρρ)
BY (225.03)
∞
3.
u
∞
4.
u
(x2
+ ρρ)
2
2
x dx
(x2
+
ρ2 ) (x2
x2 dx
+
ρ2 )
=−
√
4 ρρ
1
(ρ − ρ)
2
√ [F (α, r) − E (α, r)]
ρρ
BY (225.07)
1
<
F (α, r)
=−
2 E (α, r) +
2√
2
2
3
3
(ρ − ρ )
(x2 + ρ2 ) (x2 + ρ2 )
2 (ρ − ρ) ρρ
2u u − ρρ
−
2
2
(ρ + ρ) (u + ρρ) (u2 + ρ2 ) (u2 + ρ2 )
BY (225.05)
∞
5.
u
6.
7.
8.
2
x2 − ρρ dx
√
4 ρρ
<
=−
2 [F (α, r) − E (α, r)]
3
3
(ρ − ρ)
(x2 + ρ2 ) (x2 + ρ2 )
2u u2 − ρρ
+
(u2 + ρρ) (u2 + ρ2 ) (u2 + ρ2 )
∞
(x2 + ρ2 ) (x2 + ρ2 )
1
dx = √ E (α, r)
ρρ
+ ρρ)
u
√
∞
2
2
x2 − dx
4 ( + )
=
−
F (α, r)
E
(α,
r)
+
√
2
2
2
2
(x2 + 2 ) (x2 + 2 )
( − )
( − ) u (x + )
2
2
∞
x + dx
1
+
,
= √ Π α, p2 , r
2
u
(x2 + ) − 4p2 x2
(x2 + 2 ) (x2 + 2 )
(x2
2
BY (225.06)
BY(225.01)
BY (225.08)
BY (225.02)
294
Power and Algebraic Functions
3.165
√
a2 − b 2
u 2 − a2
√
.
3.165 Notation: α = arccos 2
, r=
2
u +a
a 2
√
a
dx
2
√
√
1.
= √
4
2
2
4
x + 2b x + a
a 2⎡+ a2 + b2
u
⎤
< √
√
2 − b2 )
2
2
2
a
2
(a
a 2+ a −b a−u
⎦
√
√
, √
× F ⎣arctan
a+u
a2 + b 2
a 2 + a2 − b 2
[a > b,
∞
2.
u
dx
1
√
F (α, r)
=
2a
x4 + 2b2 x2 + a4
a > u ≥ 0]
a2 > b2 > −∞,
BY (264.00)
a2 > 0,
u≥0
BY (263.00, 266.00)
√
∞
dx
u4 + 2b2 u2 + a4
1
√
= 3 [F (α, r) − 2 E (α, r)] +
2 x4 + 2b2 x2 + a4
2a
a2 u (u2 + a2 )
u x
[a > b > 0, u > 0]
∞
x2 dx
1
[F (α, r) − E (α, r)]
=
√
2
2
2
2
4a (a − b2 )
x4 + 2b2 x2 + a4
u (x + a )
2
a > b2 > −∞, a2 > 0,
3.
4.
BY (263.06)
u≥0
BY (263.03, 266.05)
√
∞
1
x2 dx
u u4 + 2b2 u2 + a4
−
E (α, r)
=
2√ 4
2 + b2 ) (u4 − a4 )
2 + b2 )
2
2
2
2
4
2
(a
4a
(a
x + 2b x + a
u (x − a )
2
a > b2 > −∞, u2 > a2 > 0
5.
BY (263.05, 266.02)
∞
6.
u
7.
8.
9.
∞
2
x dx
1
a
<
E (α, r) −
F (α, r)
=
4
4
2
2 (a − b )
− b2 )
3
2 4a (a
(x4 + 2b2 x2 + a4 )
u u − a2
√
−
2 (a2 + b2 ) (u2 + a2 ) u4 + 2b2 u2 + a4
a2 > b2 > −∞, a2 > 0, u ≥ 0
BY (263.08, 266.03)
2
2
x − a2 dx
u
a
u 2 − a2
<
√
= 2
[F (α, r) − E (α, r)] + 2
2
2
4 + 2b2 u2 + a4
a
−
b
u
+
a
3
u
u
(x4 + 2b2 x2 + a4 )
! 2 !
!b ! < a2 , u ≥ 0
BY (266.08)
2
∞
2
x + a2 dx
a
a2 − b 2 u 2 − a2
u
<
= 2
E
(α,
r)
−
· 2
·√
2
2
2
2
4
a +b
a +b u +a
3
u + 2b2 u2 + a4
u
(x2 + 2b2 x2 + a4 )
! 2 !
!b ! < a2 , u ≥ 0
BY (266.06)a
2
∞
2
x − a2 dx
a
a2 + b 2
F (α, r)
= 2
E (α, r) −
√
2
2
2
2
a −b
2a (a2 − b2 )
x4 + 2b2 x2 + a4
u (x + a )
2
a > b2 > −∞, a2 > 0, u ≥ 0
BY (263.04, 266.07)
3.166
Square roots of polynomials
∞
10.
√
x4 + 2b2 x2 + a4
(x2
u
11.
+
2
a2 )
√
∞
x4 + 2b2 x2 + a4
(x2 −
u
2
a2 )
dx =
1
E (α, r)
2a
295
a2 > b2 > −∞,
a2 > 0,
u≥0
BY (263.01, 266.01)
dx =
u 4
1
[F (α, r) − E (α, r)] + 4
u + 2b2 u2 + a4
2a
u − a4
[a > b > 0,
2
∞
x2 + a2 dx
1 +
,√
Π α, p2 , r
=
2
2a
u
(x2 + a2 ) − 4a2 p2 x2
x4 + 2b2 x2 + a4
u > a]
BY (263)
[a > b > 0,
u ≥ 0]
BY (263.02)
12.
3.166
Notation:
α = arccos
u2 − 1
,
u2 + 1
β = arctan
1+
√
2
1−u
,
1+u
1 − u2
1
,
γ = arccos u, δ = arccos , ε = arccos
u
1 + u2
√
<
√
√
√
2
4
, q =2 3 2−4=2 2
2 − 1 ≈ 0.985171
r=
2
∞
1.
u
∞
2.
u
dx
1
√
= F (α, r)
4
2
x +1
[u ≥ 0]
H (287), BY (263.50)
√
dx
u4 + 1
1
√
= [F (α, r) − 2 E (α, r)] +
2
u (u2 + 1)
x2 x4 + 1
[u > 0]
∞
u u2 − 1
x2 dx
1
1
√
√
= E (α, r) − F (α, r) −
4
4
2
4
2 (u2 + 1) u4 + 1
u (x + 1) x + 1
BY (263.57)
3.
4.
5.
6.
7.
8.
∞
2
1
x dx
= [F (α, r) − E (α, r)]
2√ 4
4
+ 1) x + 1
u
√
∞
u u4 + 1 1
x2 dx
=
− E (α, r)
2√ 4
2
2 (u4 − 1) 4
x +1
u (x − 1)
√
∞√ 4
u u4 + 1
x +1
1
2 dx = 2 [F (α, r) − E (α, r)] + u4 − 1
2
u (x − 1)
(x2
2
∞ 2
x − 1 dx
1
= E (α, r) − F (α, r)
2√ 4
2
2
x +1
u (x + 1)
∞√ 4
x + 1 dx
1
= E (α, r)
2
2
(x2 + 1)
u
[u ≥ 0]
BY (263.59)
[u ≥ 0]
BY (263.53)
[u > 1]
BY (263.55)
[u > 1]
BY (263.58)
[u ≥ 0]
BY (263.54)
[u ≥ 0]
BY (263.51)
296
Power and Algebraic Functions
∞
9.
u
10.
11.
12.
13.
1
14.
u
1
15.
u
17.
18.8
19.
20.
21.3
22.
H 66(288)
BY (264.50)
[0 ≤ u < 1]
BY (264.51)
[0 ≤ u < 1]
√
√
(1 + x)2 dx
3 2−4
3 2+4
√
√
E (β, q) −
F (β, q)
=
2
4
2
2
x −x 2+1 x +1
BY (264.55)
[0 ≤ u < 1]
BY (264.56)
dx
1
√
= √ F (γ, r)
2
1 − x4
[u < 1]
H 66 (290), BY (259.75)
[u > 1]
H 66 (289), BY (260.75)
1
[u < 1]
[u = 0]
BY (259.76)
√
x dx
1 4
1
√
u −1
[u > 1]
= √ F (δ, r) − 2 E (δ, r) +
4
u
2
x −1
1
1 4
x dx
u
1
√
= √ F (γ, r) +
1 − u4
[u < 1]
3
3 2
1 − x4
u
u 4
x dx
1 1
√
[u > 1]
= √ F (δ, r) + u u4 − 1
3
3 2
x4 − 1
1
√ √ u
1+ 1− 3 u
dx
2+ 3
1
√ ,
F arccos
= √
4
2
3
1+ 1+ 3 u
x (1 + x3 )
0
23.
0
BY (263.52)
[0 ≤ u < 1]
2
dx
1
1
√
= √
Γ
4
4 2π
1 − x4
0
u
dx
1
√
= √ F (δ, r)
2
x4 − 1
1
1 2
√
x dx
1
√
= 2 E (γ, r) − √ F (γ, r)
4
1−x
u
22
1
3
=√
Γ
4
2π
[u ≥ 0]
u
dx
1
√
= F (ε, r)
4
2
x +1
0
1
√
dx
√
= 2 − 2 F (β, q)
4
x +1
u
√
1 2
√
x + x 2 + 1 dx
√
√
= 2 + 2 E (β, q)
2
4
x +1
u x −x 2+1
1
2
(1 − x) dx
1
√
√
= √ [F (β, q) − E (β, q)]
2
4
2
x +1
u x −x 2+1
16.
2
2
x + 1 dx
1 +
,√
= Π α, p2 , r
2
2
2
2
2
4
(x + 1) − 4p x
x +1
3.166
u
u
2
dx
x (1 −
x3 )
1
= √
F
4
3
[u > 0]
√ √ 1− 1+ 3 u
2− 3
√
,
arccos
2
1+
3−1 u
[1 ≥ u > 0]
BY (260.77)
BY (259.76)
BY (260.77)
BY (260.50)
BY (259.50)
3.167
Square roots of polynomials
.
3.167
Notation: In 3.167 and 3.168 we set:
.
β = arcsin
.
δ = arcsin
α = arcsin
297
(a − c)(d − u)
,
(a − d)(c − u)
.
(a − c)(u − d)
,
(c − d)(a − u)
γ = arcsin
(b − d)(u − c)
,
(b − c)(u − d)
κ = arcsin
.
(b − d)(c − u)
,
(c − d)(b − u)
(a − c)(b − u)
,
(b − c)(a − u)
.
(a − c)(u − b)
(b − d)(a − u)
,
μ = arcsin
,
λ = arcsin
(a − b)(u − c)
(a − b)(u − d)
.
.
.
(b − d)(u − a)
(b − c)(a − d)
(a − b)(c − d)
ν = arcsin
,
q=
,
r=
.
(a − d)(u − b)
(a − c)(b − d)
(a − c)(b − d)
.
d.
2(c − d)
d−x
dx = (a − x)(b − x)(c − x)
(a − c)(b − d)
1.
u
u.
2.
d
u
u.
4.
c
u
b
a.
7.
u
BY (251.05)
d−c
, r − F (β, r)
Π β,
a−c
(c − b) Π γ,
BY (252.14)
c−d
, r + (b − d) F (γ, r)
b−d
[a > b > c > u ≥ d]
BY (253.14)
2(c − d)
x−d
b−c
dx = ,q
Π δ,
(a − x)(b − x)(x − c)
b
−d
(a − c)(b − d)
[a > b ≥ u > c > d]
2
x−d
dx = (a − x)(b − x)(x − c)
(a − c)(b − d)
u.
6.
2(d − a)
x−d
dx = (a − x)(b − x)(c − x)
(a − c)(b − d)
2
x−d
dx = (a − x)(b − x)(c − x)
(a − c)(b − d)
3.
5.
[a > b > c > d > u]
[a > b > c ≥ u > d]
c.
b.
a−d
, q − F (α, q)
Π α,
a−c
b−c
, q + (a − d) F (κ, q)
(b − a) Π κ,
a−c
[a > b > u ≥ c > d]
2
x−d
dx = (a − x)(x − b)(x − c)
(a − c)(b − d)
BY (254.02)
(b − c) Π λ,
BY (255.20)
a−b
, r + (c − d) F (λ, r)
a−c
[a ≥ u > b > c > d]
BY (256.13)
x−d
2(a − d)
b−a
dx = ,r
Π μ,
(a − x)(x − b)(x − c)
b−d
(a − c)(b − d)
[a > u ≥ b > c > d]
BY (257.02)
298
Power and Algebraic Functions
u.
8.
a
u
u
10.
d
c
11.
u
u
12.
c
u
14.
b
a
15.
u
u
16.
a
d.
17.
u
u.
d
[a > b > c > d > u]
BY (251.02)
2
c−x
d−c
dx = , r − (a − c) F (β, r)
(a − d) Π β,
(a − x)(b − x)(x − d)
a−c
(a − c)(b − d)
[a > b > c ≥ u > d]
2(b − c)
c−x
c−d
dx = , r − F (γ, r)
Π γ,
(a − x)(b − x)(x − d)
b−d
(a − c)(b − d)
BY (252.13)
[a > b > c > u ≥ d]
2(c − d)
x−c
b−c
dx = , q − F (δ, q)
Π δ,
(a − x)(b − x)(x − d)
b−d
(a − c)(b − d)
BY (253.13)
[a > b ≥ u > c > d]
BY (254.12)
2
x−c
b−c
dx = , q + (a − c) F (κ, q)
(b − a) Π κ,
(a − x)(b − x)(x − d)
a−c
(a − c)(b − d)
u
18.
BY (258.14)
2(c − d)
c−x
a−d
dx = ,q
Π α,
(a − x)(b − x)(d − x)
a−c
(a − c)(b − d)
9.
13.
a−d
, q + (b − d) F (ν, q)
(a − b) Π ν,
b−d
[u > a > b > c > d]
d
b
2
x−d
dx = (x − a)(x − b)(x − c)
(a − c)(b − d)
3.167
[a > b > u ≥ c > d]
BY (259.19)
2(b − c)
x−c
a−b
dx = ,r
Π λ,
(a − x)(x − b)(x − d)
a−c
(a − c)(b − d)
[a ≥ u > b > c > d]
BY (256.02)
2
x−c
b−a
dx = , r + (d − c) F (μ, r)
(a − d) Π μ,
(a − x)(x − b)(x − d)
b−d
(a − c)(b − d)
[a > u ≥ b > c > d]
BY (257.13)
2
x−c
a−d
dx =
, q + (b − c) F (ν, q)
(a − b) Π ν,
(x − a)(x − b)(x − d)
b−d
(a − c)(b − d)
[u > a > b > c > d]
BY (258.13)
2
b−x
a−d
dx = , q + (b − c) F (α, q)
(c − d) Π α,
(a − x)(c − x)(d − x)
a−c
(a − c)(b − d)
[a > b > c > d > u]
BY (251.07)
2
b−x
d−c
dx = , r − (a − b) F (β, r)
(a − d) Π β,
(a − x)(c − x)(x − d)
a−c
(a − c)(b − d)
[a > b > c ≥ u > d]
BY (252.15)
3.167
Square roots of polynomials
c.
2(b − c)
b−x
c−d
dx = ,r
Π γ,
(a − x)(c − x)(x − d)
b−d
(a − c)(b − d)
19.
u
u.
20.
c
b.
21.
u
b
23.8
u
u.
24.
a
25.
u
u
26.
d
[a > b > u ≥ c > d]
2(b − c)
x−b
a−b
dx = , r − F (λ, r)
Π λ,
(a − x)(x − c)(x − d)
a−c
(a − c)(b − d)
BY (255.21)
[a > u ≥ b > c > d]
BY (257.15)
2(a − b)
x−b
a−d
dx = ,q
Π ν,
(x − a)(x − c)(x − d)
b−d
(a − c)(b − d)
[a > b > c > d > u]
BY (251.06)
2(a − d)
a−x
d−c
dx = ,r
Π β,
(b − x)(c − x)(x − d)
a−c
(a − c)(b − d)
[a > b > c ≥ u > d]
BY (252.02)
2
a−x
c−d
dx = , r + (a − b) F (γ, r)
(b − c) Π γ,
(b − x)(c − x)(x − d)
b−d
(a − c)(b − d)
c
27.
u
u
u
BY (254.14)
[u > a > b > c > d]
BY (258.02)
2
a−x
a−d
dx = , q + (a − c) F (α, q)
(c − d) Π α,
(b − x)(c − x)(d − x)
a−c
(a − c)(b − d)
d
29.
2
b−x
b−c
dx = , q + (b − d) F (δ, q)
(d − c) Π δ,
(a − x)(x − c)(x − d)
b−d
(a − c)(b − d)
[a ≥ u > b > c > d]
BY (256.15)
2
x−b
b−a
dx = , r − (b − d) F (μ, r)
(a − d) Π μ,
(a − x)(x − c)(x − d)
b−d
(a − c)(b − d)
a.
b
BY (253.02)
b−x
2(a − b)
b−c
dx = , q − F (κ, q)
Π κ,
(a − x)(x − c)(x − d)
a−c
(a − c)(b − d)
22.
c
[a > b > c > u ≥ d]
[a > b ≥ u > c > d]
u.
28.
299
[a > b > c > u ≥ d]
BY (253.15)
a−x
2
b−c
dx = , q + (a − d) F (δ, q)
(d − c) Π δ,
(b − x)(x − c)(x − d)
b−d
(a − c)(b − d)
[a > b ≥ u > c > d]
BY (254.13)
2(a − b)
a−x
b−c
dx = ,q
Π κ,
(b − x)(x − c)(x − d)
a−c
(a − c)(b − d)
[a > b > u ≥ c > d]
BY (255.02)
300
Power and Algebraic Functions
u
30.
b
a
31.
u
u
32.
a
3.168
a−x
2
a−b
dx = , r + (a − c) F (λ, r)
(c − b) Π λ,
(x − b)(x − c)(x − d)
a−c
(a − c)(b − d)
[a ≥ u > b > c > d]
2(d − a)
a−x
b−a
dx = , r − F (μ, r)
Π μ,
(x − b)(x − c)(x − d)
b−d
(a − c)(b − d)
BY (256.14)
[a > u ≥ b > c > d]
2(a − b)
x−a
a−d
dx = , q − F (ν, q)
Π ν,
(x − b)(x − c)(x − d)
b−d
(a − c)(b − d)
BY (257.14)
[u > a > b > c > d]
3.168
c
1.
u
u
2.
c
b
3.
4.
%
c−x
2
dx =
(a − x)(b − x)(x − d)3
d−a
x−c
2
dx =
3
(a − x)(b − x)(x − d)
a−d
x−c
2
dx =
(a − x)(b − x)(x − d)3
a−d
a−c
E (γ, r) −
b−d
.
BY (258.15)
(a − u)(c − u)
(b − u)(u − d)
&
[a > b > c > u > d]
BY (253.06)
a−c
[F (δ, q) − E (δ, q)]
b−d
[a > b ≥ u > c > d]
.
BY (254.04)
2
a−c
(b − u)(u − c)
[F (κ, q) − E (κ, q)] +
b
−
d
b
−
d
(a
− u)(u − d)
u
BY (255.09)
[a > b > u ≥ c > d]
.
%
&
u
c − d (a − u)(u − b)
x−c
a−c
2
E (λ, r) −
dx =
3
(a
−
x)(x
−
b)(x
−
d)
a
−
d
b
−
d
b − d (u − c)(u − d)
b
a
5.
u
u
6.
a
c.
7.
u
[a ≥ u > b > c > d]
BY (256.06)
[a > u ≥ b > c > d]
BY (257.01)
x−c
a−c
2
[F (ν, q) − E (ν, q)]
dx =
(x − a)(x − b)(x − d)3
a−d .
b−d
2
(u − a)(u − c)
+
a − d (u − b)(u − d)
[u > a > b > c > d]
BY (258.10)
x−c
2
dx =
3
(a − x)(x − b)(x − d)
a−d
a−c
E (μ, r)
b−d
b−x
2
dx =
3
(a − x)(c − x)(x − d)
(a − d)(c − d) (a − c)(b − d)
× [(b − c)(a − d) F (γ, r) − (a − c)(b − d) E (γ, r)]
.
(a − u)(c − u)
2(b − d)
+
(a − d)(c − d) (b − u)(u − d)
[a > b > c > u > d]
BY (253.03)
3.168
Square roots of polynomials
u.
8.
c
301
b−x
2
dx =
(a − x)(x − c)(x − d)3
(a − d)(c − d) (a − c)(b − d)
× [(a − c)(b − d) E (δ, q) − (a − b)(c − d) F (δ, q)]
b.
9.
u
[a > b ≥ u > c > d]
b−x
2
dx =
3
(a − x)(x − c)(x − d)
(a − d)(c − d) (a − c)(b − d)
u.
10.
b
11.
12.
13.
14.
BY (254.15)
× [(a − c)(b − d) E (κ, q) − (a − b)(c − d) F (κ, q)]
.
(b − u)(u − c)
2
−
c − d (a − u)(u − d)
[a > b > u ≥ c > d]
BY (255.06)
x−b
2
dx =
(a − x)(x − c)(x − d)3
(a − d)(c − d) (a − c)(b − d)
× [(a − c)(b − d) E (λ, r) − (a − d)(b − c) F (λ, r)]
.
(a − u)(u − b)
2
−
a − d (u − c)(u − d)
BY (256.03)
[a ≥ u > b > c > d]
a.
(a − c)(b − d)
x−b
E (μ, r)
dx = 2
3
(a
−
x)(x
−
c)(x
−
d)
(a
− d)(c − d)
u
2(b − c)
−
F (μ, r)
(c − d) (a − c)(b − d)
[a > u ≥ b > c > d]
BY (257.09)
u.
x−b
dx
(x − a)(x − c)(x − d)3
a
.
2(b − d)
2(a − b)
(u − a)(u − c)
=
+
F (ν, q)
(a − d)(c − d) (u − b)(u − d) (a − d) (a − c)(b − d)
(a − c)(b − d)
E (ν, q)
+2
(a − d)(c − d)
[u > a > b > c > d]
BY (258.09)
.
c
2
a−x
a−c
(a − u)(c − u)
2
[F (γ, r) − E (γ, r)] +
dx =
3
(b
−
x)(c
−
x)(x
−
d)
c
−
d
b
−
d
c
−
d
(b − u)(u − d)
u
[a > b > c > u > d]
BY (253.04)
u
a−x
a−c
2
E (δ, q)
dx =
(b − x)(x − c)(x − d)3
c−d b−d
c
[a > b ≥ u > c > d]
BY (254.01)
302
Power and Algebraic Functions
b
15.
u
a−x
2
dx =
3
(b − x)(x − c)(x − d)
c−d
u
16.
b
a
17.
u
18.
a−x
2
dx =
3
(x − b)(x − c)(x − d)
c−d
a−x
2
dx =
(x − b)(x − c)(x − d)3
c−d
a−c
2(a − d)
E (κ, q) −
b−d
(b − d)(c − d)
3.168
.
(b − u)(u − c)
(a − u)(u − d)
BY (255.08)
[a > b > u ≥ c > d]
.
a−c
(a − u)(u − b)
2
[F (λ, r) − E (λ, r)] +
b−d
b − d (u − c)(u − d)
BY (256.05)
[a ≥ u > b > c > d]
a−c
[F (μ, r) − E (μ, r)]
b−d
[a > u ≥ b > c > d]
.
u
2
x−a
a−c
(u − a)(u − c)
−2
E (ν, q) +
dx =
3
(x
−
b)(x
−
c)(x
−
d)
c
−
d
b
−
d
c
−
d
(u − b)(u − d)
a
d.
19.
u
u.
20.
d
[u > a > b > c > d]
d−x
2
dx =
(a − x)(b − x)(c − x)3
b−c
x−d
−2
dx =
3
(a − x)(b − x)(c − x)
b−c
BY (257.06)
BY (258.05)
b−d
[F (α, q) − E (α, q)]
a−c
[a > b > c > d > u]
.
2
b−d
(b − u)(u − d)
E (β, r) +
a−c
b − c (a − u)(c − u)
BY (251.01)
BY (252.06)
[a > b > c ≥ u > d]
.
2
x−d
b−d
(b − u)(u − d)
2
[F (κ, q) − E (κ, q)] +
dx
=
3
(a
−
x)(b
−
x)(x
−
c)
b
−
c
a
−
c
b
−
c
(a
− u)(u − c)
u
BY (255.05)
[a > b > u > c > d]
u.
x−d
b−d
2
E (λ, r)
dx =
3
(a
−
x)(x
−
b)(x
−
c)
b
−
c
a
−c
b
b.
21.
22.
a.
23.
u
x−d
2
dx =
3
(a − x)(x − b)(x − c)
b−c
[a ≥ u > b > c > d]
BY (256.01)
.
2(c − d)
b−d
(a − u)(u − b)
E (μ, r) −
a−c
(a − c)(b − c) (u − c)(u − d)
BY (257.06)
[a > u ≥ b > c > d]
.
2
x−d
b−d
(u − a)(u − d)
2
[F (ν, q) − E (ν, q)] +
dx
=
3
(x
−
a)(x
−
b)(x
−
c)
b
−
c
a
−
c
a
−
c
(u − b)(u − c)
a
BY (258.06)
[u > a > b > c > d]
a.
2
b−x
b−d
dx =
E (α, q)
3 (d − x)
(a
−
x)(c
−
x)
c
−
d
a
−c
u
u.
24.
25.
[a > b > c > d > u]
BY (251.01)
3.168
Square roots of polynomials
.
b−d
(b − u)(u − d)
2
[F (β, r) − E (β, r)] +
a−c
c − d (a − u)(c − u)
d
BY (252.03)
[a > b > c > u > d]
.
.
b
2
2
b−x
b−d
(b − u)(u − d)
dx =
E (κ, q) +
3
(a − x)(x − c) (x − d)
d−c a−c
c − d (a − u)(u − c)
u
u.
26.
27.
u.
28.
b
30.
32.
33.
34.
35.
[a > b > u > c > d]
2
x−b
dx =
3
(a − x)(x − c) (x − d)
c−d
BY (255.03)
b−d
[F (λ, r) − E (λ, r)]
a−c
BY (258.03)
[u > a > b > c > d]
2 (a − c)(b − d)
a−b
2
a−x
dx =
E (α, q) −
F (α, q)
3
(b − x)(c − x) (d − x)
(b − c)(c − d)
b − c (a − c)(b − d)
u
[a > b > c > d > u]
BY (251.08)
u
(a − c)(b − d)
2(a − d)
a−x
dx =
E (β, r)
F (β, r) − 2
3 (x − d)
(b
−
x)(c
−
x)
(b
− c)(c − d)
(c
−
d)
(a
−
c)(b
−
d)
d
.
a−c
(b − u)(u − d)
+2
(b − c)(c − d) (a − u)(c − u)
BY (252.04)
[a > b > c > u > d]
.
b
2(a − b)
a−x
(a − c)(b − d)
dx =
E (κ, q)
F (κ, q) − 2
(b − x)(x − c)3 (x − d)
(b − c)(c − d)
(b − c) (a − c)(b − c)
u
.
2(a − c)
(b − u)(u − d)
+
(b − c)(c − d) (a − u)(u − c)
BY (255.04)
[a > b > u > c > d]
u
2 (a − c)(b − d)
2(a − d)
a−x
dx =
E (λ, r) −
F (λ, r)
3 (x − d)
(x
−
b)(x
−
c)
(b
−
c)(c
−
d)
(c − d) (a − c)(b − d)
b
[a ≥ u > b > c > d]
BY (256.09)
a
2 (a − c)(b − d)
2(a − d)
a−x
dx =
E (μ, r) −
F (μ, r)
3
(x − b)(x − c) (x − d)
(b − c)(c − d)
(c − d) (a − c)(b − d)
u
.
2
(a − u)(u − b)
−
b − c (u − c)(u − d)
BY (257.04)
[a > u ≥ b > c > d]
d
31.
b−x
2
dx =
3
(a − x)(c − x) (x − d)
c−d
[a ≥ u > b > c > d]
BY (256.08)
.
2
2
x−b
b−d
(a − u)(u − b)
dx =
[F (μ, r) − E (μ, r)] +
3
(a − x)(x − c) (x − d)
c−d a−c
a − c (u − c)(u − d)
u
BY (257.03)
[a > u ≥ b > c > d]
.
.
u
2
2(b − c)
x−b
b−d
(u − a)(u − d)
dx =
E (ν, q) −
3 (x − d)
(x
−
a)(x
−
c)
c
−
d
a
−
c
(a
−
c)(c
−
d)
(u − b)(u − c)
a
a.
29.
303
304
Power and Algebraic Functions
2 (a − c)(b − d)
x−a
2(a − b)
dx =
E (ν, q) −
F (ν, q)
(x − b)(x − c)3 (x − d)
(b − c)(c − d)
(a − c)(b − d)
(b
−
c)
a
.
2
(u − a)(u − d)
−
c − d (u − b)(u − c)
BY (258.04)
[u > a > b > c > d]
.
d
2 (a − c)(b − d)
2(c − d)
d−x
dx =
E (α, q) −
F (α, q)
3
(a − x)(b − x) (c − x)
(a − b)(b − c)
(b − c) (a − c)(b − d)
u
.
2
(a − u)(d − u)
−
a − b (b − u)(c − u)
BY (251.11)
[a > b > c > d > u]
.
u
2 (a − c)(b − d)
2(a − d)
x−d
dx =
E (β, r) −
F (β, r)
3
(a − x)(b − x) (c − x)
(a − b)(b − c)
(a − b) (a − c)(b − d)
d
.
2
(c − u)(u − d)
+
b − c (a − u)(b − u)
BY (252.07)
[a > b > c ≥ u > d]
.
c
2 (a − c)(b − d)
2(a − d)
x−d
dx =
E (γ, r) −
F (γ, r)
3
(a − x)(b − x) (c − x)
(a − b)(b − c)
(a − b) (a − c)(b − d)
u
BY (253.07)
[a > b > c > u ≥ d]
.
u
2 (a − c)(b − d)
2(c − d)
x−d
dx
=
E (δ, q)
F
(δ,
q)
−
(a − x)(b − x)3 (x − c)
(a − b)(b − c)
(b − c) (a − c)(b
−
d)
c
.
2(b − d)
(a − u)(u − c)
+
(a − b)(b − c) (b − u)(u − d)
BY (254.05)
[a > b > u > c > d]
.
a
2 (a − c)(b − d)
2(a − d)
x−d
dx
=
E (μ, r)
F
(μ,
r)
−
(a − x)(x − b)3 (x − c)
(a − b)(b − c)
(a − b) (a − c)(b
−
d)
u
.
2(b − d)
(a − u)(u − c)
+
(a − b)(b − c) (u − b)(u − d)
[a > u > b > c > d]
BY (257.07)
.
u
2 (a − c)(b − d)
2(c − d)
x−d
dx
=
E (ν, q) −
F (ν, q)
3 (x − c)
(x
−
a)(x
−
b)
(a
−
b)(b
−
c)
(b
−
c)
(a − c)(b − d)
a
BY (258.07)
[u > a > b > c > d]
.
d
c−x
a−c
(a − u)(d − u)
2
2(b − c)
dx =
E (α, q) −
3 (d − x)
(a
−
x)(b
−
x)
a
−
b
b
−
d
(a
−
b)(b
−
d)
(b − u)(c − u)
u
[a > b > c > d > u]
u
36.
37.
38.
39.
40.
41.
42.
43.
3.168
3.168
Square roots of polynomials
u
44.
d
c
45.
u
u
46.
c
47.
48.
49.
50.
x−c
2
dx =
3
(a − x)(b − x) (x − d)
b−a
a−c
[F (γ, r) − E (γ, r)]
b−d
[a > b > c > u ≥ d]
.
a−c
(a − u)(u − c)
2
E (δ, q) +
b−d
a − b (b − u)(u − d)
BY (254.08)
[u > a > b > c > d]
BY (258.01)
.
2
2
a−x
a−c
(a − u)(d − u)
dx =
[F (α, q) − E (α, q)] +
3 (c − x)(d − x)
(b
−
x)
b
−
c
b
−
d
b
−
d
(b − u)(c − u)
u
BY (251.12)
[a > b > c > d > u]
.
u
2
2(a − b)
a−x
a−c
(u − d)(c − u)
dx =
E (β, r) −
3
(b − x) (c − x)(x − d)
b−c b−d
(b − c)(b − d) (a − u)(b − u)
d
d
c
u
53.
c−x
2
dx =
(a − x)(b − x)3 (x − d)
a−b
BY (254.08)
[a > b ≥ u > c > d]
.
a
2
2
x−c
a−c
(a − u)(u − c)
dx
=
[F (μ, r) − E (μ, r)] +
3 (x − d)
(a
−
x)(x
−
b)
a
−
b
b
−
d
a
−
b
(u − b)(u − d)
u
BY (257.10)
[a > u ≥ b > c > d]
u
2
x−c
a−c
dx =
E (ν, q)
3
(x − a)(x − b) (x − d)
a−b b−d
a
51.
52.
.
a−c
(c − u)(u − d)
2
[F (β, r) − E (β, r)] +
b−d
b − d (a − u)(b − u)
BY (252.10)
[a > b > c ≥ u > d]
c−x
2
dx =
3
(a − x)(b − x) (x − d)
a−b
305
2
a−x
dx =
(b − x)3 (c − x)(x − d)
b−c
[a > b > c ≥ u > d]
BY (252.09)
a−c
E (γ, r)
b−d
[a > b > c > u ≥ d]
BY (253.01)
.
u
2
2
a−x
a−c
(a − u)(u − c)
dx
=
[F (δ, q) − E (δ, q)] +
3 (x − c)(x − d)
(b
−
x)
b
−
c
b
−
d
b
−
c
(b − u)(u − d)
c
BY (254.06)
[a > b > u > c > d]
.
a
2
2
a−x
a−c
(a − u)(u − c)
dx =
E (μ, r) +
3 (x − c)(x − d)
(x
−
b)
c
−
b
b
−
d
b
−
c
(u − b)(u − d)
u
u
54.
a
2
x−a
dx =
(x − b)3 (x − c)(x − d)
b−c
[a > u > b > c > d]
BY (257.08)
a−c
[F (ν, q) − E (ν, q)]
b−d
[u > a > b > c > d]
BY (258.08)
306
Power and Algebraic Functions
d.
55.
u
d−x
2
dx =
3
(a − x) (b − x)(c − x)
b−a
d
58.
59.
61.
62.
63.
64.
(b − u)(d − u)
(a − u)(c − u)
(a −
2
x−d
dx =
− x)(c − x)
a−b
x)3 (b
BY (251.09)
b−d
[F (β, q) − E (β, q)]
a−c
[a > b > u ≥ c > d]
BY (255.01)
.
x−d
b−d
(u − b)(u − d)
2
2
dx
=
[F (λ, r) − E (λ, r)] +
3 (x − b)(x − c)
(a
−
x)
a
−
b
a
−
c
a
−
b
(a
− u)(u − c)
b
BY (256.10)
[a > u > b > c > d]
d
2 (a − c)(b − d)
c−x
2(c − d)
dx =
E (α, q)
F (α, q) −
3
(a − x) (b − x)(d − x)
(a − b)(a − d)
(a − d) (a − c)(b
u
. − d)
2(a − c)
(b − u)(d − u)
+
(a − b)(a − d) (a − u)(c − u)
[a > b > c > d > u]
BY (251.15)
u
2 (a − c)(b − d)
2(b − c)
c−x
dx =
E (β, r) −
F (β, r)
3 (b − x)(x − d)
(a
−
x)
(a
−
b)(a
−
d)
(a
−
b)
(a − c)(b − d)
d
[a > b > c ≥ u > d]
BY (252.08)
c
2 (a − c)(b − d)
c−x
2(b − c)
dx =
E (γ, r) −
F (γ, r)
3 (b − x)(x − d)
(a
−
x)
(a
−
b)(a
−
d)
(a − b) (a − c)(b − d)
u
.
2
(c − u)(u − d)
−
a − d (a − u)(b − u)
BY (253.10)
[a > b > c > u ≥ d]
u
2 (a − c)(b − d)
2(c − d)
x−c
dx =
E (δ, q) −
F (δ, q)
3
(a − x) (b − x)(x − d)
(a − b)(a − d)
(a − d) (a − c)(b − d)
c
.
2
(b − u)(u − c)
−
a − b (a − u)(u − d)
BY (254.09)
[a > b ≥ u > c > d]
u.
60.
.
[a > b > c ≥ u > d]
BY (252.05)
.
x−d
b−d
(c − u)(u − d)
2
2
dx =
[F (γ, r) − E (γ, r)] +
3
(a − x) (b − x)(c − x)
a−b a−c
a − c (a − u)(b − u)
u
BY (253.05)
[a > b > c > u ≥ d]
.
.
u
2
2(a − d)
x−d
b−d
(b − u)(u − c)
dx =
E (δ, q) −
3
(a − x) (b − x)(x − c)
a−b a−c
(a − b)(a − c) (a − u)(u − d)
c
BY (254.03)
[a > b ≥ u > c > d]
.
b
2
x−d
b−d
dx =
E (κ, q)
3 (b − x)(x − c)
(a
−
x)
a
−
b
a
−c
u
c.
57.
b−d
2
E (α, q) +
a−c
a−b
[a > b > c > d > u]
u.
56.
3.168
3.169
Square roots of polynomials
307
2 (a − c)(b − d)
x−c
2(c − d)
dx =
E (κ, q) −
F (κ, q)
(a − x)3 (b − x)(x − d)
(a − b)(a − d)
(a − d) (a − c)(b − d)
u
[a > b > u ≥ c > d]
BY (255.10)
u
2 (a − c)(b − d)
x−c
2(b − c)
dx =
E (λ, r)
F (λ, r) −
3 (x − b)(x − d)
(a
−
x)
(a − b)(a − d)
(a − b) (a − c)(b
b
. − d)
2(a − c)
(u − b)(u − d)
+
(a − b)(a − d) (a − u)(u − c)
BY (256.07)
[a > u > b > c > d]
.
d.
2
2
b−x
b−d
(b − u)(d − u)
dx =
[F (α, q) − E (α, q)] +
3 (c − x)(d − x)
(a
−
x)
a
−
d
a
−
c
a
−
d
(a
− u)(c − u)
u
BY (251.13)
[a > b > c > d > u]
u.
2
b−x
b−d
dx =
E (β, r)
3 (c − x)(x − d)
(a
−
x)
a
−
d
a
−c
d
b
65.
66.
67.
68.
c.
2
b−x
dx =
(a − x)3 (c − x)(x − d)
a−d
69.
u
u.
70.
c
b.
71.
u
u.
72.
b
(a −
x−b
−2
dx =
− c)(x − d)
a−d
x)3 (x
[a > b > c > u ≥ d]
BY (253.08)
.
2
b−d
(b − u)(u − c)
[F (δ, q) − E (δ, q)] +
a−c
a − c (a − u)(u − d)
BY (254.07)
[a > b ≥ u > c > d]
2
b−x
dx =
(a − x)3 (x − c)(x − d)
a−d
b−x
2
dx =
3
(a − x) (x − c)(x − d)
a−d
[a > b > c ≥ u > d]
BY (252.01)
.
2(a − b)
b−d
(c − u)(u − d)
E (γ, r) −
a−c
(a − c)(a − d) (a − u)(b − u)
b−d
[F (κ, q) − E (κ, q)]
a−c
[a > b > u ≥ c > d]
.
b−d
(u − b)(u − d)
2
E (λ, r) +
a−c
a − d (a − u)(u − c)
[a ≥ u > b > c > d]
3.169
Notation: In 3.169–3.172, we set:
a
β = arctan ,
u
BY (256.04)
b
a2 + b 2
u
ε = arccos ,
ξ = arcsin
δ = arccos ,
,
b
u
a2 + u 2
a b2 − u2
a u2 − b 2
ζ = arcsin
,
κ
=
arcsin
,
b a2 − u 2
u a2 − b 2
√
a2 − b 2
u 2 − a2
a
,
q
=
,
μ = arcsin
,
ν
=
arcsin
u2 − b 2
u
a
b
a
b
r= √
,
s= √
t= .
2
2
2
2
a
a +b
a +b
a2 + b 2
,
a2 + u 2
u
η = arcsin ,
b
2
a − u2
λ = arcsin
,
a2 − b 2
u
γ = arcsin
b
u
α = arctan ,
b
BY (255.07)
308
Power and Algebraic Functions
u
1.
0
u
2.6
0
u
3.
0
b
4.
5.
x2 + a2
dx = a {F (α, q) − E (α, q)} + u
x2 + b2
x2 + b2
b2
F (α, q) − a E (α, q) + u
dx =
2
2
x +a
a
x2 + a2
dx = a2 + b2 E (γ, r) − u
2
2
b −x
a2 + u 2
b2 + u2
6.
0
u > 0]
BY (221.03)
[a > b,
u > 0]
BY (221.04)
a2 + u 2
b2 + u2
b 2 − u2
a2 + u 2
[b ≥ u > 0]
b2 − x2
dx = a2 + b2 {F (γ, r) − E (γ, r)} + u
2
2
a +x
[u > b > 0]
BY (214.11)
b2 − x2
dx
=
a2 + b2 {F (δ, r) − E (δ, r)}
[b > u ≥ 0]
a2 + x2
u
u 2
x − b2
1 2
dx =
(a + u2 ) (u2 − b2 ) − a2 + b2 E (ε, s)
2
2
a +x
u
b
u
0
b2 − x2
a2 − b 2
F (η, t)
dx
=
a
E
(η,
t)
−
a2 − x2
a
u
b2 − x2
a2 − b 2
F (ζ, t) − u
dx = a E (ζ, t) −
2
2
a −x
a
9.
b
10.
u
11.
b
a
12.
u
u
13.
a
u
14.
0
BY (211.03)
b2 − u2
a2 + u 2
[b ≥ u > 0]
b
8.
[a > b,
a2 + x2
dx = a2 + b2 E (δ, r)
[b > u ≥ 0]
BY (213.01), ZH 64 (273)
2
2
b −x
u
u 2
a + x2
1 2
dx = a2 + b2 {F (ε, s) − E (ε, s)} +
(u + a2 ) (u2 − b2 )
2
2
x −b
u
b
u
7.
3.169
BY (214.03)
BY (213.03)
[u > b > 0]
BY (211.04)
[a > b ≥ u > 0]
BY (219.03)
b2 − u2
a2 − u 2
[a > b > u ≥ 0]
BY (220.04)
x2 − b2
1 2
b2
F
(κ,
q)
−
dx
=
a
E
(κ,
q)
−
(a − u2 ) (u2 − b2 )
a2 − x2
a
u
x2 − b2
b2
F (λ, q)
dx
=
a
E
(λ,
q)
−
a2 − x2
a
[a ≥ u > b > 0]
BY (217.04)
[a > u ≥ b > 0]
BY (218.03)
x2 − b2
u 2 − a2
a2 − b 2
F
(μ,
t)
−
a
E
(μ,
t)
+
μ
dx
=
x2 − a2
a
u2 − b 2
a2 − x2
dx = a E (η, t)
b2 − x2
[u > a > b > 0]
BY (216.03)
[a > b ≥ u > 0]
H 64 (276), BY (219.01)
3.171
Square roots of polynomials
309
a2 − x2
u b2 − u2
dx = a E (ζ, t) −
[a > b > u ≥ 0]
b2 − x2
a a2 − u 2
u
u 2
a − x2
1 2
dx = a {F (κ, q) − E (κ, q)} +
(a − u2 ) (u2 − b2 )
2
2
x −b
u
b
b
15.
16.
a
17.
18.
3.171
1.
2.
a2 − x2
dx = a {F (λ, q) − E (λ, q)}
x2 − b2
u
u 2
x − a2
u 2 − a2
dx = u
− a E (μ, t)
2
2
x −b
u2 − b 2
a
[a ≥ u > b > 0]
BY (217.03)
[a > u ≥ b > 0]
BY (218.09)
[u > a > b > 0]
BY (216.04)
√
a2 + x2
a2 + b 2
=
E (ε, s)
[u > b > 0]
2
2
x −b
b2
b
√
∞
dx a2 + x2
u2 − b 2
a2 + b 2
a2
=
E (ξ, s) − 2
2
2
2
2
x
x −b
b
b u a2 + u 2
u
u
b
3.
u
u
4.
b
a
5.
u
u
6.
a
dx
x2
dx
x2
dx
x2
dx
x2
∞
7.
u
∞
8.
u
b
9.
u
dx
x2
[u ≥ b > 0]
a2 − x2
b2 − u2
a2 − b 2
a
a2
=
F (ζ, t) − 2 E (ζ, t) + 2
2
2
2
b −x
ab
b
b u a2 − u 2
dx
x2
a2 − x2
a
1
= 2 E (κ, q) − F (κ, q)
x2 − b2
b
a
a2 − x2
a
1
= 2 E (λ, q) − f (λ, q) −
x2 − b2
b
a
dx
x2
dx
x2
BY (220.03)
BY (212.09)
[a > b > u > 0]
BY (220.12)
[a ≥ u > b > 0]
BY (217.11)
(a2 − u2 ) (u2 − b2 )
b2 u
x2 − a2
a
a2 − b 2
1
=
E
(μ,
t)
−
F (μ, t) −
x2 − b2
b2
ab2
u
BY (211.01), ZH 64 (274)
a
x2 + a2
1
a2
F
(β,
q)
−
=
E
(β,
q)
+
x2 + b2
a
b2
b2 u
x2 + b2
1
1
= {F (β, q) − E (β, q)} +
2
2
x +a
a
u
[a > u ≥ b > 0]
BY (218.10)
u 2 − a2
u2 − b 2
[u > a > b > 0]
BY (216.08)
b2 + u2
a2 + u 2
[a > b,
u > 0]
BY (222.08)
u > 0]
BY (222.09)
b2 + u2
a2 + u 2
[a > b,
√
(b2 − u2 ) (a2 + u2 )
b2 − x2
a2 + b 2
−
=
E (δ, r)
2
2
2
a +x
a u
a2
[b > u > 0]
BY (213.10)
310
Power and Algebraic Functions
11.
12.
13.
14.
15.
16.
17.
18.
√
x2 − b2
a2 + b 2
=
{F (ε, s) − E (ε, s)}
[a > b > 0]
2
2
a +x
a2
b
√
∞
dx x2 − b2
a2 + b 2
1 u2 − b 2
=
{F (ξ, s) − E (ξ, s)} +
x2 a2 + x2
a2
u a2 + u 2
u
[u ≥ b > 0]
√
b
(b2 − u2 ) (a2 + u2 )
dx a2 + x2
a2 + b 2
=
{F
(δ,
r)
−
E
(δ,
r)}
+
2
b2 − x2
b2
b2 u
u x
[b > u > 0]
∞
dx x2 − a2
a
a2 − b 2
=
E
(ν,
t)
−
F (ν, t)
[u ≥ a > b > 0]
x2 x2 − b2
b2
ab2
u
b
dx b2 − x2
1 b 2 − u2
1
[a > b > u > 0]
=
− E (ζ, t)
2
2
2
2 − u2
x
a
−
x
u
a
a
u
u
dx x2 − b2
1
[a ≥ u > b > 0]
= {F (κ, q) − E (κ, q)}
2
2
2
a −x
a
b x
a
(a2 − u2 ) (u2 − b2 )
dx x2 − b2
1
{F
(λ,
q)
−
E
(λ,
q)}
+
=
2
u2 − x2
a
a2 u
u x
[a > u ≥ b > 0]
u
1 u 2 − a2
dx x2 − b2
1
= E (μ, t) −
[u > a > b > 0]
2
2
2
x −a
a
u u2 − b 2
a x
∞
dx x2 − b2
1
[u ≥ a > b > 0]
= E (ν, t)
x2 x2 − a2
a
u
10.
3.172
u
dx
x2
BY (211.07)
BY (212.11)
BY (213.05)
BY (215.08)
BY (220.11)
BY (217.08)
BY (218.08)
BY (216.07)
BY (215.01), ZH 65 (281)
3.172
u.
1.
3
∞
.
2.
u
u.
3.
∞
.
4.
u
u.
5.
0
+
3
a2 )
x2 + a2
(x2
0
x2 + b2
(x2
+
3
b2 )
x2 + a2
(x2 + b2 )
3
a2 − b 2
u
1
E (α, q) −
2
2
2
a
a
(a + u ) (b2 + u2 )
dx =
(x2 + a2 )
0
x2 + b2
[a > b,
u > 0]
BY (221.10)
1
E (β, q)
a
[a > b,
u ≥ 0]
H 64 (271)
a
E (α, q)
b2
[a > b,
u > 0]
H 64 (270)
u ≥ 0]
BY (222.06)
dx =
dx =
dx =
u
a
a2 − b 2
E
(β,
q)
−
2
2
b2
b2
(a + u ) (b2 + u2 )
[a > b,
√
b2 − x2
a2 + b 2
1
dx
=
E (γ, r) − √
F (γ, r)
3
2
2
2
2
a
a + b2
(a + x )
[b ≥ u > 0]
BY (214.08)
3.172
Square roots of polynomials
b.
6.
u
√
a2 + b 2
1
u b2 − u2
E (δ, r) − √
F (δ, r) − 2
3 dx =
a2
a
a2 + u 2
a2 + b 2
(a2 + x2 )
b2 − x2
u.
7.
b
∞
[b > u ≥ 0]
√
x2 − b2
a2 + b 2
b2
1 u2 − b 2
E (ε, s) − √
F (ε, s) −
3 dx =
a2
u u 2 + a2
a2 a2 + b 2
(a2 + x2 )
.
8.
u.
9.
∞
10.
u
u.
11.
14.
dx =
BY (211.06)
a2 + b 2
b2
E (ξ, s) − √
F (ξ, s)
2
2
a
a a2 + b 2
a2
√
F (γ, r) −
b 2 a2 + b 2
√
[u ≥ b > 0]
BY (212.08)
2
a + b2 u
a2 + b 2
E (γ, r) + b2
b2 (a2 + u2 ) (b2 − u2 )
b2 − x2
−
3
x2 )
dx =
1
a
F (η, t) − E (η, t) +
u
a
[u > b > 0]
b2 − u2
a2 − u 2
[a > b ≥ u > 0]
b.
13.
3
x2 )
dx =
BY (213.04)
BY (214.09)
[b > u > 0]
2
√
a + b2 u
x2 + a2
a2 + b 2
1
√
dx
=
F
(ξ,
s)
−
E
(ξ,
s)
+
3
b2
a2 + b 2
b2 (a2 + u2 ) (u2 − b2 )
(x2 − b2 )
(a2
0
3
x2 )
x2 + a2
.
√
x2 − b2
(b2 −
0
12.
[u > b > 0]
(a2 +
u
311
b2 − x2
BY (212.07)
BY (219.09)
1
{F (ζ, t) − E (ζ, t)}
[a > b > u ≥ 0]
a
−
u
u. 2
x − b2
1 u2 − b 2
1
[a > u > b > 0]
dx
=
− E (κ, q)
3
2 − u2
2
2
u
a
a
(a − x )
b
∞. 2
1 u2 − b 2
x − b2
1
3 dx = a [F (ν, t) − E (ν, t)] + u
u 2 − a2
(x2 − a2 )
u
BY (217.07)
[u > a > b > 0]
BY (215.05)
(a2
u.
15.
0
a.
16.
u
u.
17.
a
3
x2 )
a2 − x2
3
(b2 − x2 )
a2 − x2
(x2 −
3
b2 )
x2 − a2
3
(x2 − b2 )
dx =
dx =
a
u
[F (η, t) − E (η, t)] + 2
2
b
b
a2 − u 2
a
− 2 E (λ, q)
2
2
u −b
b
dx =
u
b2
dx =
a
[F (μ, t) − E (μ, t)]
b2
BY (220.07)
a2 − u 2
b2 − u2
[a > b > u > 0]
BY (219.10)
[a > u > b > 0]
BY (218.05)
[u > a > b > 0]
BY (216.05)
312
Power and Algebraic Functions
∞
.
18.
u
x2 − a2
a
1
3 dx = b2 [F (ν, t) − E (ν, t)] + u
2
2
(x − b )
u 2 − a2
u2 − b 2
[u ≥ a > b > 0]
3.173
1
1.
u
u
2.
1
3.174
dx
x2
dx
x2
x2 + 1 √
= 2E
x2 − 1
√ 2
1
arccos ,
u 2
√ 1− 1+ 3 u
√
,
β = arccos
1+
3−1 u
u
1.
0
2.
3.
4.
3.175
BY (215.03)
% √ √ & √
x2 + 1 √
2
2
1 − u4
= 2 F arccos u,
− E arccos u,
+
2
1−x
2
2
u
Notation: In 3.174 and 3.175, we take:
3.173
.
dx
√ 2
1+ 1+ 3 x
[u < 1]
BY (259.77)
[u > 1]
BY (260.76)
√ 1+ 1− 3 u
√ ,
1+ 1+ 3 u
√
√
2+ 3
2− 3
,
q=
.
p=
2
2
α = arccos
1
1 − x + x2
= √
E (α, p)
4
x(1 + x)
3
[u > 0]
BY (260.51)
.
1
dx
1 + x + x2
= √
E (β, q)
[1 ≥ u > 0]
BY (259.51)
√
2
4
x(1
−
x)
3
0
1+
3−1 x
√ √ √
u
2 2+ 3 1+ 1− 3 u
dx
x(1 + x)
1
2− 3
√ √
√
F (α, p) −
E (α, p) + − √
4
4
2
1 − x + x2
27
3
1+ 1+ 3 u
27
0 1−x+x
u(1 + u)
×
1 − u + u2
[u > 0]
BY (260.54)
√ √ √
u
2 2− 3 1− 1+ 3 u
dx
x(1 − x)
4
2+ 3
√
√
√
E (β, q) − √
F (β, q) −
4
4
2
2
1
+
x
+
x
1
+
x
+
x
27
27
3
3−1 u
1+
0
u(1 − u)
×
1 + u + u2
[1 ≥ u > 0]
BY (259.55)
u
u
1.
0
2.
0
u
dx
1+x
dx
1−x
u (1 − u + u2 )
x
2
1
√ [F (α, p) − 2 E (α, p)] + √ √
= √
4
3
1+x
27
3 1+u 1+ 1+ 3 u
[u > 0]
u (1 + u + u2 )
x
2
1
√
√
√
[F
(β,
q)
−
2
E
(β,
q)]
+
=
√
4
1 − x3
27
3 1−u 1+
3−1 u
[0 < u < 1]
BY (260.55)
BY (259.52)
3.183
Fourth roots of polynomials
313
3.18 Expressions that can be reduced to fourth roots of second-degree polynomials
and their products with rational functions
3.181
1.
2.
3.182
.
&
1
1
4 4(a − u)(u − b)
= a−b 2 E √
,√
+ E arccos
4
(a − b)2
2
2
(a − x)(x − b)
b
.
&
%
1
4(a − u)(u − b) 1
,√
− K √
+ F arccos 4
(a − b)2
2
2
[a ≥ u > b]
BY (271.05)
%
u
a − b − 2 (u − a)(u − b) 1
dx
a−b
F
arccos
,√
4
2
2
(x − a)(x − b)
a − b + 2 (u − a)(u − b)
a
&
a − b − 2 (u − a)(u − b) 1
− 2 E arccos
,√
2
a − b + 2 (u − a)(u − b)
4
2(2u − a − b) (u − a)(u − b)
+
a − b + 2 (u − a)(u − b)
BY (272.05)
[u > a > b]
u
u
1.
b
%
√
dx
%
2
=√
K
4
a−b
[(a − x)(x − b)]3
dx
.
1
√
2
+F
arccos
4(a − u)(u − b) 1
,√
(a − b)2
2
&
[a ≥ u > b]
√
u
a − b − 2 (u − a)(u − b) 1
dx
2
,√
=√
F arccos
4
a−b
2
[(x − a)(x − b)]3
a − b + 2 (u − a)(u − b)
a
BY (271.01)
2.
[u > a > b]
3.183
BY (272.00)
Notation: In 3.183–3.186 we set:
1
,
α = arccos √
4
u2 + 1
u
1.
0
u
2.
0
3.
1
u
√
dx
√
= 2 F
4
x2 + 1
dx
√
=
4
1 − x2
√
dx
√
=F
4
x2 − 1
β = arccos
1
α, √
2
2 2E
1
γ, √
2
− 2E
1
β, √
2
− 2E
−F
4
1−
1
α, √
2
1
β, √
2
1
γ, √
2
+
u2 ,
√
1 − u2 − 1
√
γ = arccos
.
1 + u2 − 1
2u
+ √
4
u2 + 1
[u > 0]
[0 1]
BY (273.55)
BY (271.55)
BY (272.55)
314
3.184
Power and Algebraic Functions
u
1.
0
u
2.
x2
1
3.185
u
1.
0
u
2.
0
3.
4.
5.
dx
√
=E
4
x2 − 1
1
β, √
2
1
γ, √
2
−
1
β, √
2
−F
1
γ, √
2
1
F
2
2u
5
−
BY (272.54)
u
0
u
7.
0
u
1.
0
u
2.
0
3.
1
BY (271.59)
√
dx
<
= 2F
3
4
(x2 + 1)
1
α, √
2
[u > 0]
BY (273.50)
√
dx
<
= 2F
3
4
(1 − x2 )
1
β, √
2
[0 1]
BY (272.50)
[0 0]
BY (273.54)
u
6.
<
4
3
(1 − u2 )
[0 1]
dx
1
<
= F γ, √
3
4
2
1
(x2 − 1)
√
u
1
x2 dx
2 2
2 4
<
F β, √
=
− u 1 − u2
3
3
2
0 4 (1 − x2 )3
u
√
√
dx
1
1
<
− 2 F α, √
= 2 2 E α, √
5
4
2
2
0
(x2 + 1)
3.186
√ x2 dx
2 2
√
=
2E
4
5
1 − x2
3.184
u
x2 dx
<
5
4
(x2 + 1)
2
√
=2 2 F
x dx
1
<
= √ F
7
4
3 2
(x2 + 1)
1
α, √
2
1
α, √
2
√
√
1 + x2 + 1
√
dx = 2 2 E
4
(x2 + 1) x2 + 1
− 2E
1
α, √
2
2u
+ √
4
u2 + 1
[u > 0]
BY (273.56)
u
− <
3
4
6 (u2 + 1)
[u > 0]
BY (273.53)
1
α, √
2
[u > 0]
BY (273.51)
√
dx
√
√
= 2 F
1 + 1 − x2 4 1 − x2
dx
1
√
√
=
F
2
x2 + 2 x2 − 1 4 x2 − 1
1
β, √
2
1
γ, √
2
−E
−E
1
β, √
2
√
u 4 1 − u2
√
+
1 + 1 − u2
[0 1]
BY (271.58)
BY (272.53)
3.194
Powers of x and binomials
√
√
1 − 1 − x2
dx
√
· <
= 2 2E
1 + 1 − x2 4 (1 − x2 )3
u
4.
0
u
2
5.
1
x dx
=E
√
<
3
4
x2 + 2 x2 − 1
(x2 − 1)
1
β, √
2
1
β, √
2
−F
1
γ, √
2
315
√
2u 4 1 − u2
√
−
1 + 1 − u2
[0 1]
BY (272.51)
3.19–3.23 Combinations of powers of x and powers of binomials of the form (α+βx)
3.191
u
xν−1 (u − x)μ−1 dx = uμ+ν−1 B(μ, ν)
1.
[Re μ > 0,
Re ν > 0]
ET II 185(7)
0
∞
2.
x−ν (x − u)μ−1 dx = uμ−ν B(ν − μ, μ)
u
1
ν−1
3.
x
(1 − x)
μ−1
[Re ν > Re μ > 0]
1
xμ−1 (1 − x)ν−1 dx = B(μ, ν)
dx =
0
0
[Re μ > 0,
3.192
1
1.
0
1
2.
0
1
3.
0
[−1 0]
EH I 2
[|arg(1 + βu)| < π,
Re μ > 0]
ET I 310(20)
∞
u
3.
n!nν+n
ν(ν + 1)(ν + 2) . . . (ν + n)
xμ−1 dx
uμ
=
2 F 1 (ν, μ; 1 + μ; −βu)
ν
(1 + βx)
μ
2.6
dx
= π sec pπ
x
xν−1 (n − x)n dx =
BI (3)(4)
(1 − x)p
dx = −π cosec pπ
xp+1
3.193
0
BI (3)(5)
u
FI II 774(1)
[−1 0]
xp dx
= −π cosec pπ
(1 − x)p+1
∞
1.
xp dx
= pπ cosec pπ
(1 − x)p
4.
3.194
ET II 201(6)
xμ−1 dx
uμ−ν
= ν
2 F 1 ν, ν − μ;
ν
(1 + βx)
β (ν − μ)
ν − μ + 1;
−
1
βu
[Re ν > Re μ]
∞
ET I 310(21)
μ−1
x
dx
= β −μ B(μ, ν − μ)
(1 + βx)ν
[|arg β| < π,
Re ν > Re μ > 0]
FI II 775a, ET I 310(19)
316
Power and Algebraic Functions
4.
∞
11
0
0
μ−1
μ
u
x
dx
=
1 + βx
μ
[|arg β| < π,
0 < Re μ < n + 1]
2 F 1 (1, μ; 1
+ μ; −βu)
[|arg(1 + uβ)| < π,
Re μ > 0]
ET I 308(5)
∞
6.
0
cosec(μπ)
ET I 308(6)
u
5.
μ−1
n
xμ−1 dx
π
= (−1)n μ
n+1
(1 + βx)
β
3.195
∞
7.
xμ−1 dx
(1 − μ)π
=
cosec μπ
(1 + βx)2
βμ
BI (16)(4)
1
xm dx
1
(a + bx)n+ 2
0
[0 < Re μ < 2]
= 2m+1 m!
(2n − 2m − 3)!! am−n+ 2
(2n − 1)!!
bm+1
m < n − 12 ,
a > 0,
b>0
BI (21)(2)
∞
xn−1 dx
−n
=
2
(1 + x)m
1
8.
0
k=0
∞
3.19511
0
m − n − 1 (−2)−k
k
n+k
BI (3)(1)
(1 + x)p−1
1 − a−p
dx
=
(a + x)p+1
p(a − 1)
ln a
=
a−1
=1
[p = 0,
a > 0,
a = 1]
[p = 0,
a > 0,
a = 1]
[a = 1]
LI (19)(6)
3.196
u
(x + β)ν (u − x)μ−1 dx =
1.
0
∞
2.
β ν uμ
u
2 F 1 1, −ν; 1 + μ; −
μ
β
!
!
!
!
!arg u ! < π
!
β!
(x + β)−ν (x − u)μ−1 dx = (u + β)μ−ν B(ν − μ, μ)
u
!
!
!
!
!arg u ! < π,
!
β!
ET II 185(8)
Re ν > Re μ > 0
ET II 201(7)
b
(x − a)μ−1 (b − x)ν−1 dx = (b − a)μ+ν−1 B(μ, ν)
3.
[b > a,
Re μ > 0,
Re ν > 0]
a
EH I 10(13)
∞
4.
1
5.
1
dx
π
= − cosec νπ
(a − bx)(x − 1)ν
b
dx
π
= cosec νπ
ν
(a
−
bx)(1
−
x)
b
−∞
ν
b
b−a
b
a−b
[a < b,
b > 0,
0 < ν < 1]
LI (23)(5)
ν
[a > b > 0,
0 < ν < 1]
LI (24)(10)
3.197
3.197
Powers of x and binomials
∞
1.
|arg γ| < π,
[|arg β| < π,
∞
u
1
3.
γ
β
Re μ > Re(ν − )]
xν−1 (β + x)−μ (x + γ)− dx = β −μ γ ν− B(ν, μ − ν + ) 2 F 1 μ, ν; μ + ; 1 −
0
2.11
317
Re ν > 0,
ET II 233(9)
β
x−λ (x + β)ν (x − u)μ−1 dx = u−λ (β + u)μ+ν B(λ − μ − ν, μ) 2 F 1 λ, μ; λ − μ; −
! !
!
u
!
!β !
!
!
u
!arg ! < π or ! ! < 1, 0 < Re μ < Re(λ − ν)
ET II 201(8)
!u!
!
β!
xλ−1 (1 − x)μ−1 (1 − βx)−ν dx = B(λ, μ) 2 F 1 (ν, λ; λ + μ; β)
0
1
4.
Re μ > 0,
[Re μ > 0,
Re ν > 0,
∞
[|arg α| < π,
∞
xλ−ν (x − 1)ν−μ−1 (αx − 1)−λ dx = α−λ B(μ, ν − μ) 2 F 1 ν, μ; λ; α−1
1
[1 + Re ν > Re λ > Re μ,
∞
7.
0
xμ− 2 (x + a)−μ (x + b)−μ dx =
1
√
π
√
√
a+ b
1−2μ
|arg(α − 1)| < π]
0
EH I 115(6)
Γ μ − 12
Γ(μ)
[Re μ > 0]
u
8.
− Re(μ + ν) > Re λ > 0]
EH I 60(12), ET I 310(23)
6.
a > −1]
xλ−1 (1 + x)ν (1 + αx)μ dx = B(λ, −μ − ν − λ) 2 F 1 (−μ, λ; −μ − ν; 1 − α)
0
WH
BI(5)4, EH I 10(11)
5.
|β| < 1]
xμ−1 (1 − x)ν−1 (1 + ax)−μ−ν dx = (1 + a)−μ B(μ, ν)
0
[Re λ > 0,
u
xν−1 (x + α)λ (u − x)μ−1 dx = αλ uμ+ν−1 B(μ, ν) 2 F 1 −λ, ν; μ + ν; −
α
+!
u !!
!
<
π,
Re μ > 0,
arg
!
!
α
BI 19(5)
,
Re ν > 0
ET II 186(9)
∞
9.
xλ−1 (1 + x)−μ+ν (x + β)−ν dx = B(μ − λ, λ) 2 F 1 (ν, μ − λ; μ; 1 − β)
0
[Re μ > Re λ > 0]
10.
11.
1
EH I 205
π
xq−1 dx
=
cosec qπ
[0 < q < 1, p > −1]
BI (5)(1)
q (1 + px)
(1
−
x)
(1
+
p)q
0
√ 1
1
2 Γ p + 12 Γ(1 − p)
xp− 2 dx
√ sin (2p − 1) arctan q
2p
√ √
cos (arctan q)
=
p
p
π
(2p − 1) sin arctan q
0 (1 − x) (1 + qx)
1
BI (11)(1)
− 2 0
318
Power and Algebraic Functions
1
12.
0
3.198
√ 1−2p √ 1−2p
1
Γ p + 12 Γ(1 − p) 1 − q
− 1+ q
xp− 2 dx
√
=
√
(1 − x)p (1 − qx)p
(2p − 1) q
π
1
− 2 0,
Re μ > 0,
Re ν > 0]
FI II 787
(x − a)μ−1 (b − x)ν−1 (x − c)−μ−ν dx = (b − a)μ+ν−1 (b − c)−μ (a − c)−ν B(μ, ν)
a
[Re μ > 0,
Re ν > 0,
c < a < b]
EH I 10(14)
1
3.211
xλ−1 (1 − x)μ−1 (1 − ux)− (1 − vx)−σ dx = B(μ, λ) F 1 ((λ, , σ, λ + μ; u, v))
0
[Re λ > 0, Re μ > 0]
EH I 231(5)
q
a
+
b
(1 + ax)−p + (1 + bx)−p xq−1 dx = 2(ab)− 2 B(q, p − q) cos q arccos √
2 ab
0
BI (19)(9)
[p > q > 0]
∞
q
a
+
b
(1 + ax)−p − (1 + bx)−p xq−1 dx = −2i(ab)− 2 B(q, p − q) sin q arccos √
2 ab
0
BI (19)(10)
[p > q > 0]
1
(1 + x)μ−1 (1 − x)ν−1 + (1 + x)ν−1 (1 − x)μ−1 dx = 2μ+ν−1 B(μ, ν)
3.212
3.213
3.214
∞
0
[Re μ > 0,
Re ν > 0]
LI(1)(15), EH I 10(10)
1
3.215
μ μ−1
a x
(1 − ax)ν−1 + (1 − a)ν xν−1 [1 − (1 − a)x]μ−1 dx = B(μ, ν)
0
[Re μ > 0,
Re ν > 0,
|a| < 1]
BI (1)(16)
3.216
1
1.
0
xμ−1 + xν−1
dx = B(μ, ν)
(1 + x)μ+ν
∞
2.
1
Re ν > 0]
FI II 775
[Re μ > 0,
Re ν > 0]
FI II 775
bp xp−1
(1 + bx)p−1
−
dx = π cot pπ
[0 0]
(1 + bx)p
bp−1 xp
0 ∞ 2p−1
x
− (a + x)2p−1
dx = π cot pπ
[p < 1] (cf. 3.217)
(a + x)p xp
0 ∞
xν
xμ
−
dx = ψ(μ + 1) − ψ(ν + 1)
ν+1
(x + 1)
(x + 1)μ+1
0
[Re μ > −1, Re ν > −1]
3.217
3.218
3.219
3.221
xμ−1 + xν−1
dx = B(μ, ν)
(1 + x)μ+ν
[Re μ > 0,
1.
a
∞
∞
(x − a)p−1
dx = π(a − b)p−1 cosec pπ
x−b
[a > b,
0 < p < 1]
BI(18)(13)
BI (18)(7)
BI (19)(13)
LI (24)(8)
3.226
Powers of x and binomials
2.
3.222
(a − x)p−1
dx = −π(b − a)p−1 cosec pπ
x−b
−∞
319
a
1
1.
0
xμ−1 dx
= β(μ)
1+x
∞
2.
0
[a < b,
0 0]
xμ−1 dx
= π cosec(μπ)aμ−1
x+a
= −π cot(μπ)(−a)μ−1
WH
for a > 0
FI II 718, FI II 737
for a < 0
BI(18)(2), ET II 249(28)
[0 < Re μ < 1]
3.223
∞
1.
0
∞
2.
0
∞
3.
0
0
∞
1.
1
∞
2.
1
∞
3.
0
1
1.
0
2.
0
1
μ−1
a
x
dx
= π cot(μπ)
(a − x)(b − x)
∞
3.224
3.226
π μ−1
xμ−1 dx
=
β
cosec(μπ) + αμ−1 cot(μπ)
(β + x)(α − x)
α+β
[|arg β| < π,
0 < Re μ < 2]
α > 0,
ET I 309(7)
0 < Re μ < 2]
ET I 309(8)
3.225
π μ−1
xμ−1 dx
=
β
− γ μ−1 cosec(μπ)
(β + x)(γ + x)
γ−β
[|arg β| < π, |arg γ| < π,
−b
b−a
μ−1
(x + β)xμ−1 dx
= π cosec(μπ)
(x + γ)(x + δ)
μ−1
[a > b > 0,
0 < Re μ < 2]
ET I 309(9)
γ − β μ−1 δ − β μ−1
γ
δ
+
γ−δ
δ−γ
[|arg γ| < π, |arg δ| < π,
0 < Re μ < 1]
ET I 309(10)
(x − 1)p−1
dx = (1 − p)π cosec pπ
x2
[−1 < p < 1]
BI (23)(8)
(x − 1)1−p
1
dx = p(1 − p)π cosec pπ
x3
2
[0 < p < 1]
BI (23)(1)
xp dx
π
= p(1 − p) cosec pπ
3
(1 + x)
2
[−1 < p < 2]
BI (16)(5)
xn dx
(2n)!!
√
=2
(2n + 1)!!
1−x
BI (8)(1)
1
xn− 2 dx
(2n − 1)!!
√
π.
=
(2n)!!
1−x
BI (8)(2)
320
3.227
Power and Algebraic Functions
∞
1.
0
∞
2.
0
3.228
b
1.
a
3.227
xν−1 (β + x)1−μ
γ
dx = β 1−μ γ ν−1 B(ν, μ − ν) 2 F 1 μ − 1, ν; μ; 1 −
γ+x
β
[|arg β| < π, |arg γ| < π, 0 < Re ν < Re μ]
ET II 217(9)
x− (β − x)−σ
dx = πγ − (β − γ)−σ cosec(π) I 1−γ/β (σ, )
γ+x
[|arg β| < π, |arg γ| < π, − Re σ < Re < 1]
ET II 217(10)
(x − a)ν (b − x)−ν
dx = π cosec(νπ) 1 −
x−c
a−c
b−c
= π cosec(νπ) 1 − cos(νπ)
c−a
= π cosec(νπ) 1 −
c−b
ν
ν
c−a
b − c
ν
for c < a
for a < c b
[|Re ν| < 1]
b
2.
a
(x − a)ν−1 (b − x)−ν
x−c
!
!ν−1
π cosec(νπ) !! a − c !!
dx =
!b−c!
b−c
ν−1
π(c − a)
=−
cot(νπ)
(b − c)ν
ET II 250(31)
for c < a or c > b;
for a < c b;
= π(c − a)ν−1 (b − c)μ−1 cot μπ − (b − a)μ+ν−2 B(μ − 1, ν)
× 2 F 1 2 − μ − ν, 1; 2 − μ;
[Re μ > 0,
4.
0
5.
0
1
b−c
b−a
Re ν > 0,
π(a − b)ν−1
(1 − x)ν−1 x−ν
dx =
cosec(νπ)
a − bx
aν
∞
for a < c < b
μ + ν = 1,
μ = 1, 2, . . .]
[0 < Re ν < 1,
0 0, 0 < Re ν < Re μ]
ET II 250(33)
BI (5)(8)
for c < 0;
for c > 0
ET II 251(34)
3.237
Powers of x and binomials
∞
6.
xν−1
0
⎡
−n
(γ + x)
x+β
ν−1
dx =
π
β
⎣1 −
sin πν (γ − β)n
γ
β
[|arg β| < π,
1
3.229
0
3.231
1
1.
0
1
2.11
0
1
3.
0
1
4.
0
1
5.
0
|arg γ| < π,
xp−1 + x−p
dx = π cosec pπ
1+x
1
xp − x−p
dx = − π cot pπ
x−1
p
xp − x−p
1
dx = − π cosec pπ
1+x
p
xμ−1 − xν−1
dx = ψ(ν) − ψ(μ)
1−x
∞
x
∞
0
∞
3.233
0
1
1.11
0
1
2.
0
−x
1−x
p−1
3.232
3.237
γ−β
γ
(1 − ν)j
j!
j
⎤
⎦
0 < Re ν < n]
AS 256 (6.1.22)
xμ−1 dx
π cosec μπ
a
b
=
−
(1 − x)μ (1 + ax)(1 + bx)
a−b
(1 + a)μ
(1 + b)μ
[0 < Re μ < 1]
0
3.23610
j=0
xp−1 − x−p
dx = π cot pπ
1−x
6.
3.235
ν−1 n−1
p2 < 1
p2 < 1
p2 < 1
p2 < 1
BI (5)(7)
BI (4)(4)
BI (4)(1)
BI (4)(3)
BI (4)(2)
[Re μ > 0,
Re ν > 0]
FI II 815, BI(4)(5)
3.234
321
q−1
dx = π (cot pπ − cot qπ)
(c + ax)−μ − (c + bx)−μ
b
dx = c−μ ln
x
a
1
− (1 + x)−ν
1+x
[p > 0,
q > 0]
[Re μ > −1;
a > 0;
FI II 718
b > 0;
c > 0]
BI (18)(14)
dx
= ψ(ν) + C
x
[Re ν > 0]
EH I 17, WH
xq−1
x−q
−
1 − ax a − x
dx = πa−q cot qπ
[0 < q < 1,
a > 0]
BI (5)(11)
xq−1
x−q
+
1 + ax a + x
dx = πa−q cosec qπ
[0 < q < 1,
a > 0]
BI (5)(10)
∞
(1 + x)μ − 1 dx
= ψ(ν) − ψ(ν − μ)
[Re ν > Re μ > 0]
(1 + x)ν
x
0 1
μ
1−μ
x 2 dx
μ
(1 − a)−μ − (1 + a)−μ
√
Γ 1+
=
Γ
μ+1
2
2
2
2aμ
π
0 [(1 − x) (1 − a x)] 2
[−2 < μ < 1, |a| < 1]
u 2
n+1
∞
u Γ 2
dx
= ln (−1)n+1
[|arg u| < π]
2
x
+
u
u+1
n
n=0
2 Γ
2
BI (18)(5)
BI (12)(32)
ET II 216(1)
322
3.238
Power and Algebraic Functions
∞
ν−1
νπ ν−1
|x|
dx = −π cot
|u|
sign u
2
−∞ x − u
1.
3.238
[0 < Re ν < 1 u real,
u = 0]
ET II 249(29)
∞
ν−1
2.
−∞
νπ ν−1
|x|
sign x dx = π tan
|u|
x−u
2
[0 < Re ν < 1 u real,
u = 0]
ET II 249(30)
b
3.
(b − x)μ−1 (x − a)ν−1
|x − u|
a
μ+ν
dx =
(b − a)μ+ν−1 Γ(μ) Γ(ν)
μ
ν
|a − u| |b − u| Γ(μ + ν)
[Re μ > 0, Re ν > 0, 0 < u < a < b and 0 < a < b < u]
MO 7
3.24–3.27 Powers of x, of binomials of the form α + βxp and of polynomials in x
3.241
1
1.
0
xμ−1 dx
1
= β
p
1+x
p
∞
2.
0
4.
0
∞
5.
6.10
3.242
1.
pπ
xp−1 dx
π
= cot
1 − xq
q
q
xμ−1 dx
n+1
(p + qxν )
0
μ ν−μ
,
ν
ν
p > 0]
WH, BI (2)(13)
[Re ν > Re μ > 0]
ET I 309(15)a, BI (17)(10)
∞
PV
11
[Re μ > 0,
μπ
1
xμ−1 dx
π
= B
= cosec
1 + xν
ν
ν
ν
3.11
μ
p
1
= n+1
νp
[p < q]
p
q
μ/ν
Γ
μ μ
ν Γ 1+n− ν
Γ(1 + n)
+
μ
0 < < n + 1,
ν
BI (17)(11)
,
p = 0,
BI (17)(22)a
∞
(p − q)π
(p − q)π
[p < 2q]
=
cosec
2
q )2
q
q
(1
+
x
0
b
p
b−u
x
x 1/p
−
G(x) =
sign
du = (b − a)F
c
b−a
c
a
where
⎧
⎪
x≤0
1
⎨−1
sign(x − t) dt = 2x − 1 0 < x < 1
F (x) =
⎪
0
⎩
1
x≥1
xp−1 dx
(2n − 2m − 1)
π
(2m + 1)π
x2m dx
=
sin
t
cosec t cosec
4n
2n
+ 2x cos t + 1
n
2n
2n
−∞ x
2
m < n, t < π 2
q=0
BI (17)(18)
∞
FI II 642
3.247
Powers of x and binomials and polynomials
2.
∞
11
0
∞
3.24311
0
3.244
1
1.
0
2.
3.
4.
x2
x4 + 2ax2 + 1
c
x2 + 1
xb + 1
323
dx
1 1
= 2−1/2−c (1 + a)1/2−c B c − ,
2
x
2 2
xμ−1 dx
(1 + x2ν ) (1 + x3ν ) π
4π
2π
=
8 cosec(2ρ) + 12 cosec(3ρ) − 8 cosec 2ρ −
+ 8 cosec 2ρ −
48ν
3
3
,
π
π
cosec ρ +
sec(ρ)
−3 cosec ρ −
6
6
μπ
, [0 < Re μ < 5 Re ν] ET I 312(34)
where ρ =
6ν
pπ
xp−1 + xq−p−1
π
dx = cosec
1 + xq
q
q
pπ
xp−1 − xq−p−1
π
dx = cot
q
1−x
q
q
0
1 ν−1
+
μ−1
μ ,
x
−x
1
dx
=
C
+
ψ
1 − xν
ν
ν
0
∞ 2m
2n
2m + 1
x −x
π
π − cot
dx =
cot
2l
l
2l
−∞ 1 − x
[q > p > 0]
BI (2)(14)
[q > p > 0]
BI (2)(16)
[Re μ > Re ν > 0]
BI (2)(17)
1
2n + 1
π
2l
[m < l,
∞
n < l]
FI II 640
ν−μ
x
− xν (1 + x)−μ dx =
ν
B(ν, μ − ν)
ν−μ+1
0
[Re μ > Re ν > 0]
∞
qπ
pπ
(p + q)π
1 − xq p−1
π
cosec
cosec
x
dx = sin
r
r
r
r
r
0 1−x
[p + q < r, p > 0]
3.245
3.246
BI (16)(13)
ET I 331(33), BI (17)(12)
dx
can be transformed by the substitution x = et
Integrals of the form f x ± x , x ± x , . . .
x
1
∞
1+p
1−p −1
−t
x
or x = e . For example, instead of
+x
dx, we should seek to evaluate
sech px dx
0
0
1 n−m−1
∞
x
+ xn+m−1
−1
and, instead of
dx, we should seek to evaluate
cosh mx (cosh nx − cos a) dx
n
2n
0 1 + 2x cos a + x
0
(see 3.514 2).
3.247
1 α−1
∞
x
(1 − x)n−1
ξk
dx
=
(n
−
1)!
1.11
b
1 − ξx
(α + kb)(α + kb + 1) . . . (α + kb + n − 1)
0
p
−p
q
−q
k=0
2.
0
[b > 0,
∞
(1 − x ) x
1 − xnp
p
ν−1
dx =
|ξ| < 1]
AD (6704)
π
πν
π
(p + ν)π
sin
cosec
cosec
np
n
np
np
[0 < Re ν < (n − 1)p]
ET I 311(33)
324
3.248
Power and Algebraic Functions
∞
1.
0
1
2.
0
3.
4.3
6.∗
3.249
1.0
2.9
3.
4.
5.
xμ−1 dx
1
√
= B
ν
1 + xν
μ 1 μ
, −
ν 2 ν
3.248
[Re ν > Re 2μ > 0]
BI (21)(9)
x2n+1 dx
(2n)!!
√
=
(2n + 1)!!
1 − x2
BI (8)(14)
x2n dx
(2n − 1)!! π
√
=
2
(2n)!! 2
1−x
0
∞
dx
π
√
=
2 ) 4 + 3x2
3
(1
+
x
−∞
1
BI (8)(13)
⎧
⎪
2
b
⎪
⎪
√
−1
arctan
⎪
⎪
⎪
a
b−a
⎪
∞
⎨
dx
2
= √
2 2
⎪
a
−∞ 1 + x
⎪
b + ax2
√
⎪
√
⎪
⎪ 1
a+ a−b
⎪
⎪
√
√
ln √
⎩
a−b
a− a−b
if a b
∞
dx
π
(2n − 3)!!
=
2n−1
2 + a2 )n
2
·
(2n
−
2)!!
a
(x
0
a
2
n− 12
(2n − 1)!!
π.
a − x2
dx = a2n
2(2n)!!
0
n
1
1 − x2 dx
= 2n+1 Q n (a)
n+1
−1 (a − x)
1 μ
μ+1
x dx
1
= β
2
1
+
x
2
2
0
1
μ−1
1 − x2
dx = 22μ−2 B(μ, μ) = 12 B 12 , μ
FI II 743
FI II 156
EH II 181(31)
[Re μ > −1]
BI (2)(7)
[Re μ > 0]
FI II 784
[p > 0]
BI (7)(7)
0
1
6.
1−
√ p−1
x
dx =
0
1
− ν1
(1 − xμ )
7.
dx =
0
∞
8.11
1+
−∞
3.251
1.
0
2.
0
1
x2
n−1
2
p(p + 1)
1
1
,1 −
μ
ν
π(n − 1)
Γ
dx =
Γ n2
1
B
μ
−n/2
[Re μ > 0,
n−1
2
ν−1
μ
1
,ν
xμ−1 1 − xλ
dx = B
λ
λ
|ν| > 1]
[n > 1]
[Re μ > 0,
Re ν > 0,
λ > 0]
FI II 787
∞
ν−1
μ
μ
1
,1 − ν −
xμ−1 1 + x2
dx = B
2
2
2
Re μ > 0,
Re ν + 12 μ < 1
3.252
Powers of x and binomials and polynomials
∞
3.
ν−1
xμ−1 (xp − 1)
dx =
1
∞
0
∞
0
x2m+1 dx
m!(n − m − 2)!
n =
2(n − 1)!am+1 cn−m−1
(ax2 + c)
xμ+1
(1 +
0
1
7.
2
x2 )
xμ dx
(1 +
0
x2m dx
(2m − 1)!!(2n − 2m − 3)!!π
√
n =
2 · (2n − 2)!!am cn−m−1 ac
(ax2 + c)
∞
6.
2
x2 )
dx =
μπ
4 sin μπ
2
xq+p−1 (1 − xq )
−p
q
dx =
0
1
9.
q
x p −1 (1 − xq )
1
−p
1
−p
q
xp−1 (1 − xq )
∞
11.
0
3.252
∞
1.
0
2.
3.
c > 0,
n > m + 1]
∞
μ−1
2
pπ
pπ
cosec
2
q
q
[q > p]
BI (9)(22)
pπ
π
cosec
q
q
[q > p > 0]
μ
μ
1 − μp
β
,ν −
B
p
p
p
[|arg β| < π, p > 0,
WH
LI (3)(11)
dx =
dx =
GU (141)(8b)
[Re μ > 1]
[p > 1,
−ν
xμ−1 (1 + βxp )
n > m + 1 ≥ 1]
[ac > 0,
π
π
cosec
q
p
0
[a > 0,
dx =
0
10.
Re μ 0,
GU (141)(8a)
5.
[p > 0,
ET I 311(32)
4.
μ
1
B 1 − ν − ,ν
p
p
325
q > 0]
BI (9)(23)a
BI (9)(20)
0 < Re μ 0, ac > b2
dx
(2n − 3)!!πan−1
a > 0,
=
n
2 + 2bx + c)
n− 12
(ax
2
−∞
(2n − 2)!! (ac − b )
∞
1
dx
(−2)n ∂ n
√ √
=
n+ 32
(2n + 1)!! ∂cn
c ( ac + b)
0 (ax2 + 2bx + c)
a ≥ 0,
ac > b2
c > 0,
GW (131)(4)
GW (131)(5)
√ b > − ac
GW (213)(4)
326
Power and Algebraic Functions
∞
4.
0
x dx
n
(ax2 + 2bx + c)
b
(−1)n ∂ n−2
b
1
−
=
arccot √
for ac > b2 ;
(n − 1)! ∂cn−2 2 (ac − b2 ) 2 (ac − b2 ) 32
ac − b2
√
b
b + b2 − ac
1
(−1)n ∂ n−2
√
+
ln
for b2 > ac > 0;
=
(n − 1)! ∂cn−2 2 (ac − b2 ) 4 (b2 − ac) 32
b − b2 − ac
an−2
=
for ac = b2
2(n − 1)(2n − 1)b2n−2
[a > 0, b > 0, n ≥ 2]
GW (141)(5)
∞
5.
−∞
x dx
(2n − 3)!!πban−2
n =−
(2n−1)
+ 2bx + c)
(2n − 2)!! (ac − b2 ) 2
ac > b2 ,
n≥2
a > 0,
GW (141)(6)
m
−∞
(ax2
m
xn dx
n+ 32
(ax2 + 2bx + c)
0
n−m−1 m
x dx
b
(−1) πa
n =
n− 1
+ 2bx + c)
(2n − 2)!! (ac − b2 ) 2
[m/2]
k
m
ac − b2
(2k − 1)!!(2n − 2k − 3)!!
×
2k
b2
k=0
ac > b2 , 0 ≤ m ≤ 2n − 2
∞
7.
(ax2
∞
6.
3.252
=
n!
√ √
n+1
(2n + 1)!! c ( ac + b)
a ≥ 0,
c > 0,
GW (141)(17)
√ b > − ac
GW (213)(5a)
∞
8.
xn+1 dx
(ax2
0
n+ 32
=
+ 2bx + c)
n!
√ √
n+1
(2n + 1)!! a ( ac + b)
a > 0,
c ≥ 0,
√ b > − ac
GW (213)(5b)
∞
9.
10.6
0
x
dx
n+1
(ax2 + 2bx + c)
0
n+ 12
=
1
22n+ 2
(2n − 1)!!π
√ n+ 1 √
(b + ac) 2 n! a
a > 0,
c > 0,
b+
√
ac > 0
LI (21)(19)
∞
μ−1
1
x
dx
1
−ν
ν− 12
(sin t) 2 t Γ ν +
ν =2
2
2
(1 + 2x cos t + x )
1
−ν
2
B(μ, 2ν − μ) P μ−ν−
1 (cos t)
2
[0 < t < π,
0 < Re μ < Re 2ν]
ET I 310(22)
3.255
Powers of x and binomials and polynomials
∞
11.
327
μ− 12 −ν−1
μ
1 + 2βx + x2
x
dx = 2−μ β 2 − 1 2 Γ(1 − μ) B(ν − 2μ + 1, −ν) P μν−μ (β)
0
[Re ν < 0,
Re(2μ − ν) < 1,
|arg (β ± 1)| < π]
EH I 160(33)
1
2 −μ
= −π cosec νπ C ν (β)
−2 < Re 12 − μ < Re ν < 0,
|arg (β ± 1)| < π
EH I 178(24)
∞
12.
0
xμ−1 dx
= −πaμ−2 cosec t cosec(μπ) sin[(μ − 1)t]
x2 + 2ax cos t + a2
[a > 0, 0 < |t| < π,
0 < Re μ < 2]
FI II 738, BI(20)(3)
∞
13.
xμ−1 dx
(x2
0
2
a2 )
+ 2ax cos t +
=
πaμ−4
cosec μπ cosec3 t
2
× {(μ − 1) sin t cos [(μ − 2)t] − sin[(μ − 1)t]}
[a > 0,
∞
14.
0
0 < |t| < π,
xμ−1 dx
√
= π cosec(μπ) P μ−1 (cos t)
1 + 2x cos t + x2
0 < Re μ < 4]
[−π < t < π,
LI(20)(8)a, ET I 309(13)
0 < Re μ < 1]
ET I 310(17)
1
3.253
−1
(1 + x)2μ−1 (1 − x)2ν−1
μ+ν
(1 + x2 )
dx = 2μ+ν−2 B(μ, ν)
[Re μ > 0,
Re ν > 0]
FI II 787
3.254 u
ν
xλ−1 (u − x)μ−1 x2 + β 2 dx
1.
0
λ λ + 1 λ + μ λ + μ + 1 −u2
;
,
; 2
= β 2ν uλ+μ−1 B(λ, μ) 3 F 2 −ν, ,
2
2
2
β
2
u
Re
ET II 186(10)
> 0, λ > 0, Re μ > 0
β
∞
2.6
−λ
ν
x (x − u)μ−1 x2 + β 2
dx
u
Γ(μ) Γ(λ − μ − 2ν)
Γ(λ − 2ν)
1+λ−μ
λ
1+λ
β2
λ−μ
− ν,
− ν; − ν,
− ν; − 2
× 3 F 2 −ν,
2
2
2
2
u
β
|u| > |β| and Re
ET II 202(9)
> 0, 0 < Re μ < Re(λ − 2ν)
u
√
Γ μ + 12
π
dx = √
√ 2 μ+ 12 √
Γ(μ + 1)
a+
c + 2b − a + c
c + 2b − a
√
√ 2
c + 2b − a + c > 0, c + 2b − a > 0, Re μ > − 12
a+
= uμ−λ+2ν
1
3.255
0
1
1
xμ+ 2 (1 − x)μ− 2
μ+1
(c + 2bx − ax2 )
BI (14)(2)
328
Power and Algebraic Functions
3.256
1
1.
xp−1 + xq−1
(1 − x2 )
0
p+q
2
q−p
π sec
4
1
cos
2
dx =
3.256
q+p
p q
π B
,
4
2 2
[p > 0,
2.
0
1
xp−1 − xq−1
(1 −
∞
9
3.257
0
%
x2 )
p+q
2
b
ax +
x
q−p
π cosec
4
1
sin
2
dx =
+c
∞
1.
x−
√
π Γ p + 12
∞
x2 − a2
n
dx =
∞
n
x2 + 1 − x
b
3.
5.
q > 0,
p + q < 2]
BI (8)(26)
x2 + a2 − x
a > 0,
b < 0,
c > 0,
p > − 12
a > 0,
b > 0,
c > −4ab,
a2
b − b 2 − a2
2(n − 1)
n−1
−
n+1
n
dx =
na
n2 − 1
n ≥ 2]
a
n+1
GW (215)(5)
[n ≥ 2]
GW (214)(7)
[n ≥ 2]
GW (214)(6a)
[a > 0,
∞
0
7.
∞
6.
BI (20)(4)
p > − 12
dx
n
[n ≥ 2]
√
n = n−1 2
2
2
a
(n
− 1)
x
+
x
+
a
0
∞
n
n · m!am+n+1
xm
x2 + a2 − x dx =
(n − m − 1)(n − m + 1) . . . (m + n + 1)
0
1
b − b 2 − a2
2(n + 1)
[0 < a ≤ b,
√
√
n−1
n+1
b2 + 1 − b
b2 + 1 − b
+
dx =
2(n − 1)
2(n + 1)
0
4.
[p > 0,
q+p
p q
π B
,
4
2 2
1
2acp+ 2
b
2.
BI (8)(25)
dx
Γ(p + 1)
1 B p + 12 , 12
=
1
2
p+
a (4ab + x) 2
p + q < 2]
&−p−1
2
=
3.258
q > 0,
GW (214)(6)
x dx
n · m!
√
n =
2
2
(n − m − 1)(n − m + 1) . . . (m + n + 1)an−m−1
x+ x +a
m
[a > 0,
∞
0 ≤ m ≤ n − 2]
GW (214)(5a)
(x − a)m x −
x2 − a2
n
dx =
0 ≤ m ≤ n − 2]
GW (214)(5)
n · (n − m − 2)!(2m + 1)!am+n+1
2m (n + m + 1)!
[a > 0,
n ≥ m + 2]
GH (215)(6)
3.264
3.259
Powers of x and binomials and polynomials
1
l
xp−1 (1 − x)n−1 (1 + bxm ) dx = (n − 1)!
1.6
0
∞
k=0
l
k
329
bk Γ(p + km)
Γ(p + n + km)
[|b| < 1 unless l = 0, 1, 2, . . . ;
2.
u
p, n, p + ml > 0]
BI (1)(14)
λ
xν−1 (u − x)μ−1 (xm + β m ) dx
11
0
= β mλ uμ+ν−1 B (μ, ν)
×
∞
3.11
m+1 F m
xλ−1 (1 + αxp )
−λ,
−μ
ν+m−1 μ+ν μ+ν+1
μ + ν + m − 1 −um
ν ν+1
,
,...,
;
,
,...,
; m
m m
m
m !
m
β
!
m
!
!
u
π
!<
Re μ > 0, Re ν > 0, !!arg
ET II 186(11)
β ! m
−ν
(1 + βxp )
0
[|arg α| < π,
3.261
1.11
PV
0
2.
3.
4.
1
λ
λ
β
1 −λ/p
λ
α
,μ + ν −
B
2 F 1 ν, ; μ + ν; 1 −
p
p
p
p
α
|arg β| < π, p > 0, 0 < Re λ < 2 Re(μ + ν)] ET I 312(35)
dx =
∞
(1 − x cos t) xμ−1 dx cos kt
=
1 − 2x cos t + x2
μ+k
[Re μ > 0,
BI (6)(9)
2
(xν + x−ν ) dx
π sin νt
ν < 1, t = (2n + 1)π
=
BI (6)(8)
2
1
+
2x
cos
t
+
x
sin
t
sin
νπ
0
1 1+p
x
+ x1−p dx
π (p sin t cos pt − cos t sin pt)
=
2 )2
2 sin3 t sin pπ
(1
+
2x
cos
t
+
x
0
2
p < 1, t = (2n + 1)π
BI (6)(18)
1
μ−1
x
dx
π cosec t cosec μπ
a sin t
·
=
μ sin t − μ arctan
2
2
μ
2
(1 − x)
(1 + 2a cos t + a ) 2
1 + a cos t
0 1 + 2ax cos t + a x
1
[a > 0,
0 < Re μ < 1]
BI (6)(21)
∞
(1 − p)π
x−p dx
π
[−2 0, |arg γ| < π, −1 < Re ν < 2, ν = 0] ET II 216(7)
3.262
3.263
3.264
t = 2nπ]
k=0
1.
0
∞
π ap−2 + bp−2 cos
xp−1 dx
=
(a2 + x2 ) (b2 − x2 )
2
a2 + b 2
pπ
2
cosec
pπ
2
[0 0,
b > 0]
BI (19)(14)
330
Power and Algebraic Functions
∞
2.
0
∞
3.
0
∞
4.
0
dx
π
2 = 2
2
2
(b + x ) (a + b + x )
dx
π
3 = 4
2
2
(b + x ) (a + b + x )
dx
4
(b + x2 ) (a + b + x2 )
0
∞
6.
0
∞
7.
0
1
3.265
0
=
⎛
0
1
2
|arg γ| < π,
0 < Re μ < 4]
ET I 309(4)
MC
2
3
1
2
−
−
−
5/2
3/2
1/2
2
3
a3 b1/2
4a (a + b)
a (a + b)
a (a + b)
π⎝ 2
5
3
−
−
7/2
5/2
2
4 a4 b1/2
8a (a + b)
4a (a + b)
⎞
1
2
⎠
−
−
3/2
1/2
3
4
a (a + b)
a (a + b)
π 1
π
−
n−1/2
n
2 an b1/2
2a (a + b)
n−1
12 j
j=0
j!
1 − n, 1;
2F 1
a+b
3
− n;
2
a
AS 263 (6.6.3.2)
a
a+b
j
a + b > 0]
π
β
γ
x2 dx
π
=
−
=
2
2
2
2
2
2
2
2
(x + α ) (x + β ) (x + γ )
2α (β − γ ) β + α γ + α
2(α + β)(α + γ)(β + γ)
∞
3.266
μ=
1
1
1
−
−
3/2
1/2
2
a2 b1/2
2a (a + b)
a (a + b)
1 − xμ−1
dx = ψ(μ) + C
1−x
= ψ(1 − μ) + C − π cot(μπ)
dx
n
(b + x2 ) (a + b + x2 )
π 1
1
1 1
=
−
B n− ,
n−1/2
2 an b1/2
2
2
2a (a + b)
=
μ
π γ 2 −1 − β 2 −1
μπ
xμ−1 dx
=
cosec
2
2
(β + x ) (γ + x )
2
β−γ
2
1
π
1
√ −√
=
2(γ − β)
γ
β
[|arg β| < π,
∞
5.
μ
3.265
π
(xν − aν ) dx
=
(x − a)(β + x)
a+β
[n > 0,
[Re μ > 0]
FI II 796, WH, ET I 16(13)
[Re μ > 0]
EH I 16(15)a
β
aν
ln
π
a
[|arg β| < π, |Re ν| < 1,
β ν cosec(νπ) − aν cot(νπ) −
ν = 0]
ET II 216(8)
3.267
1.
2.
Γ n + 13
x3n dx
2π
√
√
=
3
3 3 Γ 13 Γ(n + 1)
1 − x3
0
1 3n−1
(n − 1)! Γ 23
x
dx
√
=
3
3 Γ n + 23
1 − x3
0
1
BI (9)(6)
BI (9)(7)
3.272
3.
∗
Powers of x and binomials and polynomials
1
0
3.268
Γ n − 13 Γ 23
x3n−2 dx
√
=
3
3 Γ n + 13
1 − x3
1
1
pxp−1
−
1 − x 1 − xp
1.
0
2.
3.
331
dx = ln p
BI (5)(14)
1 − xμ ν−1
x
dx = ψ(μ + ν) − ψ(ν)
0 1−x
1
n
n
xμ−1
n−k
√
−
dx
=
nC
+
ψ μ+
n
1−x 1− x
n
0
1
[Re ν > 0,
Re μ > 0]
BI (2)(3)
k=1
[Re μ > 0]
3.269
1
1.
0
1
2.
0
1
3.
0
3.271
pπ
1
xp − x−p
π
−
x dx = cot
2
1−x
2
2
p
pπ
xp − x−p
1 π
x dx = − cosec
1 + x2
p
2
2
xμ − xν
1
dx = ψ
2
1−x
2
∞
1.
0
∞
2.
0
4.
π
xp − xq dx
=
x−1 x+a
1+a
−
μ+1
2
1
ψ
2
1
a2p − 1
− ap ln a
sin(2pπ) π
BI (4)(12)
BI (4)(8)
[Re μ > −1,
ap − cos pπ aq − cos qπ
−
sin pπ
sin qπ
π
xp − ap xp − 1
dx =
x−a x−1
a−1
p2 < 1
p2 < 1,
Re ν > −1]
q 2 < 1,
1
p2 <
4
a>0
BI (2)(9)
BI (19)(2)
BI (19)(3)
1 p
2 (a − 1) cot pπ − (a + 1) ln a
π
0
2
p <1
BI (18)(9)
∞ p
ap+q − 1
ap − aq
sin pπ
x − ap 1 − x−p q
π
x dx =
+
x
−
a
1
−
x
a
−
1
sin[(p
+
q)π]
sin[(q
−
p)π]
sin qπ
0
(p + q)2 < 1, (p − q)2 < 1
3.
ν+1
2
p2 < 1
BI (13)(10)
∞
π
xp − ap x−p − 1
dx =
x−a x−1
a−1
p
BI (19)(4)
∞
5.
0
3.272
1.
0
1
2
xp − x−p
1−x
dx = 2 (1 − 2pπ cot 2pπ)
0 < p2 <
1
4
BI (16)(3)
1
xn−1 + xn− 2 − 2x2n−1
dx = 2 ln 2
1−x
BI (8)(8)
332
Power and Algebraic Functions
1
2.
0
3.273
1
1.
0
2
3.273
1
xn−1 + xn− 3 + xn− 3 − 3x3n−1
dx = 3 ln 3
1−x
BI (8)(9)
(k − 1)!ak−1 sin kt
sin t − an xn sin[(n + 1)t] + an+1 xn+1 sin nt
p−1
(1
−
x)
dx
=
Γ(p)
1 − 2ax cos t + a2 x2
Γ(p + k)
n
k=1
[p > 0]
1
2.
0
BI (6)(13)
(k − 1)!ak−1 cos kt
cos t − ax − an xn cos[(n + 1)t] + an+1 xn+1 cos nt
p−1
(1
−
x)
dx
=
Γ(p)
1 − 2ax cos t + a2 x2
Γ(p + k)
n
k=1
[p > 0]
x
0
1
0
cos kt
1 − x cos t − xn+1 cos[(n + 1)t] + xn+2 cos nt
dx
=
1 − 2x cos t + x2
k+1
n
∞
0
1
0
1
1 − xn
dx
= n+1
(1 + x)n+1 1 − x
2
∞
3.
0
3.275
1
1.
0
1
2.
0
1
3.
0
π
μπ
(μ + 1)π
xμ−1 (1 − x)
π
cosec
dx = sin cosec
1 − xn
n
n
n
n
[0 < Re μ < n − 1]
2.
xn−1
pxnp−1
−
1−x
1 − x1/p
nxn−1
xmn−1
−
n
1−x
1−x
xp−1
qxpq−1
−
1−x
1 − xq
∞
%
1.
0
∞
k=1
b
ax +
x
BI (20)(13)
2k
k
BI (5)(3)
dx = p ln p
[p > q]
BI (18)(6)
[p > 0]
BI (13)(9)
n−k
1
ψ m+
n
n
n
dx = C +
BI (5)(13)
k=1
dx = ln q
1
1
−
1 + x2n
1 + x2m
0
3.276
n
π
qπ
xq − 1 dx
=
tan
xp − x−p x
2p
2p
4.
10
BI (6)(11)
k=0
1.
BI (6)(12)
k=1
4.
3.274
sin kt
sin t − xn sin[(n + 1)t] + xn+1 sin nt
dx =
2
1 − 2x cos t + x
k+1
n
1
3.
BI (6)(14)
[q > 0]
BI (5)(12)
dx
= 0.
x
BI (18)(17)
&−p−1
2
+c
x2
dx
1
=
2|b|
B p + 12 , 12
1
p+ 2
(2a (b + |b|) + c)
a > 0,
c > −4ac,
p > − 12
3.278
Powers of x and binomials and polynomials
2.
∞
10
0
3.277
∞
11
1.
xμ−1
0
%
b
a+ 2
x
∞ xμ−1
2.
√
1 + x2 + β
√
1 + x2
+
0
ν
&−p−1
2
b
ax +
x
+c
B p + 12 , 12
1
p+ 2
(4ab + c)
a > 0,
b > 0,
c > −4ac,
p > − 12
ν+μ
μ
μ
ν+ μ
dx = 2 2 −1 β 2 − 1 2 4 Γ
Γ(1 − μ − ν) P μ −12 (β)
2
2
[Re β > −1, 0 < Re μ < 1 − Re ν]
ET I 310(25)
,ν
β 2 + x2 + x
β 2 + x2
dx =
333
dx =
β μ+ν−1
1−μ−ν
B μ,
2μ
2
[Re β > 0,
0 < Re μ < 1 − Re ν]
ET I 311(28)
√
ν
∞ μ−1
Γ μ2 Γ(1 − μ − ν)
cos t ± i sin t 1 + x2
x
1−μ
μ−1
2
2
√
dx = 2
sin
t
Γ(−ν)
1 + x2
0
μ+1
1
i 1 − μ+1
−2
2
2 −ν
2
× π Q − μ+1 −ν (cos t) ∓ π P μ−1
(cos t)
2
2
2
[Re μ > 0]
ET I 311 (27)
+
,ν
∞ xμ−1
(β 2 − 1) (x2 + 1) + β
√
dx
x2 + 1
0
μ−1
1 − μ μ−1
2 2 − 1 iπ(μ−1) Γ μ2 Γ(1 − μ − ν) 2
= √ e 2
Q −2μ+1 −ν (β)
β −1
Γ(−ν)
4
π
2
[Re β > 1, Re ν < 0, Re μ < 1 − Re ν] ET I 311(26)
3.
4.
∞
5.
(x − u)μ−1
u
6.
7.
∞ xμ−1
x−
[|arg u| < π,
3.2788
ET II 202(10)
√
√
ν −ν ,
x2 − 1 + x − x2 − 1
1−μ+ν 1−μ−ν
√
,
dx = 2−μ B
2−1
2
2
x
1
ET I 311(29)
[Re μ < 1 + Re ν]
+√
,
√
√
√
2ν
2ν
u (u − x)μ−1
<
x+2+ x
+
x+2− x
1
2μ + 1
1
2 −μ
dx = 2
π[u(u + 2)]μ− 2 P ν−
1 (u+
2
2
x(x + 2)
0
1)
+
√
2ν
√
1
2μ−1 12 −μ
x+1− x−1
1
2ν+ 2
√
dx = √ e(μ− 2 )πi u2 − 1 4 Q ν−
1 (u)
2
2
π
x −1
[|arg(u − 1)| < π, 0 < Re μ < 1 + Re ν]
1.
0
∞
xp
1 + x2p
q
dx
=0
1 − x2
[pq > 1]
Re μ > 0]
ET II 186(12)
334
Exponential Functions
3.310
3.3–3.4 Exponential Functions
3.31 Exponential functions
∞
11
3.310
e−px dx =
0
3.311
∞
1.
0
∞
2.
0
3.11
1
p
[Re p > 0]
dx
ln 2
=
px
1+e
p
LO III 284a
e−μx
dx = β(μ)
1 + e−x
[Re μ > 0]
EH I 20(3), ET I 144(7)
∞
pπ
e−px
π
cosec
dx =
−qx
|q|
q
−∞ 1 + e
[q > p > 0 or 0 > p > q]
∞
4.
0
∞
0
∞
ak
e−qx dx
=
−px
1 − ae
q + kp
[0 < a < 1]
1 − eνx
dx = ψ(ν) + C + π cot(πν)
ex − 1
[Re ν < 1]
e
0
− e−νx
dx = ψ(ν) + C
1 − e−x
9.
11.
Re ν > 0]
(cf. 3.231 5)
[b > 0,
0 < Re μ < 1]
[|arg b| < π,
ET I 120(14)a
0 < Re μ < 1]
ET I 120(15)a
∞
∞
0
0
WH, EH I 16(14)
BI (27)(8)
12.
EH I 16(16)
∞
−px
−qx
pπ
e
−e
π
cot
dx =
−(p+q)x
p
+
q
p
+q
0 1−e
∞ px
r−q
e − eqx
1
dx =
ψ
−ψ
rx
sx
−e
r−s
r−s
0 e
13.∗
[Re μ > 0,
e−μx dx
= πbμ−1 cot(μπ)
−x
b
−
e
−∞
∞ −μx
e
dx
= πbμ−1 cosec(μπ)
−x
−∞ b + e
10.11
[Re ν > 0]
− e−νx
dx = ψ(ν) − ψ(μ)
1 − e−x
e
0
8.
(cf. 3.265)
∞ −μx
7.
BI (27)(7)
∞ −x
6.
BI (28)(7)
k=0
5.
(cf. 3.241 2)
a −b
1
dx =
ψ
c x − dx
ln dc
x
x
ln
ln
c
b
c
d
−ψ
ln
ln
[p > 0,
r−p
r−s
c
a
c
d
q > 0]
GW (311)(16c)
[r > s, r > p, r > q]
[c > a > 0,
b > 0,
GW (311)(16)
d > 0]
GW (311)(16a)
∞
−px
−qx
e
+e
π
cosec
dx =
−(p+q)x
p+q
1+e
πp
p+q
3.317
3.312
Exponential functions
∞
1 − e− β
x
1.
ν−1
e−μx dx = β B(βμ, ν)
335
[Re β > 0,
Re ν > 0,
Re μ > 0]
0
LI(25)(13), EH I 11(24)
∞
2.
−1 1 − e−x
1 − e−αx 1 − e−βx e−px dx = ψ(p + α) + ψ(p + β) − ψ(p + α + β) − ψ(p)
0
[Re p > 0,
3.
∞
11
Re p > − Re α,
Re p > − Re β,
Re p > − Re(α + β)]
ν−1 − −μx
1 − e−x
1 − βe−x
e
dx = B(μ, ν) 2 F 1 (, μ; μ + ν; β)
0
[Re μ > 0,
3.313
1.
7
2.7
3.314
3.315
1.
|arg(1 − β)| < π]
EH I 116(15)
[0 < Re μ < 1]
∞
e−μx dx
= B(μ, ν − μ)
[0 < Re μ < Re ν]
−x )ν
−∞ (1 + e
∞
e−μx dx
ν
= γ exp β μ −
B(γμ, ν − γμ)
β/γ + e−x/γ ν
γ
−∞ e
ν
> Re μ > 0, |Im β| < π Re γ
Re
γ
ET I 120(21)
∞
e−μx dx
= exp[γ(μ − ) − βν] B(μ, ν + − μ) 2 F 1 ν, μ; ν + ; 1 − eν−β
β + e−x )ν (eγ + e−x )
(e
−∞
[|Im β| < π, |Im γ| < π, 0 < Re μ < Re(ν + )] ET I 121(22)
π β μ−1 − γ μ−1
e−μx dx
=
cosec(μπ)
−x ) (γ + e−x )
γ−β
−∞ (β + e
[|arg β| < π, |arg γ| < π,
2.
Re ν > 0,
∞
e−μx dx
= π cot πμ
−x
−∞ 1 − e
PV
ET I 145(15)
∞
3.316
∞
β = γ,
0 < Re μ < 2]
ET I 120(18)
ν
(1 + e−x ) − 1
dx = ψ(μ) − ψ(μ − ν)
−x )μ
−∞ (1 + e
[Re μ > Re ν > 0]
(cf. 3.235)
BI (28)(8)
3.317
∞
1.
−∞
∞
2.
−∞
1
1
−
μ
1 + e−x
(1 + e−x )
dx = C + ψ(μ)
1
1
ν −
μ
(1 + e−x )
(1 + e−x )
dx = ψ(μ) − ψ(ν)
[Re μ > 0]
[Re μ > 0,
(cf. 3.233)
Re ν > 0]
BI (28)(10)
(cf. 3.219)
BI (28)(11)
336
3.318
1.
2.7
Exponential Functions
3.318
√
√
−ν −ν
β + 1 − e−x
+ β − 1 − e−x
√
e−μx dx
1 − e−x
0
(μ−ν)/2
2μ+1 e(μ−ν)πi β 2 − 1
Γ(μ) Q ν−μ
μ−1 (β)
=
Γ(ν)
[Re μ > 0]
ET I 145(18)
∞
ν
1
√
e−u 1 − e−2x − e−x 1 − e−2u e−μx dx
1 − e−2x
u
√
1
u
− 1 (μ+ν) √
2− 2 (μ+ν) πe− 2 (μ+ν) Γ(μ) Γ(ν + 1) P − 21 (μ−ν)
1 − e−2u
2
=
Γ[(μ + ν + 1)/2]
[u > 0, Re μ > 0, Re ν > −1] ET I 145(19)
∞
3.32–3.34 Exponentials of more complicated arguments
3.321
1.
√
u
∞
2
π
π
(−1)k u2k+1
Φ(u) =
erf(u) =
e−x dx =
2
2
k!(2k + 1)
0
k=0
∞
k 2k+1
2 2 u
= e−u
(2k + 1)!!
√
11
k=0
(cf. 8.25)
√
2 2
π
Φ(qu)
[q > 0]
e−q x dx =
2q
0
√
∞
2 2
π
[q > 0]
e−q x dx =
2q
0
u
,
2 2
2 2
1 +
xe−q x dx = 2 1 − e−q u
2q
0
√
u
π
1
2 −q 2 x2
−q 2 u2
Φ(qu) − que
x e
dx = 3
2q
2
0
u
,
2 2
2 2
1 +
x3 e−q x dx = 4 1 − 1 + q 2 u2 e−q u
2q
0
√
u
2 2
2 2
3
1 3 π
Φ(qu) −
+ q 2 u2 que−q u
x4 e−q x dx = 5
2q
4
2
0
2.
3.
4.∗
5.∗
6.∗
7.∗
3.322 1.11
∞
u
2.
0
AD 6.700
u
∞
x2
− γx
exp −
4β
exp −
x2
− γx
4β
FI II 624
u
βγ 2
dx = πβe
1−Φ γ β+ √
2 β
[Re β > 0]
,
+
dx = πβ exp βγ 2 1 − Φ γ β
ET I 146(21)
[Re β > 0]
NT 27(1)a
3.326
Exponentials of more complicated arguments
3.
11
PV
∞
e
±iλx2
0
3.323
1.
11
2.10
3.11
1
dx =
2
π ±πi/4
e
λ
[λ > 0]
√
π q2 /4
e
exp −qx − x dx =
1−Φ 1+
2
1
√
∞
q2
π
exp −p2 x2 ± qx dx = exp
2
4p
p
−∞
∞
γ4
3 γ
exp −β 2 x4 − 2γ 2 x2 dx = 2− 2 e 2β2 K 14
β
0
∞
2
337
1
q
2
PBM 343 (2.3.15(2))
BI (29)(4)
γ4
2β 2
Re p2 > 0
BI (28)(1)
+
π
|arg β| < ,
4
|arg γ| <
π,
4
ET I 147(34)a
3.324
1.
2.
11
β
− γx
exp −
4x
0
%
∞
b
exp − x −
x
−∞
∞
3.325
0
3.326
∞
1.8
0
∞
2.10
0
3.
∗
.
∞
∞
dx =
2n
&
dx =
b
exp −ax − 2
x
2
exp (−xμ ) dx =
1
Γ
μ
xm exp (−βxn ) dx =
β
K1
βγ
γ
1
2n
1
Γ
n
1
dx =
2
4.∗
γ=
(x − a) exp (−β(x − b) ) dx =
(x − a) exp (−β(x − b)n ) dx =
5.∗
∞
u
ET I 146(25)
b > 0]
FI II 644
[Re μ > 0]
n
u
0
[a > 0,
1
μ
Γ(γ)
nβ γ
Re γ > 0]
[b ≥ 0]
√
π
exp −2 ab
a
Γ
0
[Re β ≥ 0,
m+1
n
[Re β > 0,
2
n
n , β(−b)
nβ 2/n
2
− (a − b)
2
Γ
BI (26)(4)
Re m > 0,
1
n
n , β(−b)
nβ 1/n
[Re n > 0,
Re n > 0]
Re β > 0,
|arg b| < π]
− Γ n , β(u − b)
2/n
1 nβ
Γ n , β(−b)n − Γ n1 , β(u − b)n
−(a − b)
nβ 1/n
[Re n > 0, Re β > 0, |arg b| < π, |arg(u − b)| < π]
Γ
(x − a) exp (−β(x − b)n ) dx =
Γ
n
n , β(−b)
2
n
n , β(−b)
2/n
nβ
− (a − b)
n
Γ
1
n , β(u −
nβ 1/n
[Re n > 0,
b)n
Re β > 0,
|arg(u − b)| < π]
338
Exponential Functions
Exponentials of exponentials
∞
1
3.327
exp (−aenx ) dx = − Ei(−a)
n
0 ∞
3.328
exp (−ex ) eμx dx = Γ(μ)
−∞ %
&
∞
b
a exp (−ceax ) b exp −cebx
3.329
−
dx = e−c ln
−ax
−bx
1
−
e
1
−
e
a
0
3.331 1.
∞
3.327
[n ≥ 1,
Re a ≥ 0,
a = 0]
LI (26)(5)
[Re μ > 0]
[a > 0,
NH 145(14)
b > 0,
c > 0]
BI (27)(12)
exp −βe−x − μx dx = β −μ γ(μ, β)
[Re μ > 0]
ET I 147(36)
exp (−βex − μx) dx = β μ Γ(−μ, β)
[Re β > 0]
ET I 147(37)
0
∞
2.
0
∞
3.11
ν−1
μ+ν
β
1 − e−x
exp βe−x − μx dx = B(μ, ν)β − 2 e 2 M ν−μ , ν+μ−1 (β)
2
0
2
[Re μ > 0,
∞
4.
ν−1
μ−1
β
1 − e−x
exp (−βex − μx) dx = Γ(ν)β 2 e− 2 W
0
1−μ−2ν
2
∞
3.332
[Re μ > 0,
1.
3
2.3
3.∗
3.33411
ET I 147(39)
Re ν > 0,
|arg(1 − λ)| < π]
ET I 147(40)
∞
e−μx dx
= Γ(μ) ζ(μ)
−x ) − 1
−∞ exp (e
∞
e−μx dx
= 1 − 21−μ Γ(μ) ζ(μ)
−x
)+1
−∞ exp (e
= ln 2
Re ν > 0]
ν−1 −
1 − e−x
1 − λe−x
exp βe−x − μx dx = B(μ, ν) Φ1 (μ, , ν, λ, β)
0
3.333
ET I 147(38)
(β)
, −μ
2
[Re β > 0,
Re ν > 0]
[Re μ > 1]
[Re μ > 0,
ET I 121(24)
μ = 1]
[μ = 1]
ET I 121(25)
∞
tanh(x)
1
7 ζ(3)
− 2
dx =
2
3
x
π2
x cosh (x)
0
∞
β
ν−1
β
ν−1
− μx dx = Γ(μ − ν + 1)e 2 β 2 W ν−2μ−1 ,− ν (β)
(ex − 1)
exp − x
2
2
e −1
0
[Re β > 0, Re μ > Re ν − 1]
ET I 137(41)
Exponentials of hyperbolic functions
∞
νx
e + e−νx cos νπ exp (−β sinh x) dx = −π [Eν (β) + Y ν (β)]
3.335
0
[Re β > 0]
EH II 35(34)
3.338
3.336
Exponentials of more complicated arguments
∞
1.
0
∞
2.
0
exp (−νx − β sinh x) dx = π cosec νπ [Jν (β) − J ν (β)]
+
π
π
|arg β| < and |arg β| = for Re ν > 0;
2
2
exp (nx − β sinh x) dx =
,
ν is not an integer
3.
0
WA 341(2)
1
[S n (β) − π En (β) − π Y n (β)]
2
[Re β > 0;
∞
339
exp (−nx − β sinh x) dx =
n = 0, 1, 2, . . .]
WA 342(6)
1
(−1)n+1 [S n (β) + π En (β) + π Y n (β)]
2
[Re β > 0; n = 0, 1, 2, . . .]
EH II 84(47)
3.337
∞
1.
−∞
∞
2.
exp (−νx + iβ cosh x) dx = iπe
−∞
∞
3.
−∞
+
π,
|arg β| <
2
exp (−αx − β cosh x) dx = 2 K α (β)
iνπ
2
exp (−νx − iβ cosh x) dx = −iπe−
H (1)
ν (β)
iνπ
2
H (2)
ν (β)
WA 201(7)
[0 < arg z < π]
EH II 21(27)
[−π < arg z < 0]
EH II 21(30)
Exponentials of trigonometric functions and logarithms
3.338
π
1.
{exp i [(ν − 1)x − β sin x] − exp i [(ν + 1)x − β sin x]} dx = 2π Jν (β) + i Eν (β)
0
exp [±i (νx − β sin x)] dx = π [Jν (β) ± i Eν (β)]
2.
[Re β > 0]
EH II 36
[Re β > 0]
EH II 35(32)
π
0
3.
10
4.6
5.∗
∞
γ J k (kβ)
1
exp [−γ (x − β sin x)] dx = + 2
[Re γ > 0]
γ
γ 2 + k2
0
k=1
a + b sin x + c cos x
π exp
2π
1 + p sin x + q cos x
dx = e−α I 0 (β),
2 − q2
1
+
p
sin
x
+
q
cos
x
1
−
p
−π
.
a2 − b 2 − c 2
bp + cq − a
; β = α2 −
;
with α =
2
2
1−p −q
1 − p2 − q 2
% ∞
&
π/4
tan2n x
√
exp −
dx = ln 2
1
n+ 2
0
n=1
∞
WA 619(4)
2
p + q2 < 1
340
Exponential Functions
3.339
π
6
3.339
exp (z cos x) dx = π I 0 (z)
0
exp (−p tan x) dx = ci(p) sin p − si(p) cos(p)
1
∞
pk−1
exp (−px ln x) dx =
x−px dx =
kk
0
3.341
BI (277)(2)a
π
2
[p > 0]
BI (271)(2)a
0
1
3.34211
0
BI (29)(1)
k=1
3.35 Combinations of exponentials and rational functions
3.351
u
1.8
xn e−μx dx =
0
n!
μn+1
− e−uμ
n
n!
k=0
uk
k! μn−k+1
= μ−n−1 γ(n + 1, μu)
[u > 0,
Re μ > 0, n = 0, 1, 2, . . .]
ET I 134(5)
∞
2.11
xn e−μx dx = e−uμ
u
n
n!
k=0
uk
k! μn−k+1
= μ−n−1 Γ(n + 1, μu)
[u > 0,
Re μ > 0, n = 0, 1, 2, . . .]
ET I 33(4)
∞
3.
xn e−μx dx = n!μ−n−1
[Re μ > 0]
ET I 133(3)
0
∞ −px
e
4.
xn+1
u
e
x
1
6.
7.9
8.11
9.7
= (−1)n+1
n−1
pn Ei(−pu) e−pu (−1)k pk uk
+ n
n!
u
n(n − 1) . . . (n − k)
k=0
[p > 0]
∞ −μx
5.
dx
dx
= − Ei(−μ)
NT 21(3)
[Re μ > 0]
BI (104)(10)
u
ex
dx = li (eu ) = Ei(u)
[u < 0]
−∞ x
u
1
1
xe−μx dx = 2 − 2 e−μu (1 + μu)
[u > 0]
μ
μ
0
u
2
1
[u > 0]
x2 e−μx dx = 3 − 3 e−μu 2 + 2μu + μ2 u2
μ
μ
0
u
6
1
x3 e−μx dx = 4 − 4 e−μu 6 + 6μu + 3μ2 u2 + μ3 u3
μ
μ
0
[u > 0]
3.352 1.
u −μx
e
dx
= eμβ [Ei(−μu − μβ) − Ei(−μβ)]
x+β
0
∞ −μx
2.
u
e
dx
= −eβμ Ei(−μu − μβ)
x+β
[u ≥ 0,
|arg β| < π]
[u ≥ 0,
|arg(u + β)| < π,
ET II 217(12)
Re μ > 0]
ET I 134(6), JA
3.354
Exponentials and rational functions
v −μx
3.
u
e
dx
= eαμ {Ei[−(α + v)μ] − Ei[−(α + u)μ]}
x+α
e
dx
= −eβμ Ei(−μβ)
x+β
∞ −px
∞ −μx
Re μ > 0]
ET II 217(11)
e
ET II 251(37)
e
0
3.353
[|arg β| < π,
dx
= e−pa Ei(pa − pu)
a
−
x
u
p > 0, a < u; for a > u, one should replace Ei(pa − pu) in this formula with Ei(pa − pu)
6.8
7.
Re μ > 0]
∞ −μx
0
5.
[−α < n, and − α > v,
ET I 134 (7)
4.
7
341
dx
= e−μa Ei(aμ)
a−x
[a < 0,
BI (91)(4)
∞
eipx dx
= iπeiap
x
−
a
−∞
Re μ > 0]
∞
1.
u
[p > 0]
ET II 251(38)
(k − 1)!(−μ)n−k−1
e−μx dx
(−μ)n−1 βμ
−uμ
e Ei[−(u + β)μ]
=
e
−
(x + β)n
(n − 1)!(u + β)k
(n − 1)!
k=1
[n ≥ 2, |arg(u + β)| < π,
n−1
Re μ > 0]
ET I 134(10)
∞
2.7
0
n−1
e−μx dx
1
(−μ)n−1 βμ
e Ei(−βμ)
=
(k − 1)!(−μ)n−k−1 β −k −
n
(x + β)
(n − 1)!
(n − 1)!
k=1
[n ≥ 2, |arg β| < π,
Re μ > 0]
ET I 134(9), BI (92)(2)
∞
3.
0
e−px dx
1
= peαp Ei(−ap) +
(a + x)2
a
[p > 0,
a > 0]
LI (281)(28), LI (281)(29)
4.
0
1
xex
e
dx = − 1.
(1 + x)2
2
∞
5.7
0
BI (80)(6)
xn e−μx
dx = (−1)n−1 β n eβμ Ei(−βμ) +
(k − 1)!(−β)n−k μ−k
x+β
k=1
[|arg β| < π,
n
Re μ > 0]
BI (91)(3)a, LET I 135(11)
3.354
∞ −μx
e
dx
1
= [ci(βμ) sin βμ − si(βμ) cos βμ]
2
2
β +x
β
1.
0
2.
3.7
[Re β > 0,
Re μ > 0]
BI (91)(7)
[Re β > 0,
Re μ > 0]
BI (91)(8)
∞
xe−μx dx
= − ci(βμ) cos βμ − si(βμ) sin βμ
β 2 + x2
0
∞ −μx
e
dx
1 −βμ
e
=
Ei(βμ) − eβμ Ei(−βμ)
2
2
2β
0 β −x
[|arg (±β)| < π,
Re μ > 0]
BI (91)(14)
342
Exponential Functions
4.
3.355
∞
xe−μx dx
1 −βμ
e
=
Ei(βμ) + eβμ Ei(−βμ)
2
2
2
0 β − x
|arg (±β)| < π, Re μ > 0; for β > 0 one should replace Ei(βμ) in this formula with Ei(βμ)
5.8
BI (91)(15)
∞
e−ipx dx
π
= e−|ap|
2 + x2
a
a
−∞
∞
1.
0
−μx
xe
dx
(β 2
2
x2 )
+
e
−px
dx
−
2
x2 )
(a2
0
∞
4.3
0
=
p real]
ET I 118(1)a
1
{ci(βμ) sin βμ − si(βμ) cos βμ} − βμ [ci(βμ) cos βμ + si(βμ) sin βμ]
2β 3
=
1
{−βμ [ci(βμ) sin βμ − si(βμ) cos βμ]}
2β 2
[Re β > 0,
∞
3.3
+
2
x2 )
[a = 0,
LI (92)(6)
∞
2.
e−μx dx
(β 2
0
3.356
3.355
∞
1.
0
−px
xe
dx
(a2
2
x2 )
−
=
=
Re μ > 0]
1 (ap − 1)eap Ei(−ap) + (1 + ap)e−ap Ei(ap)
3
4a
2
Im a > 0,
p>0
1 −2 + ap e−ap Ei(ap) − eap Ei(−ap)
2
4a
2
Im a > 0,
p>0
BI (92)(7)
BI (92)(8)
LI (92)(9)
x2n+1 e−px
dx = (−1)n−1 a2n [ci(ap) cos ap + si(ap) sin ap]
a2 + x2
n
k−1
1 (2n − 2k + 1)! −a2 p2
+ 2n
p
k=1
[p > 0]
∞
2.
0
BI (91)(12)
2n −px
x e
1
dx = (−1)n a2n−1 [ci(ap) sin ap − si(ap) cos ap] + 2n−1
2
2
a +x
p
[p > 0]
∞
3.
0
x2n+1 e−px
1
1
dx = a2n eap Ei(−ap) + e−ap Ei(ap) − 2n
2
2
a −x
2
p
4.
0
∞
k−1
(2n − 2k)! −a2 p2
k=1
BI (91)(11)
n
k−1
(2n − 2k + 1)! a2 p2
k=1
[p > 0]
n
x2n e−px
1
1
dx = a2n−1 e−ap Ei(ap) − eap Ei(−ap) − 2n−1
2
2
a −x
2
p
[p > 0]
BI (91)(17)
n
k−1
(2n − 2k)! a2 p2
k=1
BI (91)(16)
3.358
Exponentials and rational functions
3.357 1.
∞
a3
0
0
a3
∞
a3
0
6.
0
∞
0
0
a > 0]
x e
dx
1 −ap
e
=
Ei(ap) − eap Ei(−ap) − 2 ci(ap) sin ap + 2 si(ap) cos ap
4
4
a −x
4a
[p > 0,
BI (92)(23)
a > 0]
BI (91)(18)
BI (91)(19)
2 −px
[p > 0,
∞
BI (92)(22)
e−px
1 dx = 3 e−ap Ei(ap) − eap Ei(−ap) + 2 ci(ap) sin ap − 2 si(ap) cos ap
4
4
a −x
4a
∞
0
4.
a > 0]
xe−px dx
1 = 2 eap Ei(−ap) + e−ap Ei(ap) − 2 ci(ap) cos ap − 2 si(ap) sin ap
4
4
a −x
4a
3.
a > 0]
∞
0
BI (92)(21)
x e
dx
1
= {ci(aμ) (cos aμ − sin aμ)
a3 − a2 x + ax2 − x3
2
+ si(aμ) (cos aμ + sin aμ) + e−aμ Ei(aμ)
[p > 0,
2.
a > 0]
2 −μx
[Re μ > 0,
3.358 1.
BI (92)(20)
xe−μx dx
1
{− ci(aμ) (sin aμ + cos aμ)
=
− a2 x + ax2 − x3
2a
− si(aμ) (sin aμ − cos aμ) + e−aμ Ei(aμ)
[Re μ > 0,
∞
a > 0]
e−μx dx
1
= 2 {ci(aμ) (sin aμ − cos aμ)
2
2
3
− a x + ax − x
2a
− si(aμ) (sin aμ + cos aμ) + e−aμ Ei(aμ)
[Re μ > 0,
5.
BI (92)(19)
x e
dx
1
= {− ci(aμ) (sin aμ + cos aμ)
2
2
3
+ a x + ax + x
2
− si(aμ) (sin aμ − cos aμ) − eaμ Ei(−aμ)}
[Re μ > 0,
∞
0
a > 0]
2 −μx
a3
4.
BI (92)(18)
xe
dx
1
{ci(aμ) (sin aμ − cos aμ)
=
a3 + a2 x + ax2 + x3
2a
− si(aμ) (sin aμ + cos aμ) − eaμ Ei(−aμ)}
[Re μ > 0,
0
a > 0]
−μx
∞
3.
e−μx dx
1
= 2 {ci(aμ) (sin aμ + cos aμ)
+ a2 x + ax2 + x3
2a
+ si(aμ) (sin aμ − cos aμ) − eaμ Ei(−aμ)}
[Re μ > 0,
∞
2.
343
a > 0]
BI (91)(20)
x3 e−px dx
1 ap
e Ei(−ap) + e−ap Ei(ap) + 2 ci(ap) cos ap + 2 si(ap) sin ap
=
4
4
a −x
4
[p > 0,
a > 0]
BI (91)(21)
344
Exponential Functions
∞
5.
0
3.359
x4n e−px
1
dx = a4n−3 e−ap Ei(ap) − eap Ei(−ap) + 2 ci(ap) sin ap − 2 si(ap) cos ap
4
4
a −x
4
n
k−1
1 (4n − 4k)! a4 p4
− 4n−3
p
k=1
[p > 0,
∞
6.
0
a > 0]
BI (91)(22)
e
1
dx = a4n−2 eap Ei(−ap) + e−ap Ei(ap) − 2 ci(ap) cos ap − 2 si(ap) sin ap
a4 − x4
4
n
k−1
1 (4n − 4k + 1)! a4 p4
− 4n−2
p
4n+1 −px
x
k=1
[p > 0,
∞
7.
0
a > 0]
BI (91)(23)
x4n+2 e−px
1
dx = a4n−1 e−ap Ei(ap) − eap Ei(−ap) − 2 ci(ap) sin ap + 2 si(ap) cos ap
4
4
a −x
4
n
k−1
1 (4n − 4k + 2)! a4 p4
− 4n−1
p
k=1
[p > 0,
∞
8.
0
a > 0]
BI (91)(24)
x4n+3 e−px
1
dx = a4n eap Ei(−ap) + e−ap Ei(ap) + 2 ci(ap) cos ap + 2 si(ap) sin ap
4
4
a −x
4
n
k−1
1 (4n − 4k + 3)! a4 p4
− 4n
p
k=1
[p > 0,
3.359
a > 0]
BI (91)(25)
∞
(i − x)n e−ipx
dx = (−1)n−1 2πpe−p Ln−1 (2p)
n
2
−∞ (i + x) i + x
=0
for p > 0;
for p < 0.
ET I 118(2)
3.36–3.37 Combinations of exponentials and algebraic functions
3.361
1.8
2.8
3.8
3.362
u −qx
π
√
Φ ( qu)
q
0
∞ −qx
e
π
√ dx =
q
x
0
∞ −qx
e
π
q
√
dx = e
q
1
+
x
−1
1.
1
2.
0
e
√ dx =
x
∞ −μx
e
dx
√
=
x−1
∞ −μx
e
dx
√
=
x+β
[q > 0]
[q > 0]
BI(98)(10)
[q > 0]
BI (104)(16)
π −μ
e
μ
[Re μ > 0]
,
π βμ +
1−Φ
e
βμ
μ
[Re μ > 0,
BI (104)(11)a
|arg β| < π]
ET I 135(18)
3.371
3.363
Exponentials and algebraic functions
∞√
x − u −μx
e
dx =
x
1.
u
∞
2.
u
2
1.
0
2.
3.
3.365
e2x dx
√
= π I 0 (2)
2
−1 1 − x
∞ −px
ap
ap
e
dx
= e 2 K0
2
x(x + a)
0
u
∞
u
2u
1.
0
∞
2.
0
3.367
ET I 136(26)
[p > 0]
GW (312)(7a)
[u > 0,
Re μ > 0]
ET I 136(28)
xe−μx dx
√
= u K 1 (uμ)
x2 − u2
[u > 0,
Re μ > 0]
ET I 136(29)
(u − x)e−μx dx
√
= πue−uμ I 1 (uμ)
2ux − x2
[Re μ > 0]
(x + β)e−μx dx
= βeβμ K 1 (βμ)
x2 + 2βx
[Re μ > 0,
xe−μx dx
βπ
[H1 (βμ) − Y 1 (βμ)] − β
=
2
x2 + β 2
0
∞
exp 2μ cos2 2t
e−μx dx
√
=
2
sin t
0 (1 + cos t + x) x + 2x
∞
3.368
0
BI (277)(2)a
p > 0]
∞
t − sin t
u
|arg β| < π]
ET I 136(30)
,
Re μ > 0
ET I 136(27)
K 0 (v)e−v cos t dv
0
|arg β| <
∞
GW (312)(8a)
ET I 136(31)
+
π
|arg β| < ,
2
[Re μ > 0]
e−μx dx
π
1
[H1 (βμ) − Y 1 (βμ)] − 2 2
=
2βμ
β μ
x + x2 + β 2
+
−μx
e
dx
2
√
√
= √ − 2 πμeaμ (1 − Φ ( aμ))
3
a
(x + a)
0
∞
√
1
2n − 1 −n− 1
1 3
2
μ
3.37111
xn− 2 e−μx dx = π · · . . .
2
2
2
0
√
= π2−n μ−n−1/2 (2n − 1)!!
3.36911
Re μ ≥ 0]
[a > 0,
xe−μx dx
πu
√
[L1 (μu) − I 1 (μu)] + u
=
2
2
2
u −x
2.
[u > 0,
ET I 136(23)
1
0
3.
e−μx dx
π
√
√
= √ [1 − Φ ( uμ)]
u
x x−u
e−px dx
= πe−p I 0 (p)
x(2 − x)
1.
3.366
√
π −uμ
√
e
− π u [1 − Φ ( uμ)]
μ
[Re μ > 0]
3.364
345
ET I 136(33)
π
,
2
[|arg a| < π,
,
Re μ > 0
ET I 136(32)
Re μ > 0]
ET I 135(20)
[n ≥ 0]
[Re μ > 0]
ET I 135(17)
346
Exponential Functions
3.372
∞
1
1
(2n − 1)!! p
xn− 2 (2 + x)n− 2 e−px dx =
e K n (p)
[p > 0, n = 0, 1, 2, . . .]
pn
0 ∞ +
n
n,
x + x2 + β 2 + x − x2 + β 2
e−μx dx = 2β n+1 O n (βμ)
3.372
3.373
GW (312)(8)
0
3.374
1.
2.
[Re μ > 0]
WA 05(1)
√
n
x + 1 + x2
1
√
e−μx dx = [S n (μ) − π En (μ) − π Y n (μ)]
2
2
1+x
0
[Re μ > 0]
√
n
∞
x − 1 + x2
1
√
e−μx dx = − [S n (μ) + π En (μ) + π Y n (μ)]
2
2
1+x
0
[Re μ > 0]
ET I 37(35)
∞
ET I 137(36)
3.38–3.39 Combinations of exponentials and arbitrary powers
3.381 u
1.
xν−1 e−μx dx = μ−ν γ(ν, μu)
[Re ν > 0]
EH I 266(22), EH II 133(1)
0
u
2.
p−1 −x
x
e
dx =
0
∞
(−1)k
k=0
= e−u
∞
k=0
up+k
k!(p + k)
up+k
p(p + 1) . . . (p + k)
AD 6.705
∞
3.8
xν−1 e−μx dx = μ−ν Γ(ν, μu)
[u > 0,
Re μ > 0]
u
EH I 256(21), EH II 133(2)
∞
4.
xν−1 e−μx dx =
0
∞
5.
0
u
∞
7.
ν
u
e
dx = u− 2 e− 2 W − ν , (1−ν) (u)
ν
2
2
x
xk−1 eiμx dx =
FI II 779
0 < Re ν < 1]
EH I 12(32)
[u > 0]
Γ(k)
(−iμ)k
WH
[0 < Re(k) < 1,
μ = 0]
GH2 62 (313.14)
u
xm e−βx dx =
n
0
9.∗
Re ν > 0]
− ν
q
xν−1 e−(p+iq)x dx = Γ(ν) p2 + q 2 2 exp −iν arctan
p
[p > 0, Re ν > 0 and p = 0,
0
8.∗
[Re μ > 0,
∞ −x
6.
1
Γ(ν)
μν
∞
u
xm e−βx dx =
n
n
γ (v, βu )
nβ v
Γ (v, βun )
nβ v
v=
v=
m+1
n
m+1
n
[u > 0,
[u > 0,
Re v > 0,
Re v > 0,
Re n > 0,
Re n > 0,
Re β > 0]
Re β > 0]
3.383
10.
∗
Exponentials and arbitrary powers
∞
xm e−βx dx =
n
0
11.∗
γ (v, βun ) + Γ (v, βun )
nβ v
v=
∞
2m −βx2n
x e
dx = 2
−∞
347
m+1
n
∞
[u > 0,
x2m e−βx
2n
Re v > 0,
dx =
0
Re β > 0]
See also 3.326 1
2 (γ (v, βun ) + Γ (v, βun ))
Γ(v)
=
nβ v
nβ v
2m + 1
2n
v=
Re n > 0,
[u > 0,
3.382 u
(u − x)ν e−μx dx = (−μ)−ν−1 e−uμ γ(ν + 1, −uμ)
1.6
Re v > 0,
[Re ν > −1,
Re n > 0,
u > 0]
Re β > 0]
ET I 137(6)
0
∞
2.
(x − u)ν e−μx dx = μ−ν−1 e−uμ Γ(ν + 1)
[u > 0,
Re ν > −1,
Re μ > 0]
u
ET I 137(5), ET II 202(11)
∞
3.
[Re μ > 0]
(x + β)ν e−μx dx = μ−ν−1 eβμ Γ(ν + 1, βμ)
[|arg β| < π,
ν
2
0
μ
(1 + x)−ν e−μx dx = μ 2 −1 e 2 W − ν , (1−ν) (μ)
∞
4.
WH
2
Re μ > 0]
0
ET I 137(4), ET II 233(10)
u
5.
(a + x)μ−1 e−x dx = ea [γ(μ, a + u) − γ(μ, a)]
[Re μ > 0]
EH II 139
0
∞
(β + ix)−ν e−ipx dx = 0
6.
−∞
=
[for p > 0]
ν−1 βp
e
2π(−p)
Γ(ν)
[for p < 0]
[Re ν > 0,
∞
7.
−∞
(β − ix)−ν e−ipx dx =
Re β > 0]
ET I 118(4)
Re β > 0]
ET I 118(3)
Re ν > 0]
ET II 187(14)
ν−1 −βp
e
2πp
Γ(ν)
[for p > 0]
=0
[for p < 0]
[Re ν > 0,
3.383 u
xν−1 (u − x)μ−1 eβx dx = B(μ, ν)uμ+ν−1 1 F 1 (ν; μ + ν; βu)
1.11
0
[Re μ > 0,
u
xμ−1 (u − x)μ−1 eβx dx =
2.11
0
√
π
u
β
μ− 12
exp
βu
2
Γ(μ) I μ− 12
βu
2
[Re μ > 0]
∞
3.
u
1
xμ−1 (x − u)μ−1 e−βx dx = √
π
u
β
μ− 12
Γ(μ) exp −
βu
2
K μ− 12
[Re μ > 0,
ET II 187(13)
βu
2
Re βu > 0]
ET II 202(12)
348
Exponential Functions
4.
∞
11
xν−1 (x − u)μ−1 e−βx dx = β −
μ+ν
2
u
μ+ν−2
2
3.384
Γ(μ) exp −
u
5.11
ν−μ 1−μ−ν
2 ,
2
(βu)
Re βu > 0]
ET II 202(13)
∞
∞
6.
ν
p
Lν−q
p
−ν
a
−
sin(πν) Γ(1 − q)
a
xν−1 (x + β)−ν+ 2 e−μx dx = 2ν− 2 Γ(ν)μ− 2 e
1
1
[|arg β| < π,
∞
7.
q
&
q−ν p L−q
a
sin(πq) Γ(1 − ν)
[ν = ±1, ±2, . . .]
[ν = 0]
1
βμ
2
0
W
[Re μ > 0,
e−px xq−1 (1 + ax)−ν dx
0
%
p
π2
= q
p Γ(ν) sin [π(q − ν)]
a
Γ(q)
= q
p
βu
2
Re ν > 0,
xν−1 (x + β)−ν− 2 e−μx dx = 2ν Γ(ν)β − 2 e
1
1
βμ
2
[Re q > 0,
D 1−2ν
2βμ
Re μ ≥ 0,
D −2ν
0
Re p > 0,
μ = 0]
Re a > 0]
ET I 39(20), EH II 119(2)a
2βμ
[|arg β| < π,
Re ν > 0,
Re μ ≥ 0]
ET I 139(21), EH II 119(1)a
∞
8.
ν−1
x
0
(x + β)
e
1
dx = √
π
β
μ
ν− 12
e
βμ
2
βμ
2
[|arg β| < π,
Γ(ν) K 12 −ν
Re μ > 0,
Re ν > 0]
ET II 233(11), EH II 19(16)a, EH II 82(22)a
∞
9.
u
ν−1 −μx
(x − u)ν e−μx
dx = uν Γ(ν + 1) Γ(−ν, uμ)
x
[u > 0,
Re ν > −1,
Re μ > 0]
ET I 138(8)
∞
10.
0
xν−1 e−μx
dx = β ν−1 eβμ Γ(ν) Γ(1 − ν, βμ)
x+β
[|arg β| < π,
Re μ > 0,
Re ν > 0]
EH II 137(3)
3.384
1
1.
−1
(1 − x)ν−1 (1 + x)μ−1 e−ipx dx = 2μ+ν−1 B(μ, ν)eip 1 F 1 (μ; ν + μ; −2ip)
[Re ν > 0,
v
2.
Re μ > 0]
ET I 119(13)
(x − u)2μ−1 (v − x)2ν−1 e−px dx
u
u+v
M μ−ν,μ+ν− 12 (vp − up)
2
Re μ > 0, Re ν > 0] ET I 139(23)
= B(2μ, 2ν)(v − u)μ+ν−1 p−μ−ν exp −p
[v > u > 0,
3.385
Exponentials and arbitrary powers
∞
3.
(x + β)2ν−1 (x − u)2−1 e−μx dx
(β − u)μ
(u + β)ν+−1
=
exp
Γ(2) W ν−,ν+− 12 (uμ + βμ)
μν+
2
[u > 0, |arg(β + u)| < π, Re μ > 0, Re > 0] ET I 139(22)
u
∞
4.
∞
5.
u
(x + β) (x − u)
1
1
1
√
(x − u)ν−1 (x + u)−ν+ 2 e−μx dx = √ 2ν− 2 Γ(ν) D 1−2ν (2 uμ)
μ
[u > 0, Re μ > 0,
u
1
1
1
√
(x − u)ν−1 (x + u)−ν− 2 e−μx dx = √ 2ν− 2 Γ(ν) D −2ν (2 uμ)
u
[u > 0, Re μ ≥ 0,
∞
−μ
(β − ix)
−∞
(γ − ix)−ν e−ipx dx =
2πe−βp pμ+ν−1
1 F 1 (ν; μ + ν; (β − γ)p)
Γ(μ + ν)
=0
∞
8.6
=
∞
6
−∞
Re γ > 0,
[for p > 0]
Re(μ + ν) > 1]
(β + ix)−μ (γ + ix)−ν e−ipx dx = 0
−∞
9.
Re ν > 0]
[for p < 0]
[Re β > 0,
Re ν > 0]
ET I 139(19)
7.6
ET I 139(17)
ET I 139(18)
∞
6.
(β + u)μ
1
− (β+u)μ
2
e
dx = νπ cosec(νπ)e
k2ν
μ
2
[ν = 0, u > 0, |arg(u + β)| < π, Re μ > 0, Re ν < 1]
−ν −μx
ν
u
349
ET I 119(10)
[for p > 0]
2πeγp (−p)μ+ν−1
[for p < 0]
1 F 1 [μ; μ + ν; (β − γ)p]
Γ(μ + ν)
[Re β > 0, Re γ > 0, Re(μ + ν) > 1] ET I 19(11)
(β + ix)−2μ (γ − ix)−2ν e−ipx dx
β−γ
pμ+ν−1
exp
p W ν−μ, 12 −ν−μ (βp + γp)
[for p > 0]
Γ(2ν)
2
β−γ
(−p)μ+ν−1
exp
p W μ−ν, 12 −ν−μ (−βp − γp) [for p < 0]
= 2π(β + γ)−μ−ν
Γ(2μ)
2
Re β > 0, Re γ > 0, Re(μ + ν) > 12
ET I 19(12)
= 2π(β + γ)−μ−ν
11
3.385
1
xν−1 (1 − x)λ−1 (1 − βx)− e−μx dx = B(ν, λ)Φ1 (ν, , λ + ν, −μ, β)
0
[Re λ > 0,
Re ν > 0,
|arg(1 − β)| < π]
ET I 39(24)
350
Exponential Functions
3.386
3.386
∞
(ix)ν0
n
-
νk
(βk + ix)
e−ipx dx
= 2πe−β0 p β0ν0
k=1
1.
−∞
β0 − ix
%
Re ν0 > −1,
n
Re βk > 0,
n
-
∞
n
-
νk
(βk + ix)
Re νk < 1,
π
arg ix = sign x,
2
Re νk < 1,
π
arg ix = sign x,
2
=0
β0 + ix
%
−∞
Re ν0 > −1,
n
Re βk > 0,
k=0
3.387
1
1.6
−1
2.
1
6
−1
1 − x2
ν−1
e−μx dx =
2 ν−1 iμx
1−x
e
√
∞
1
∞
4.
ET I 118(8)
√
dx = π
Γ(ν) I ν− 12 (μ)
p>0
ET I 119(9)
+
Re ν > 0,
π,
2
WA 172(2)a
|arg μ| <
ν− 12
2
μ
2
ν−1 −μx
1
x −1
e
dx = √
π
&
ν− 12
2
μ
π
Γ(ν) J ν− 12 (μ)
[Re ν > 0]
3.
p>0
e−ipx dx
k=1
2.
&
k=1
k=0
(ix)ν0
νk
(β0 + βk )
2
μ
WA 25(3), WA 48(4)a
ν− 12
Γ(ν) K ν− 12 (μ)
+
π
|arg μ| < ,
2
,
Re ν > 0
WA 190(4)a
2
ν−1 iμx
x −1
e dx
1
√
ν− 12
π 2
(1)
[Im μ > 0, Re ν > 0] EH II 83(28)a
Γ(ν) H 1 −ν (μ)
2
2
μ
√
ν− 12
π
2
(2)
[Im μ < 0, Re ν > 0] EH II 83(29)a
= −i
Γ(ν) H 1 −ν (−μ)
−
2
2
μ
√
u
ν− 12
+
,
2
π 2u
2 ν−1 μx
u −x
e dx =
Γ(ν) I ν− 12 (uμ) + Lν− 12 (uμ)
2
μ
0
=i
5.
[u > 0,
∞
6.
u
2
ν−1 −μx
1
x − u2
e
dx = √
π
2u
μ
Re ν > 0]
ET II 188(20)a
ν− 12
Γ(ν) K ν− 12 (uμ)
[u > 0,
Re μ > 0,
Re ν > 0]
ET II 203(17)a
3.389
Exponentials and arbitrary powers
7.
∞
2
ν−1 −μx
x + u2
e
dx =
11
0
√
π
2
ν− 12
2u
μ
351
+
,
Γ(ν) Hν− 12 (uμ) − Y ν− 12 (uμ)
[|arg u| < π,
3.388
2u
1.
ν−1 −μx
√
2ux − x2
e
dx = π
0
2u
μ
ν− 12
∞
0
ν−1 −μx
1
2βx + x2
e
dx = √
π
2β
μ
ν− 12
ET I 138(10)
e−uμ Γ(ν) I ν− 12 (uμ)
[u > 0,
2.
Re μ > 0]
Re ν > 0]
ET I 138(14)
eβμ Γ(ν) K ν− 12 (βμ)
[|arg β| < π,
Re ν > 0,
Re μ > 0]
ET I 138(13)
∞
3.
0
∞
4.
√
2
ν−1 −μx
i πe
(2)
x + ix
e
dx = − ν− 1 Γ(ν) H ν− 1
2
2μ 2
2
ν−1 −μx
x − ix
e
dx =
0
3.389
1.
2.7
iμ
2
√
iμ
i πe− 2
1
2μν− 2
Re ν > 0]
ET I 138(15)
[Re μ > 0,
Re ν > 0]
ET I 138(16)
μ
2
−1 μx
μ2 u2
1
1
x2ν−1 u2 − x2
e dx = B(ν, )u2ν+2−2 1 F 2 ν; , ν + ;
2
2
4
0
1
3
1
μ
+ B ν + , u2ν+2−1 1 F 2 ν + ; , ν + +
2
2
2 2
[Re > 0, Re ν > 0]
!
∞
2
μ2 u2 !! 1 − ν
u2ν+2−2
2ν−1
2 −1 −μx
31
√
u +x
x
e
dx =
G
2 π Γ(1 − ) 13 +4 ! 1 − − ν, 0, 12
0
π
|arg u| < , Re μ > 0,
2
√
u ν−1 μx
π
u2ν
+
x u2 − x2
e dx =
2ν
2
0
u
μ
2
ET II 188(21)
,
Re ν > 0
+
,
1
uν+ 2 Γ(ν) I ν+ 12 (μu) + Lν+ 12 (μu)
ET II 188(19)a
ν−1 −μx
1 √ −1
1
1
π
x x2 − u2
e
dx = 2ν− 2
μ 2 −ν uν+ 2 Γ(ν) K ν+ 12 (uμ)
[Re(uμ) > 0]
∞
1 μ2 u2
;
2 4
ET II 234(15)a
1
2 −ν
[Re ν > 0]
∞
4.
5.
2
[Re μ > 0,
u
3.7
(1)
Γ(ν) H ν− 1
μ
2
ET II 203(16)a
−ν −ipx
(ix) e
dx
= πβ −ν−1 e−|p|β
2 + x2
β
−∞
+
|ν| < 1,
Re β > 0,
arg ix =
,
π
sign x
2
ET I 118(5)
352
Exponential Functions
∞
6.
0
∞
7.
0
8.
xν e−μx
1
(ν − 1)π
ν−1
dx = Γ(ν)β
exp iμβ + i
2
2
β +x
2
2
(ν − 1)π
× Γ(1 − ν, iβμ) + exp −iβμ − i
Γ (1 − ν, −iβμ)
2
[Re β > 0, Re μ > 0, Re ν > −1]
xν−1 e−μx dx
= π cosec(νπ)Vν (2μ, 0)
1 + x2
(β + ix)−ν e−ipx
π
dx = (β + γ)−ν e−pγ
2 + x2
γ
γ
−∞
[Re ν > −1,
Re ν > 0]
ET I 138(9)
p > 0,
Re β > 0,
Re γ > 0]
ET I 118(6)
Re ν > −1]
ET I 118(7)
∞
(β − ix)−ν e−ipx
π
dx = (β + γ)−ν eγp
2 + x2
γ
γ
−∞
[p < 0,
∞
3.391
x + 2β +
√
x
2ν
−
x + 2β −
0
3.392
[Re μ > 0,
ET II 218(22)
∞
9.6
3.391
∞
x+
1.
1 + x2
ν
e−μx dx =
0
√
x
Re β > 0,
2ν
Re γ > 0,
ν
e−μx dx = 2ν+1 β ν eβμ K ν (βμ)
μ
ET I 140(30)
[|arg β| < π, Re μ > 0]
1
ν
S 1,ν (μ) + S 0,ν (μ)
μ
μ
[Re μ > 0]
ET I 140(25)
[Re μ > 0]
√
ν
∞
x + 1 + x2
√
e−μx dx = π cosec νπ [J−ν (μ) − J −ν (μ)]
1 + x2
0
[Re μ > 0]
√
ν
∞
1 + x2 − x
√
e−μx dx = S 0,ν (μ) − ν S −1,ν (μ)
[Re μ > 0]
1 + x2
0
ET I 140(26)
∞
2.
1 + x2 − x
ν
e−μx dx =
0
1
ν
S 1,ν (μ) − S 0,ν (μ)
μ
μ
3.
4.
∞
3.393
0
x + x2 + 4β 2
x3 + 4β 2 x
3.394
0
∞
ET I 140(28)
2ν
e−μx dx
=
ET I 140(27), EH II 35(33)
μπ 3
22ν+3/2 β 2ν
J ν+1/4 (βμ) Y ν−1/4 (βμ) − J ν−1/4 (βμ) Y ν+1/4 (βμ)
[Re β > 0, Re μ > 0]
√
ν+1/2
2
√
1+ 1+x
√
−2iμ
e−μx dx = 2 Γ(−ν) D ν
2iμ D ν
xν+1 1 + x2
[Re μ ≥ 0, Re ν < 0]
ET I 140(33)
ET I 140(32)
3.411
3.395
1.
2.
3.
Rational functions of powers and exponentials
353
√
ν √
−ν
x2 − 1 + x
x2 − 1 + x +
√
e−μx dx = 2 K ν (μ)
2−1
x
1
[Re μ > 0]
√
√
2ν
2ν
∞
x + x2 − 1
+ x − x2 − 1
μ
μ
2μ
K ν+1/4
K ν−1/4
e−μx dx =
2
π
2
2
x (x − 1)
1
∞
ET I 140(29)
[Re μ > 0]
√
√
ν
−ν
∞
x + x2 + 1 + cos νπ x + x2 + 1
√
e−μx dx = −π [Eν (μ) + Y ν (μ)]
2+1
x
0
[Re μ > 0]
ET I 140(34)
EH II 35(34)
3.41–3.44 Combinations of rational functions of powers and exponentials
3.411
∞
1.
0
∞
2.
0
∞
3.
0
∞
4.
0
x2n−1 dx
= (−1)n−1
epx − 1
0
∞
6.8
0
[Re μ > 0,
2n
2π
p
B2n
4n
x2n−1 dx = 1 − 21−2n
px
e +1
[n = 1, 2, . . .]
2π
p
2n
∞
0
FI II 792a
[n = 1, 2, . . .]
[Re μ > 0,
|B2n |
4n
FI II 721a
Re ν > 0]
FI II 792a, WH
BI(83)(2), EH I 39(25)
x dx
π2
=
−x
1−e
12
BI (104)(5)
∞
xν−1 e−μx
dx
=
Γ(ν)
(μ + n)−ν β n = Γ(ν) Φ(β, ν, μ)
1 − βe−x
n=0
[Re μ > 0 and either |β| ≤ 1,
7.11
Re ν > 1]
1 xν−1 dx
= ν 1 − 21−ν Γ(ν) ζ(ν)
eμx + 1
μ
ln 2
5.
1
xν−1 dx
= ν Γ(ν) ζ(ν)
μx
e −1
μ
xν−1 e−μx
1
μ
dx = ν Γ(ν) ζ ν,
−βx
1−e
β
β
β = 1,
Re ν > 0; or β = 1,
[Re β > 0,
Re ν > 1]
Re μ > 0,
EH I 27(3)
Re ν > 1]
ET I 144(10)
∞
8.
0
∞
9.
0
10.
0
∞
xn−1 e−px
dx = (n − 1)!
1 + ex
∞
k=1
(−1)k−1
(p + k)n
[p > −1;
n = 1, 2, . . .]
BI (83)(9)
π2
xe−x dx
=
−1
ex − 1
6
(cf. 4.231 3)
BI (82)(1)
π2
xe−2x dx
=
1
−
e−x + 1
12
(cf. 4.251 6)
BI (82)(2)
354
Exponential Functions
∞
11.
0
∞
12.11
0
13.11
14.7
xe−3x
π2
3
dx
=
−
−x
e +1
12 4
2n−1
(−1)k−1
xe−(2n−1)x
π2
+
dx
=
−
1 + ex
12
k2
∞
∞
15.7
0
17.
18.
(cf. 4.251 6)
BI (82)(5)
(cf. 4.251 5)
BI (82)(4)
x2 e−nx
dx = 2
1 + e−x
∞
k=n
[n = 1, 2, . . .]
(cf. 4.261 12)
k=1
(−1)n+k
= (−1)n+1
k3
BI (82)(9)
n−1
(−1)k
3
ζ(3) + 2
2
k3
k=1
[n = 1, 2, . . .]
(cf. 4.261 11)
LI (82)(10)
∞
x e
dx = π 3 csc3 μπ 2 − sin2 μπ
−x
−∞ 1 + e
∞ 3 −nx
n−1
1
x e
π4
−
6
dx
=
−x
15
k4
0 1−e
16.
BI (82)(3)
2n
k=n
(cf. 4.251 5)
k=1
xe−2nx
π 2 (−1)k
+
dx
=
1 + ex
12
k2
0
k=1
∞ 2 −nx
∞
n−1
1
x e
1
dx = 2
= 2 ζ(3) −
−x
k3
k3
0 1−e
3.411
2 −μx
k=1
∞
11
0
∞
(−1)n+k
x3 e−nx
dx = 6
= (−1)n+1
−x
1+e
k4
k=n
[0 < Re μ < 1]
ET I 120(17)a
(cf. 4.262 5)
(−1)k
7 4
π +6
120
k4
n−1
BI (82)(12)
k=1
(cf. 4.262 4)
∞
19.9
0
∞
20.9
0
n dx
=−
e−px e−x − 1
x
(−1)k
k=0
n
ln(p + n − k)
k
∞
∞
0
1
1 − e−mx
dx
=
(n
−
1)!
1 − ex
kn
m
xn−1
(cf. 4.272 11)
LI (83)(8)
∞
qk
1
xp−1
dx = p Γ(p)
= Γ(p)r−p Φ(q, p, 1)
rx
e −q
qr
kp
k=1
[p > 0,
−1 < q < 1]
r > 0,
BI (83)(5)
∞
μx
xe dx
= πβ μ−1 cosec(μπ) [ln β − π cot(μπ)]
x
−∞ β + e
LI (89)(15)
k=1
22.7
23.
LI (89)(10)
n
n dx n
(p + n − k) ln(p + n − k)
e−px e−x − 1
=
(−1)k
2
k
x
0
LI (82)(13)
k=0
21.
n
[|arg β| < π,
0 < Re μ < 1]
BI (101)(5), ET I 120(16)a
∞
24.
−∞
π
μπ
xeμx
dx =
cosec
−1
ν
ν
eνx
2
[Re ν > Re μ > 0]
(cf. 4.254 2)
LI (101)(3)
3.414
Rational functions of powers and exponentials
∞
25.
x
0
26.
27.
28.
29.
(cf. 4.231 4)
BI (82)(6)
∞
1 − e−x −x
2π 2
e dx =
−3x
1+e
27
0
% &
∞
Γ μ2 + 1 √
1 − e−μx dx
π
= ln
1 + ex x
Γ μ+1
0
2
∞ −νx
Γ ν2 Γ μ+1
e
− e−μx dx
2
= ln μ ν+1 e−x + 1
x
Γ 2 Γ 2
0
∞ px
+
qx
pπ
qπ ,
e − e dx
=
ln
cot
tan
rx
x
2r
2r
−∞ 1 + e
x
LI (82)(7)a
[Re μ > −1]
[Re μ > 0,
[|r| > |p|,
BI (93)(4)
Re ν > 0]
|r| > |q|,
BI (93)(6)
rp > 0,
rq > 0]
BI (103)(3)
30.
π2
1 + e−x
dx
=
−1
ex − 1
3
355
∞
+
qπ ,
pπ
e − e dx
= ln sin
cosec
rx
1
−
e
x
r
r
−∞
px
qx
[|r| > |p|,
|r| > |q|,
rp > 0,
rq > 0]
BI (103)(4)
∞ −qx
e
31.
0
π
qπ
cosec
p
p
2
[0 < q < p]
BI (82)(8)
[0 0, q > 0]
1 − e−βx (1 − e−γx ) e−μx dx
Γ(μ) Γ (β + γ + μ)
= ln
1 − e−x
x
Γ(μ + β) Γ(μ + γ)
[Re μ > 0, Re μ > − Re β, Re μ > − Re γ, Re μ > − Re(β + γ)]
BI (96)(7)
(cf. 4.267 25)
BI (93)(13)
∞
1−e
qπ
dx
= ln cosec
qx
(q−2p)x
x
2p
e −e
2.
0
(q−p)x 2
e
0
− e−qx 1 + e−(2n+1)x
dx
1 + e−x
x
= ln
3.414
0
BI (95)(6)
∞ −px
3.
[0 < q < p]
∞
q(q + 2)(q + 4) · · · (q + 2n)(p + 1)(p + 3) · · · (p + 2n − 1)
p(p + 2)(p + 4) · · · (p + 2n)(q + 1)(q + 3) · · · (q + 2n − 1)
[Re p > −2n, Re q > −2n]
(cf. 4.267 14) BI (93)(11)
1 − e−βx (1 − e−γx ) 1 − e−δx e−μx dx
Γ(μ) Γ(μ + β + γ) Γ(μ + β + δ) Γ(μ + γ + δ)
= ln
1 − e−x
x
Γ(μ + β) Γ(μ + γ) Γ(μ + δ) Γ(μ + β + γ + δ)
[2 Re μ > |Re β| + |Re γ|+| Re δ|]
(cf. 4.267 31) BI (93)(14), ET I 145(17)
356
3.415
Exponential Functions
∞
1.
0
1
x dx
=
ln
(x2 + β 2 ) (eμx − 1)
2
βμ
2π
−
π
−ψ
βμ
3.415
βμ
2π
[Re β > 0,
Re μ > 0]
BI (97)(20), EH I 18(27)
∞
2.11
x dx
(x2
0
+
2
β2)
(e2πx
1
1
1 ψ (β)
− 2+
3
8β
4β
4β
− 1)
∞
1 |B2k+2 |
∼
4β 4
β 2k
=−
k=0
[asymptotic expansion for Re β > 0]
∞
3.11
0
∞
4.8
∞
1.
0
∞
2.
0
3.
x dx
(x2
0
3.416
1
x dx
=
ψ
(x2 + β 2 ) (eμx + 1)
2
∞
8
0
+
2
β2)
(e2πx
+ 1)
=
βμ 1
+
2π
2
− ln
1
1 1
ψ β+
−
2
4β
4β
2
βμ
2π
BI(97)(22), EH I 22(12)
[Re β > 0,
Re μ > 0]
[Re β > 0,
Re μ > 0]
1 2n − 1
dx
(1 + ix)2n − (1 − ix)2n
=
i
e2πx − 1
2 2n + 1
[n = 1, 2, . . .]
BI (88)(4)
1
(1 + ix)2n − (1 − ix)2n dx
=
πx
i
e +1
2n + 1
[n = 1, 2, . . .]
BI (87)(1)
,
1 +
(1 + ix)2n−1 − (1 − ix)2n−1 dx
2n
=
1
−
2
B
2n
i
eπx + 1
2n
[n = 1, 2, . . .]
3.417
1.
2.
3.418
1.6
2.6
3.
∞
b
x dx
π
ln
=
2 ex + b2 e−x
a
2ab
a
−∞
∞
x dx
π2
=
2
x
2
−x
4ab
−∞ a e − b e
1
1
x dx
=
ψ
x + e−x − 1
e
3
3
0
∞
−x
1
1
xe dx
=
ψ
x + e−x − 1
e
6
3
0
ln 2
π
x dx
= ln 2
x + 2e−x − 2
e
8
0
BI (87)(2)
∞
[ab > 0]
(cf. 4.231 8)
(cf. 4.231 10)
2
− π 2 = 1.1719536193 . . .
3
5 2
− π = 0.3118211319 . . .
6
BI (101)(1)
LI (101)(2)
LI (88)(1)
LI (88)(2)
BI (104)(7)
3.421
3.419
Rational functions of powers and exponentials
∞
2
(ln β)
x dx
=
x
−x
)
2(β − 1)
−∞ (β + e ) (1 + e
1.
357
[|arg β| < π]
(cf. 4.232 2)
BI (101)(16)
2.
∞
x dx
−∞ (β
2
+
ex ) (1
−
e−x )
=
π + (ln β)
2(β + 1)
2
+
[|arg β| < π]
2
BI (101)(17)
,
π 2 + (ln β) ln β
x2 dx
=
x
−x )
3(β + 1)
−∞ (β + e ) (1 − e
3.
∞
(cf. 4.232 3)
[|arg β| < π]
(cf. 4.261 4)
BI (102)(6)
+
2
π 2 + (ln β)
x3 dx
=
x
−x )
4(β + 1)
−∞ (β + e ) (1 − e
4.
∞
,2
[|arg β| < π]
(cf. 4.262 3)
BI (102)(9)
+
2
π 2 + (ln β)
x4 dx
=
x
−x )
15(β + 1)
−∞ (β + e ) (1 − e
5.
∞
,2
,
+
2
7π 2 + 3 (ln β) ln β
(cf. 4.263 1)
6.
11
2
π 2 + (ln β)
x5 dx
=
x
−x )
6(β + 1)
−∞ (β + e ) (1 − e
7.
+
∞
+
,2
− 4π 2 + (ln β)
(x − ln β) x dx
=
x
−x )
6(β − 1)
−∞ (β − e ) (1 − e
∞
BI (102)(10)
,
+
2
3π 2 + (ln β)
2
(cf. 4.264 3)
,
BI (102)(11)
ln β
[|arg β| < π]
(cf. 4.257 4)
BI (102)(7)
3.421
∞
1.
0
−νx
n −ρx
m
dx
e
e
−1
− 1 e−μx 2
x
=
n
k=0
(−1)k
m
n m
(−1)l
k
l
l=0
× {(m − l)ρ + (n − k)ν + μ} ln [(m − l)ρ + (n − k)ν + μ]
[Re ν > 0,
2.
0
∞
Re μ > 0,
Re ρ > 0]
BI (89)(17)
n
n n
dx 1 (ρ + kν + 1)2
1 − e−νx
1 − e−ρx e−x 3 =
(−1)k
k
x
2
k=0
n
n
1
(kν + 1)2 ln(kν + 1)
× ln(ρ + kν + 1) +
(−1)k−1
k
2
k=1
[n ≥ 2,
Re ν > 0,
Re ρ > 0]
BI (89)(31)
358
Exponential Functions
3.422
π β μ−1 ln β − γ μ−1 ln γ
π 2 β μ−1 − γ μ−1 cos μπ
xe−μx dx
=
+
−x ) (γ + e−x )
(β − γ) sin μπ
(γ − β) sin2 μπ
−∞ (β + e
[|arg β| < π, |arg γ| < π, β = γ. 0 < Re μ < 2]
3.
∞
∞
4.
0
∞
5.
0
ET I 120(19)
−px
(p + s + 1)(q + r + 1)
dx
e
= ln
− e−qx e−rx − e−sx e−x
x
(p + r + 1)(q + s + 1)
[p + s > −1, p + r > −1, q > p]
(cf. 4.267 24)
BI (89)(11)
dx
1 − e−px 1 − e−qx 1 − e−rx e−x
x
= (p + q + 1) ln(p + q + 1)
+(p + r + 1) ln(p + r + 1) + (q + r + 1) ln(q + r + 1)
−(p + 1) ln(p + 1) − (q + 1) ln(q + 1) − (r + 1) ln(r + 1)
−(p + q + r) ln(p + q + r)
[p > 0,
3.423 1.
(cf. 4.268 3)
∞
2
∞
xν−1 e−μx
2
(ex − 1)
0
dx = Γ(ν) [ζ(ν − 1) − ζ(ν)]
q −px
x e
(1 −
0
∞
4.7
dx
2
ae−px )
xν−1 e−μx
=
∞
Γ(q + 1) ak
apq+1
k=1
−∞ (β
+
2
ex )
[a < 1,
kq
[Re ν > 0,
xex dx
ET I 313(10)
Re ν > 2]
q > −1,
ET I 313(11)
p > 0]
BI (85)(13)
dx = Γ(ν) [Φ(β; ν − 1; μ) − (μ − 1) Φ(β; ν; μ)]
2
∞
BI (102)(8)a
dx = Γ(ν) [ζ(ν − 1, μ + 2) − (μ + 1) ζ(ν, μ + 2)]
(1 − βe−x )
0
[Re ν > 2]
[Re μ > −2,
∞
3.8
xν−1
(ex − 1)
6
BI (89)(14)
∞
0
5.
r > 0]
−π 2
x(x − a)eμx dx
=
cosec2 μπ [(eαμ + 1) ln μ − 2π cot μπ (eαμ − 1)]
x
−x )
ea − 1
−∞ (β − e ) (1 − e
[a > 0, |arg β| < π, |Re μ| < 1]
(cf. 4.257 5)
3.422
2.
q > 0,
=
1
ln β
β
Re μ > 0,
|arg(1 − β)| < π]
[|arg β| < π]
(cf. 9.550)
ET I 313(12)
(cf. 4.231 5)
BI (101)(10)
6.∗
t
x5
0
e−x
(1 −
2
e−x )
∞
e−kt 5
y + 5y 4 + 20y 3 + 60y 2 + 120y + 120
5
k
k=1
∞
5 −t/2
e−kt 4
t e
−5
= 120 ζ(5) −
y + 4y 3 + 12y 2 + 24y + 24
2 sinh(t/2)
k5
k=1
y = kt
dx = 120 ζ(5) −
3.427
Rational functions of powers and exponentials
3.424 1.7
∞
(1 + a)ex − a
(1 − ex )
0
∞
2.
(1 + a)ex + a
(1 + ex )
0
3.
∞
∞
−∞
1.
7
2
∞
[a > −1,
n = 1, 2, . . .]
BI (85)(14)
dx =
π2
2ab
[ab > 0]
BI (102)(3)a
dx =
b
π
ln
ab a
[ab > 0]
BI (102)(1)
x2 dx =
2 2
π −2
3
BI (85)(7)
∞
(a2 ex
∞
∞
0
∞
7
0
∞
3.
0
∞
4.
0
0
0
+
2
ex )
(1 +
(ex − ae−x ) x2 dx
2
=
(ln a)
a−1
=
π 2 + (ln a)
a+1
1
1
−
1 − e−x
x
e−x dx = C
1
1
−
2 1 + e−x
e−2x
1 π
dx = ln
x
2 4
∞
1 −2x
1
e
− x
2
e +1
eμx − 1
−μ
1 − e−x
BI (102)(5)
BI (102)(12)
dx = ψ(μ)
1 1
1
− +
2 x ex − 1
p > 0]
2
+ ex ) (1 − e−x )
e−x
e−μx
+ −x
x
e −1
[a > 0,
2
2
e−x )
2
1
p p
,
ln a
B
ap+1
2 2
e−μx
1
dx = ln Γ(μ) − μ −
x
2
BI (102)(13)
[Re μ > 0]
(cf. 4.281 4)
(cf. 4.281 1)
WH
BI (94)(1)
BI (94)(5)
ln μ + μ −
[Re μ > 0]
∞
5.
6.
=−
p+1
e−x )
(ex − ae−x ) x2 dx
−∞ (a
3.427 1.
+
n > 0]
BI(101)(13), LI(101)(13)
2 x
a e − e−x x2 dx
−∞ (a
2.
BI (85)(15)
∞
−∞
2.
+
2
2x
2
−x
b e )
(ex − 1)
2.7
1.
(a2 ex
n = 1, 2, . . .]
√
π Γ n − 12
1
xex dx
a
−C−ψ n−
=
2 ln
2
2 2x n
4a2n−1 b Γ(n)
2b
2
−∞ (a + b e )
[ab > 0,
3.426
−
e − e−x + 2
0
3.425
2 ex
∞
(−1)k
(a + k)n
[a > −1,
k=1
2
2x
b2 e−x )
∞ x
5.
e−ax xn dx = n!
a2 ex − b2 e−x
4.
2
e−ax xn dx = n! ζ(n, a)
a2 ex + b2 e−x
−∞ (a
2
359
dx
1
= − ln π
x
2
1
ln(2π)
2
WH
BI (94)(6)
e−x
dx = − ln Γ(μ) − ln sin(πμ) + ln π
x
[Re μ < 1]
EH I 21(6)
360
Exponential Functions
∞
e−νx
e−μx
−
1 − e−x
x
7.
0
∞
n
e−μx
−
x 1 − e−x/n
8.
0
∞
μ−
9.
0
∞
1−e
1 − e−x
νe−x −
10.
0
∞
11.
−μx
dx = ln μ − ψ(ν)
3.428
(cf. 4.281 5)
BI (94)(3)
[Re μ > 0,
BI (94)(4)
e−x dx = n ψ(nμ + n) − n ln n
e
−x
x
[Re μ > −1]
dx = ln Γ(μ + 1)
e−μx − e−(μ+ν)x
ex − 1
n = 1, 2, . . .]
WH
dx
Γ(μ + ν + 1)
= ln
x
Γ(μ + 1)
[Re μ > −1,
,
+
−1
(1 − ex ) + x−1 − 1 e−xz dx = ψ(z) − ln z
Re ν > 0]
[Re z > 0]
BI (94)(8)
EH I 18(24)
0
3.428 1.
∞
νe−μx −
0
1 −x 1 e−1 − e−μνx
e −
μ
μ 1 − e−x
dx
1
= ln Γ(μν) − ν ln μ
x
μ
[Re μ > 0,
∞
n−1
n−1
e(1−μ)x
e−nμx
+
+
+
−x
x/n
2
1−e
1 − e−x
1−e
2.
0
e−x
n−1
1
dx
=
ln 2π − nμ +
x
2
2
[Re μ > 0,
∞
nμ −
3.
0
−νx
−μνx
x
μx
e
e
e
e
−
−
+
1 − ex
1 − eμx
1 − ex
1 − eμx
0
6.
e−x
dx =
x
n−1
ln Γ μ −
k=0
[Re μ > 0,
∞
4.
5.
n
e(1−μ)x
n−1
−
−
−x
2
1−e
1 − ex/n
Re ν > 0]
n = 1, 2, . . .]
BI (94)(18)
ln n
BI (94)(14)
k
+1
n
n = 1, 2, . . .]
BI (94)(13)
dx
= ν ln μ
x
[Re μ > 0, Re ν > 0]
LI (94)(15)
1
dx
μe−μx
μ + 1 −μx
−
+ aμ −
+ (1 − aμ)e−x
e
x
−μx
e −1 1−e
2
x
0
1
μ−1
ln(2π) +
− aμ ln μ
=
2
2
[Re μ > 0]
BI (94)(16)
∞ −νx
−μνx
−μx
1
dx
e
μ−1
e
(μ − 1)e
μ − 1 −μ
e x
=
ln(2π) +
− μν ln μ
−
−
−
−x
−μx
−μx
1−e
1−e
1−e
2
x
2
2
0
[Re μ > 0, Re ν > 0]
(cf. 4.267 37) BI (94)(17)
∞
∞
7.
0
3.429
0
(1 − e−νx ) (1 − e−μx ) dx
−x
= ln B(μ, ν)
1−e −
1 − e−x
x
[Re μ > 0,
∞
−x
dx
= ψ(μ)
e − (1 + x)−μ
x
[Re μ > 0]
Re ν > 0]
BI (94)(12)
NH 184(7)
3.437
3.431
Rational functions of powers and exponentials
∞
1
−1
Γ(ν + 3)
e−μx − 1 + μx − μ2 x2 xν−1 dx =
2
ν(ν + 1)(ν + 2)μν
[Re μ > 0, −2 > Re ν > −3]
1.
0
∞
2.
0
361
−px
1 −2
1
−1
−x
dx = −1 + p +
x − x (x + 2) 1 − e
e
2
2
LI (90)(5)
ln 1 +
1
p
[Re p > 0]
3.432
∞
1.
0
n
n
n
1
xν−1 e−mx e−x − 1 dx = Γ(ν)
(−1)k
k (n + m − k)ν
k=0
[n = 0, 1, . . . , Re ν > 0]
∞
2.
+
ν−1 ,
xν−1 e−x − e−μx 1 − e−x
dx = Γ(ν) −
0
∞
3.433
%
xp−1 e−x +
0
n
(−1)k
k=1
xk−1
(k − 1)!
LI (90)(10)
Γ(μ)
Γ(μ + ν)
[Re μ > 0,
ET I 144(6)
Re ν > 0]
LI (81)(14)
&
dx = Γ(p)
[−n 0,
Re ν > 0,
Re ρ < 1]
BI (90)(6)
2.
3.435
− e−μx
e−μx −
1
1 + ax
Re ν > 0]
dx
= 1 − ln 2
x
1
1 − e−μx
dx =
ln (βμ) + C − eβμ Ei(−βμ)
x(x + β)
β
1
− e−x
1+x
[Re μ > 0,
LI (89)(19)
[|arg β| < π,
Re μ > 0]
dx
=C
x
dx
a
= ln − C
x
μ
FI II 634
ET II 217 (18)
FI II 7 95, 802
[a > 0,
e−npx − e−nqx
e−mpx − e−mqx dx
m
−
= (q − p) ln
2
n
m
x
n
−px
dx
1
−
e
= p ln p − p
[p > 0]
pe−x −
x
x
Re μ > 0]
[p > 0,
BI (92)(10)
q > 0]
BI (89)(28)
BI (89)(24)
362
3.438
Exponential Functions
∞
dx
ln 2 − 1
=
x
2
2
2
−px
p −x p
dx
p
p2
11
1−e
e −
− 2−
=
ln p − p3
3
6
2x x
x
x
6
36
1 1
+
2 x
1.
0
3.438
∞
2.7
0
e−x −
1 −x
e 2
x
BI (89)(19)
[p > 0]
∞
0
dx
= 1 − ln 2
x
x
dx
x + 2 −px
=
e
− e− 2
e−x +
2x
x
e−x − e−2x −
3.
∞
p−
4.
0
1
2
BI (89)(33)
1 −2x
e
x
BI (89)(25)
1
2
p−
(ln p − 1)
[p > 0]
∞
(p − q)e−rx +
3.439
0
∞
3.441
0
3.442
∞
1.
0
0
∞
3.
0
3.443
∞
1.
0
∞
2.
0
0
r
dx
= p ln p − q ln q − (p − q) 1 + ln
x
m
[p > 0, q > 0, r > 0] LI(89)(26), LI(89)(27)
dx
(p − r)e−qx + (r − q)e−px + (q − p)e−rx
= (r − q)p ln p + (p − r)q ln q + (q − p)r ln r
x2
[p > 0, q > 0, r > 0]
(cf. 4.268 6) BI (89)(18)
1−
1
dx
x+2
= −1 + q +
1 − e−x e−qx
2x
x
2
e−x − 1
1
+
x
1+x
ln
q+1
q
e−px −
1
1 + a2 x2
BI (89)(23)
dx
=C−1
x
BI (92)(16)
dx
a
= −C + ln
x
p
e−x p2
p 1 − e−px
− +
2
x
x2
dx
p2
3
=
ln p − p2
x
2
4
[p > 0]
BI (92)(11)
[p > 0]
BI (89)(32)
n
(1 − e−px ) e−qx
1
dx =
(−1)k−1
(q + kp)2 ln(q + kp)
3
k
x
2
n
n
k=2
[n > 2,
3.
BI (89)(22)
[q > 0]
∞
2.
1 −mpx
e
− e−mqx
mx
∞
q > 0,
pn + q > 0]
(cf. 4.268 4)
2
dx
1 − e−px e−qx 2 = (2p + q) ln(2p + q) − 2(p + q) ln(p + q) + q ln q
x
[q > 0, 2p > −q]
BI (89)(30)
(cf. 4.268 2)
BI (89)(13)
3.456
Powers and algebraic functions of exponentials
363
3.45 Combinations of powers and algebraic functions of exponentials
3.451
1.
2.
3.452
√
4 4
− ln 2
xe−x 1 − e−x dx =
3 3
0
∞
π 1
+ ln 2
xe−x 1 − e−2x dx =
4 2
0
∞
∞
1.
0
∞
2.
0
∞
3.
0
∞
4.
0
∞
5.
0
3.453
∞
1.
0
1.
∞
11
0
∞
2.
0
3.455
1.
2.
3.456
BI (99)(5)
BI (99)(6)
xe−x dx
√
= 1 − ln 2
e2x − 1
BI (99)(8)
7
xe−2x dx
3
√ x
= π ln 2 −
4
12
e −1
BI (99)(7)
dx
b
xex
2π
√ x
ln 1 +
=
2
x
2
2
a e − (a − b ) e − 1
ab
a
[ab > 0]
(cf. 4.298 17)
BI (99)(16)
b
xex dx
2π
√
arctan
=
ab
a
− (a2 + b2 )] ex − 1
[ab > 0]
(cf. 4.298 18)
BI (99)(17)
xe−2nx dx
(2n − 1)!! π
√
=
2x
(2n)!! 2
e −1
ln 2 +
xe−(2n−1)x dx
(2n − 2)!!
√
=−
(2n − 1)!!
e2x − 1
0
2n
(−1)k
k=1
ln 2 +
LI (99)(10)
k
2n−1
k=1
(−1)k
k
LI (99)(9)
∞
x2 ex dx
<
= 8π ln 2
3
0
(ex − 1)
∞
x3 ex dx
π2
2
<
= 24π (ln 2) +
12
3
0
(ex − 1)
1.
BI (99)(2)
xe−x dx
π
√ x
= [2 ln 2 − 1]
2
e −1
[a2 ex
0
(cf. 4.241 9)
FI II 643a,BI(99)(4)
x2 dx
π2
2
√ x
= 4π (ln 2) +
12
e −1
∞
2.
3.454
x dx
√ x
= 2π ln 2
e −1
BI (99)(1)
∞
x dx
π
π
√
√
√
=
ln
3
+
3 3x
3 3
3 3
e −1
BI (99)(11)
BI (99)(12)
BI (99)(13)
364
Exponential Functions
π
π
<
= √ ln 3 − √
2
3
3 3
3 3
(e3x − 1)
∞
x dx
2.
0
3.457
∞
1.
0
∞
x
xe dx
−∞ (a
−∞
+
∞
3.
3.458
n+3/2
ex )
=
BI (99)(3)
2
[ln(4a) − 3C − 2 ψ(2n) − ψ(n)]
(2n + 1)an+1/2
BI (101)(12)
x dx
1
μ μ
,
ln a
B
μ =−
μ
−x
2a
2 2
+e )
p−1
xex (ex − 1)
0
BI (99)(14)
(cf. 4.241 5)
(a2 ex
ln 2
1.7
2.
(cf. 4.244 3)
n−1/2
(2n − 1)!!
π [C + ψ(n + 1) + 2 ln 2]
xe−x 1 − e−2x
dx =
4 · (2n)!!
2.
3.457
[a > 0,
Re μ > 0]
BI (101)(14)
%
&
∞
(−1)k−1
1
ln 2 +
dx =
p
p+k+1
BI (104)(4)
k=0
∞
xex dx
−∞ (a
+
ν+1
ex )
1
[ln a − C − ψ(ν)]
νaν %
&
ν−1
1
1
= ν ln a −
νa
k
=
[a > 0]
[a > 0,
ν = 1, 2, . . .]
k=1
BI (101)(11)
3.46–3.48 Combinations of exponentials of more complicated arguments and powers
3.461
∞ −p2 x2
e
1.
x2n
u
√
(−1)n 2n−1 p2n−1 π
[1 − Φ(pu)]
dx =
(2n − 1)!!
2 2 n−1
e−p u (−1)k 2k+1 (pu)2k
+ 2n−1
2u
(2n − 1)(2n − 3) · · · (2n − 2k − 1)
k=0
∞
2.
x2n e−px dx =
2
0
∞
3.
x2n+1 e−px dx =
2
0
∞
6.∗
∞
π
p
[p > 0,
n = 0, 1, . . .]
[p > 0]
FI II 743
BI (81)(7)
(2a)2k n!
(2n − 1)!! √ k
π
(−1)
2n
(2k)!(n − k)!
k=0
2
dx
1
√
√
= e−μu − μπ [1 − Φ (u μ)]
2
x
u
u
∞
exp −a x2 + b2 dx = b K 1 (ab)
0
NT 21(4)
n
2
−∞
[p > 0]
n!
2pn+1
(x + ai)2n e−x dx =
4.
5.11
(2n − 1)!!
2(2p)n
e−μx
2
BI (100)(12)
+
π
|arg μ| < ,
2
[Re a > 0,
,
u>0
Re b > 0]
ET I 135(19)a
3.462
7.
∗
Exponentials of complicated arguments and powers
∞
0
8.∗
9.∗
[Re a > 0,
2
3
12b
3b
x4 exp −a x2 + b2 dx = 3 K 2 (ab) + 2 K 1 (ab)
a
a
Re b > 0]
[Re a > 0,
3
4
90b
15b
x6 exp −a x2 + b2 dx = 4 K 3 (ab) + 3 K 2 (ab)
a
a
Re b > 0]
∞
[Re a > 0,
Re b > 0]
γ
√
2β
[Re β > 0,
Re ν > 0]
0
3.462
2b
b2
K 0 (ab)
x2 exp −a x2 + b2 dx = 2 K 1 (ab) +
a
a
∞
0
∞
1.
xν−1 e−βx
2
−γx
dx = (2β)−ν/2 Γ(ν) exp
0
∞
xn e−px
2
2.8
+2qx
1
2n−1 p
∞
(ix)ν e−β
2
x2 −iqx
dx = 2− 2
ν
−∞
∞
4.
−∞
∞
2
−2νx
dx =
0
6.
7.11
8.
9.∗
EH II 119(3)a, ET I 313(13)
n−1
∞
p
4q 2
[p > 0]
BI (100)(8)
[p > 0]
LI (100)(8)
k
√ −ν−1
q
q2
√
πβ
exp − 2 D ν
8β
β 2
,
+
π
2
Re β > 0, Re ν > −1, arg ix = sign x
2
√
xn exp −(x − β)2 dx = (2i)−n π H n (iβ)
xe−μx
5.11
D −ν
k=0
3.11
γ2
8β
2
π d
qeq /p
p dq n−1
n/2
n 2
π q
1
= n!eq /p
p p
(n − 2k)!(k)!
dx =
−∞
365
ν
1
−
2μ 2μ
π νμ2
e
1 − erf
μ
EH II 195(31)
ν
√
μ
+
π
|arg ν| < ,
2
,
Re μ > 0
q2
π
exp
[Re p > 0]
p
p
−∞
∞
2
ν
π 2ν 2 + μ νμ2
ν
e
x2 e−μx −2νx dx = − 2 +
1
−
erf
√
2μ
μ5
4
μ
0
,
+
π
|arg ν| < , Re μ > 0
2
∞
2
2
ν2
π
1
ν
x2 e−μx +2νx dx =
[|arg ν| < π, Re μ > 0]
1+2
eμ
2μ μ
μ
−∞
∞
n
1
e±a
e−βx ±a dx =
Γ
[Re β > 0, Re n > 0]
1/n
n
nβ
0
xe−px
2
+2qx
dx =
q
p
ET I 121(23)
ET I 146(31)a
BI (100)(7)
ET I 146(32)
BI (100)(8)a
366
10.
Exponential Functions
∗
∞
(x − a)e−β(x−a) dx = eaβ
0
11.∗
∞
(x − a)e−β(x+a) dx = e−aβ
0
12.∗
∞
13.∗
∞
u
15.
am e±pb/a
pb
Γ m + 1, pu ±
m+1
p
a
(ax ± b)m e−px dx =
16.∗
17.
∗
0
18.∗
e−px
pn−1 e±pb/a
pb
dx =
Γ −n + 1, pu ±
n
(ax ± b)
an
a
u
e−px
pn−1 e±pb/a
pb
dx =
Γ −n + 1, ±
(ax ± b)n
an
a
∞
x−a
b
19.
20.∗
21.∗
22.∗
23.∗
j
exp −β
x−a
b
k
bΓ
dx =
!
!
!arg
!
p > 0,
!
!
!arg
!
!
!
!arg
!
p > 0,
∞
0
∗
am e±pb/a
pb
Γ m + 1, ±
pm+1
a
e−px
pn−1 e±pb/a
pb
dx
=
Γ −n + 1, ±
(ax ± b)n
an
a
u
p > 0,
!
b !!
<
π
a !
b
±u
a
pb
− Γ m + 1, pu ±
a
!
!
u > 0, p > 0, !!arg
∞
0
[Re β > 0]
(ax ± b)m e−px dx =
0
∗
(1 − aβ)
β2
am e±pb/a
pb
Γ m + 1, ±
pm+1
a
u
14.∗
[Re β > 0]
(ax ± b)m e−px dx =
0
(1 − aβ)
β2
3.462
u > 0,
!
b !!
<π
a !
!
!
!arg
!
p > 0,
pb
− Γ −n + 1, pu ±
a
u > 0, p > 0,
j+1
k ,β
!
!
!arg
!
a k
−b
kβ (j+1)/k
+
a
arg −
> 0,
b
!
!
!<π
!
!
!
b
± u !! < π
a
!
!
b
± u !! < π
a
!
!
b
± u !! < π
a
,
Re b > 0,
Re β > 0,
Re k > 0
∞ −βxn
Γ(z, βun )
1−m
[u > 0,
z=
m
z
x
nβ
n
u
√
∞
exp −a x + b2
√
dx = K 0 (ab)
x2 + b2
0
√
∞ 2
x exp −a x + b2
b
√
dx = K 1 (ab)
2 + b2
a
x
0
√
∞ 4
x exp −a x + b2
3b2
√
dx = 2 K 1 (ab)
2
2
a
x +b
0
√
∞ 6
2
x exp −a x + b
15b3
√
dx = 3 K 3 (ab)
a
x2 + b2
0
e
dx =
Re β > 0,
Re n > 0,
Re z > 0]
[Re a > 0,
Re b > 0]
[Re a > 0,
Re b > 0]
[Re a > 0,
Re b > 0]
[Re a > 0,
Re b > 0]
3.471
24.
∗
Exponentials of complicated arguments and powers
∞
0
25.∗
∞
0
∞
√
x2n exp −a x + b2
√
dx = (2n − 1)!!
x2 + b2
exp −px2
1
√
dx = exp
2
2
2
a +x
a2 p
2
K0
b
a
367
n
K n (ab)
a2 p
2
[Re a > 0,
Re b > 0]
[Re a > 0,
Re b > 0]
dx
1
= C
x
2
0 ∞
dx √ √
√ −μx2
−νx2
e
3.464
−e
= π ν− μ
[Re μ > 0, Re ν > 0]
2
x
0 ∞
2
π
μ+β
1 + 2βx2 e−μx dx =
3.465
[Re μ > 0]
2
μ3
0
3.466
∞ −μ2 x2
+
e
π β 2 μ2
Re β > 0, |arg μ| <
e
dx = [1 − Φ(βμ)]
1.
2
2
2β
0 x +β
√
∞ 2 −μ2 x2
+
x e
π πβ μ2 β 2
−
e
2.
dx
=
[1
−
Φ(βμ)]
Re β > 0, |arg μ| <
x2 + β 2
2μ
2
0
1 x2
∞
e −1
1
3.
dx
=
2
x
k!(2k − 1)
0
3.463
e−x − e−x
2
∞
e−x −
2
0
1.
2.
3.469
∞
dx
1
=− C
x
2
1
1 + x2
∞
1.
∞
2.
e−μx
4
−2νx2
dx =
e−x − e−x
4
0
∞
3.
e−x − e−x
4
2
0
3.471 1.
u
exp −
0
2.
0
π,
4
π,
4
NT 19(13)
ET II 217(16)
FI II 683
2
0
ET I 136(24)a
BI (92)(12)
π
e−x
dx
2
√
=
[1 − Φ(u)]
√
2 − u2 x
4u
x
u 2
∞ −μx2
xe
dx
1 π a2 μ
√
√
e
=
[1 − Φ (a μ)]
2 + x2
2
μ
a
0
FI II 645
k=1
3.467
3.468
BI (89)(5)
u
β
x
1
4
2ν
exp
μ
ν2
2μ
[u > 0]
NT 33(17)
[Re μ > 0,
K 14
ν2
2μ
a > 0]
[Re μ ≥ 0]
NT 19(11)
ET I 146(23)
dx
3
= C
x
4
BI (89)(7)
dx
1
= C
x
4
BI (89)(6)
dx
β
1
= exp −
2
x
β
u
β
xν−1 (u − x)μ−1 e− x dx = β
ν−1
2
ET II 188(22)
u
2μ+ν−1
2
exp −
β
2u
Γ(μ) W
β
u
Re β > 0,
1−2μ−ν
2
[Re μ > 0,
, ν2
u > 0]
ET II 187(18)
368
Exponential Functions
u
3.
β
x−μ−1 (u − x)μ−1 e− x dx = β −μ uμ−1 Γ(μ) exp −
0
β
u
[Re μ > 0,
u
4.
0
3.471
β
β
1
β
1
x−2μ (u − x)μ−1 e− x dx = √ β 2 −μ e− 2u Γ(μ) K μ− 12
2u
πu
[u > 0,
Re μ > 0]
β
u
[0 < Re μ < Re(1 − ν),
β
xν−1 (x − u)μ−1 e x dx = B(1 − μ − ν, μ)uμ+ν−1 1 F 1 1 − μ − ν; 1 − ν;
u
∞
6.
β
x−2μ (x − u)μ−1 e x dx =
u
Re β > 0,
ET II 187(16)
ET II 187(17)
∞
5.
u > 0]
ET II 203(15)
π 1 −μ
β 2 Γ(μ) exp
u
β
2u
I μ− 12
β
2u
[Re μ > 0,
∞
7.
β
u > 0]
xν−1 (x + γ)μ−1 e− x dx = β
ν−1
2
γ
ν−1
2 +μ
β
Γ(1 − μ − ν)e 2γ W
0
u > 0]
ET II 202(14)
β
γ
Re(1 − μ) > Re ν > 0]
ν−1
ν
2 +μ,− 2
[|arg γ| < π,
ET II 234(13)a
8.
0
u
μ−1 − β
1
x−2μ u2 − x2
e x dx = √
π
2
β
μ− 12
3
β
u
[Re β > 0, u > 0,
uμ− 2 Γ(μ) K μ− 12
Re μ > 0]
ET II 188(23)a
∞
9.
β
xν−1 e− x −γx dx = 2
0
∞
10.
xν−1 exp
0
∞
11.
ν−1
x
0
iμ
2
iμ
exp
2
x−
β2
x
β2
x+
x
ν
2
β
γ
K ν 2 βγ
[Re β > 0,
ET II 82(23)a, LET I 146(29)
iνπ
dx = 2β ν e 2 K −ν (βμ)
Im μ > 0, Im β 2 μ < 0; note that K −ν ≡ K ν
dx = iπβ ν e−
iνπ
2
∞
xν−1 exp −x −
0
μ2
4x
dx = 2
μ
2+
0
(1)
H −ν (βμ)
Im μ > 0,
Im β 2 μ > 0
ν
K −ν (μ)
|arg μ| <
13.
EH II 82(24)
EH II 21(33)
12.
Re γ > 0]
∞
,
π
, Re μ2 > 0; note that K −ν ≡ K ν
2
WA 203(15)
β
β
xν−1 e− x
β
dx = γ ν−1 e γ Γ(1 − ν) Γ ν,
x+γ
γ
[|arg γ| < π,
Re β > 0,
Re ν < 1]
ET II 218(19)
3.475
Exponentials of complicated arguments and powers
exp 1 − x1 − xν
dx = ψ(ν)
x(1 − x)
0
∞
1
π −2√βγ
e
x− 2 e−γx−β/x dx =
γ
0
∞
√
√ ∂n
1
xn− 2 e−px−q/x dx = (−1)n π n p−1/2 e−2 pq
∂p
0
14.
15.
16.
369
1
[Re ν > 0]
BI (80)(7)
[Re β ≥ 0,
Re γ > 0]
[Re p > 0,
Re q > 0]
ET 245 (5.6.1)
PBM 344 (2.3.16(2))
3.472
∞
exp −
1.
0
∞
2.
x2 exp −
0
3.
4.
5.
∞
2
1
a
− 1 e−μx dx =
2
x
2
1
a
− μx2 dx =
x2
4
π
√
[exp (−2 aμ) − 1]
μ
[Re μ > 0,
a
− μx2
x2
ET I 146(30)
[Re μ > 0,
Re a > 0]
ET I 146(26)
[Re μ > 0,
a > 0]
π
√
√
(1 + 2 aμ) exp (−2 aμ)
μ3
dx
1 π
√
exp (−2 aμ)
=
2
x
2
a
0
∞
dx
aπ
1
1
(1 + a)e−1/a
exp −
=
x2 + 2
4
2a
x
x
2
0
∞
π ∂ n −2√pq
−n−1/2 −px−q/x
n
x
e
dx = (−1)
e
p ∂q n
0
exp −
Re a > 0]
[a > 0]
[Re p > 0,
ET I 146(28)a
BI (98)(14)
Re q > 0]
PBM 344 (2.3.16(3))
∞
3.473
(2m − 1)!! √
π
2m n
exp (−xn ) x(m+1/2)n−1 dx =
0
3.474
n exp (1 − x−n )
xnp
−
n
1−x
1−x
1
1.
0
1
2.
0
3.475
∞
1.7
∞
2.
0
3.
0
∞
dx
1
k−1
=
ψ p+
x
n
n
n exp (1 − x−n ) exp 1 −
−
1 − xn
1−x
exp −x2
n
exp −x2
n
0
n
exp −x2
n
k=1
1
x
[p > 0]
dx
= − ln n
x
dx
1
1
= − nC
n+1
2
x
2
1+x
dx
1
−
= −2−n C
1 + x2
x
dx = 1 − 2−n C
− e−x
x
−
BI (98)(6)
BI (80)(8)
BI (80)(9)
[n ∈ Z]
BI (92)(14)
BI (92)(13)
BI (89)(8)
370
3.476
Exponential Functions
∞
[exp (−νxp ) − exp (−μxp )]
1.
0
∞
[exp (−xp ) − exp (−xq )]
2.
0
3.477
1.
10
2.8
3.476
1 μ
dx
= ln
x
p ν
[Re μ > 0,
p−q
dx
=
C
x
pq
[p > 0,
Re ν > 0]
BI (89)(3)
q > 0]
BI (89)(9)
∞
e−a|x|
dx = e−au γ(0, −au) − eau γ(0, au)
[Re a > 0, Im u = 0,
−∞ x − u
∞
sign x exp (−a|x|)
dx = − [exp (a|u|) Ei (−a|u|) − exp (−a|u|) Ei (a|u|)]
x−u
−∞
arg u = 0]
[a > 0]
3.478
∞
1.
xν−1 exp (−μxp ) dx =
0
ν
1 − νp
μ Γ
|p|
p
[Re μ > 0 and − p < Re ν < 0 for p > 0,
0 < Re ν < −p for p < 0]
ET I 313(18, 19)
ν ν+1
ν+n−1
,
,...,
;
n
n
n
μ+ν μ+ν+1
μ+ν+n−1
,
,...,
; βun
n
n
n
[Re μ > 0, Re ν > 0, n = 2, 3, . . .]
u
0
∞
4.
0
2
xν−1 exp −βxp − γx−p dx =
p
γ
β
ν
2p
∞
1.
0
∞
2.11
√
xν−1 exp −β 1 + x
2
√
dx = √
π
1+x
ν−1
x
0
√
√
exp iμ 1 + x2
π
√
dx = i
2
2
1+x
β
2
∞
−∞
xex exp (−μex ) dx = −
1
(C + ln μ)
μ
ET I 313(17)
Re ν > 0]
ET I 313(14)
Re ν > 0]
EH II 83(30)
Γ(ν) K 12 −ν (β)
1−ν
2
Γ
ν
(1)
H 1−ν (μ)
2
2
[Im μ > 0,
1.
Re γ > 0]
1
2 −ν
[Re β > 0,
μ
2
ET II 187(15)
K νp 2 βγ
[Re β > 0,
3.481
p > 0]
xν−1 (u − x)μ−1 exp (βxn ) dx = B(μ, ν)uμ+ν−1 n F n
11
3.479
Re ν > 0,
xν−1 [1 − exp (−μxp )] dx = −
0
3.
[Re μ > 0,
ET II 251(36)
BI(81)(8)a, ET I 313(15, 16)
∞
2.
ν
p
1 − νp
μ Γ
p
MC
[Re μ > 0]
BI (100)(13)
3.511
Hyperbolic functions
∞
2.
−∞
3.482
∞
1.3
xe exp −μe
x
2x
1
dx = − [C + ln(4μ)]
4
[Re μ > 0]
BI (100)(14)
[Re β > 0]
∞
2.
0
π
μ
1
[S n (β) − π En (β) − π Y n (β)]
2
exp (nx − β sinh x) dx =
0
371
ET I 168(11)
1
exp (−nx − β sinh x) dx = (−1)n+1 [S n (β) + π En (β) + π Y n (β)]
2
[Re β > 0]
∞
3.
exp (−νx − β sinh x) dx =
0
ET I 168(12)
π
[Jν (β) − J ν (β)]
sin νπ
[Re β > 0]
⎧
iνπ
⎪
⎪
∞
K ν (a) for a > 0,
⎨2 exp −
exp (ν arcsinh x − iax)
2
√
3.483
dx =
iνπ
⎪
1 + x2
−∞
⎪
K ν (−a) for a < 0
⎩2 exp
2
qx
px dx
a
q
a
= (ea − 1) ln
[p > 0,
3.484
− 1+
1+
qx
px
x
p
0 π/2
πe
[1 − Φ(1)]
3.485
exp − tan2 x dx =
2
1
0 1
∞
3.4866
x−x dx =
e−x ln x dx =
k −k = 1.2912859970627 . . .
∞
0
3.487
1.
∗
π/4
%
exp −
0
∞
k=0
[|Re ν| < 1]
ET I 122(32)
0
ET I 168(13)
q > 0]
BI (89)(34)
FI II 483
k=1
tan2k+1 x
k + 12
&
dx = ln 2
3.5 Hyperbolic Functions
3.51 Hyperbolic functions
3.511
∞
1.
0
∞
2.
0
∞
3.
0
4.
0
∞
π
dx
=
cosh ax
2a
[a > 0]
π
aπ
sinh ax
dx =
tan
sinh bx
2b
2b
[b > |a|]
BI (27)(10)a
[b > |a|]
GW (351)(3b)
[b > |a|]
BI (4)(14)a
π
aπ 1
sinh ax
dx =
sec
− β
cosh bx
2b
2b
b
π
aπ
cosh ax
dx =
sec
cosh bx
2b
2b
a+b
2b
372
Hyperbolic Functions
∞
5.
0
∞
6.
0
∞
7.
0
∞
8.11
0
9.
10.
3.512
sin aπ
π
sinh ax cosh bx
c
dx =
bπ
sinh cx
2c cos aπ
c + cos c
[c > |a| + |b|]
BI (27)(11)
bπ
π cos aπ
cosh ax cosh bx
2c cos 2c
dx =
bπ
cosh cx
c cos aπ
c + cos c
[c > |a| + |b|]
BI (27)(5)a
bπ
sinh ax sinh bx
π sin aπ
2c sin 2c
dx =
bπ
cosh cx
c cos aπ
c + cos c
[c > |a| + |b|]
BI (27)(6)a
dx
=1
cosh2 x
BI (98)(25)
∞
sinh2 ax
dx = 1 − aπ cot aπ
2
−∞ sinh x
∞
sinh ax sinh bx
aπ
aπ
dx = 2 sec
2
2b
2b
cosh
bx
0
∞
1.
0
3.512
β
β
cosh 2βx
4ν−1
B ν + ,ν −
dx =
2ν
a
a
a
cosh ax
a2 < 1
BI (16)(3)a
[b > |a|]
BI (27)(16)a
[Re (ν ± β) > 0,
a > 0,
β > 0]
LI(27)(17)a, EH I 11(26)
∞
2.
0
μ
1
sinh x
dx = B
coshν x
2
μ+1 ν−μ
,
2
2
[Re μ > −1,
Re(μ − ν) < 0]
EH I 11(23)
3.513
1.
2.
√
1
dx
a + b + a2 + b 2
√
=√
ln
a2 + b 2 a + b − a2 + b 2
0 a + b sinh x
√
∞
dx
b 2 − a2
2
=√
arctan
2
2
a√
+b
b −a
0 a + b cosh x
1
a + b + a2 − b 2
√
=√
ln
a2 − b 2 a + b − a2 − b 2
√
∞
dx
2
b 2 − a2
=√
arctan
a√
+b
b 2 − a2
0 a sinh x + b cosh x
1
a + b + a2 − b 2
√
=√
ln
a2 − b 2 a + b − a2 − b 2
3.
∞
[ab = 0]
b 2 > a2
b 2 < a2
GW (351)(8)
GW (351)(7)
2
b > a2
2
a > b2
GW (351)(9)
3.516
Hyperbolic functions
∞
4.
0
373
&
%
√
2
b 2 − a2 − c 2
dx
=√
+ π
arctan
a + b cosh x + c sinh x
a+b+c
b 2 − a2 − c 2
⎧
⎡
⎤
=0
for (b − a)(a + b + c) > 0
⎪
⎪
⎪
⎢
⎨|| = 1 for (b − a)(a + b + c) < 0 ⎥
⎢
⎥
2
2
2
⎢when b > a + c ; and
⎥
⎪
⎣
⎦
=1
for a b + c
√
1
a + b + c + a2 − b 2 + c 2
√
=√
ln
a2 − b2 + c2 a + b + c − a2 − b2 +c2
b 2 < a2 + c 2 , a 2 = b 2
1 a+c
= ln
c
a
[a = b = 0, c = 0]
2(a − b)
=
c(a − b − c)
2
b = a2 + c2 , c(a − b − c) < 0
GW (351)(6)
3.514
∞
1.
0
∞
2.
0
[0 < t < π,
cosh ax dx
2
(cosh x + cos t)
0
∞
a > 0]
BI (27)(22)a
a(πt )
cosh ax − cos t1
πt2
π sin b 2
−
dx =
cos t1
cosh bx − cos t2
b sin t2 sin ab π b sin t2
[0 < |a| < b,
∞
3.
dx
t
= cosec t
cosh ax + cos t
a
=
sinh ax sinh bx
π (− cos t sin at + a sin t cos at)
sin3 t sin aπ
0 < a2 < 1,
bt
bπ
bπ
sin
cosec t cosec
2
a
a
a
0 < t2 < π]
0<t<π
[0 < |b| < a, 0 < t < π]
(cosh ax + cos t)
√
∞
2 cosh x
dx = − ln 2
3.515
1− √
cosh 2x
−∞
3.516
∞
dx
dx
1 ∞
1.
√
√
μ =
μ = Q μ−1 (z)
2
2
2
z
+
z
+
z
−
1
cosh
x
z
−
1 cosh x
0
−∞
4.
0
2
dx =
BI (6)(20)a
BI (6)(18)a
BI (27)(27)a
BI (21)(12)a
[Re μ > −1]
For a suitable choice of a single-valued branch of the integrand, this formula is valid for arbitrary
values of z in the z-plane cut from −1 to +1 provided μ <0. If μ > 0, this formula ceases to be valid for
points at which the denominator vanishes.
CO, WH
∞
dx
1.
EH II 181(32)
n+1 = Q n (β)
0
β + β 2 − 1 cosh x
374
Hyperbolic Functions
∞
2.
0
cosh γx dx
β + β 2 − 1 cosh x
=
ν+1
3.517
e−iγπ Γ(ν − γ + 1) Q γν (β)
Γ(ν + 1)
[Re (ν ± γ) > −1,
∞
3.
0
μ −iμπ
2μ
sinh x dx
β + β 2 − 1 cosh x
=
ν+1
2 e
Γ(ν − 2μ + 1) Γ μ + 12
μ
√
π (β 2 − 1) 2 Γ(ν + 1)
ν = −1, −2, −3, . . .]
EH I 157(12)
Q μν−μ (β)
[Re(ν − 2μ + 1) > 0,
Re(ν + 1) > 0]
EH I 155(2)
3.517
∞
1.
ν+ 12
2.
0
3.518
a
cosh γ +
∞
∞
∞
∞
7
0
5.6
0
ν+ 12
=
1
sinh2μ+1 x dx
∞
Re(ν + γ + 1) > 0]
EH I 156(11)
πΓ 2 −ν
P νγ (cosh a)
2 sinhν a
sinh2μ x dx
ν+1
0
4.
x dx
Re ν < 12 ,
a>0
EH I 156(8)
2μ e−iμπ Γ(ν − 2μ + 1) Γ μ + 12
Q μν−μ (cosh a)
=√
Γ(ν + 1)
π sinhμ a
[Re(ν + 1) > 0, Re(ν − 2μ + 1) > 0, a > 0] EH I 155(3)a
μ−ν
= 2μ β 2 − 1 2 Γ(ν − 2μ) Γ(μ + 1) P μ−ν
(β)
μ
(β + cosh x)
[Re(ν − μ) > Re μ > −1,
3.
ν+1
10
0
− ν Γ(ν + γ + 1) Γ(ν − γ) P −ν
π 2
γ (β)
β −1 2
2
Γ ν + 12
[Re(ν − γ) > 0,
(cosh a + sinh a cosh x)
0
2.
1
2
(cosh a − cosh x)
1.
=
(β + cosh x)
0
cosh γ + 12 x dx
β does not lie on the ray (−∞, +1) of the real axis]
sinh2μ−1 x cosh x dx
1
ν = a−μ B(μ, ν − μ)
2
1 + a sinh2 x
ν−1
sinhμ−1 x (cosh x + 1)
(β + cosh x)
ν−1
sinhμ−1 x (cosh x − 1)
(β + cosh x)
a > 0]
EH I 11(22)
1 1
1
× 2 F 1 , + 2 − μ − ν; 2 − μ − ν; − β
2
2 2
[Re μ > 0, Re( − μ − ν) > −2, |arg(1 + β)| < π]
EH I 115(11)
dx
dx
= 2μ+ν−ρ B
[Re ν > Re μ > 0,
EH I 155(1)
1
μ, + 2 − μ − ν
2
= 2−(2−μ−ν+) 2 F 1 , 2 − μ − ν + ; 1 + −
μ 1−β
;
2
2
μ
2
Re(2 + ) Re(μ + ν), Re(2ν + μ) > 2]
× B 2 − μ − ν + , −1 + ν +
[β ∈ (−∞, −1) ,
EH I 115(10)
3.522
Hyperbolic and algebraic functions
∞
6.7
0
sinhμ−1 x coshν−1 x
dx =
cosh2 x − β
π/2
3.519
0
2F 1
, 1 + −
[β ∈ (1, ∞) ,
375
μ
ν
μ+ν
μ+ν
;1 + − ;β 2B
,1 + −
2
2
2
2
Re μ > 0, 2 Re(1 + ) > Re(μ + ν)] EH I 115(9)
∞
1
pkπ
sinh [(r − p)] tan x
dx = π
sin
sinh (r tan x)
kπ + r
r
p2 < r2
BI (274)(13)
k=1
3.52–3.53 Combinations of hyperbolic functions and algebraic functions
3.521
∞
1.
0
∞
2.
0
∞
3.
1
∞
1
∞
0
∞
0
∞
0
∞
5.
0
∞
7.
0
0
[a > 0,
b > 0]
[b > 0]
[a > 0,
1
1
dx
= β b+
(b2 + x2 ) cosh πx
b
2
[b > 0]
1
x dx
= ln 2 −
(1 + x2 ) sinh πx
2
BI(97)(16), GW(352)(8)
b > 0]
BI (97)(5)
∞
BI (97)(4)
BI (97)(7)
π
2
BI (97)(1)
x dx
π
= −1
(1 + x2 ) sinh πx
2
2
BI (97)(8)
dx
x2 ) cosh πx
dx
(1 +
0
9.
LI (104)(13)
∞
2π dx
(−1)k−1
=
(b2 + x2 ) cosh ax
b
2ab + (2k − 1)π
∞
8.
x dx
1
=
− β(b + 1)
2
+ x ) sinh πx
2b
(1 +
0
[a > 0]
∞
(−1)k
π
x dx
=
+
π
(b2 + x2 ) sinh ax
2ab
ab + kπ
∞
6.
LI (104)(14)
k=1
4.
[a > 0]
∞
dx
=2
(−1)k+1 Ei[−(2k + 1)a]
x cosh ax
(b2
3.
LI III 225(103a), BI(84)(1)a
∞
dx
= −2
Ei[−(2k + 1)a]
x sinh ax
∞
0
= 1.831931188 . . .
k=1
2.
π
x dx
= 2G = π ln 2 − 4 L
cosh x
4
GW (352)(2b)
k=0
1.
[a > 0]
k=0
4.
3.522
π2
x dx
= 2
sinh ax
4a
x2 ) cosh πx
2
=2−
= ln 2
,
√
x dx
1 +
√
=
2
+
1
−2
π
+
2
ln
πx
(1 + x2 ) sinh 4
2
BI (97)(2)
BI (97)(9)
376
Hyperbolic Functions
∞
10.
0
3.523
∞
1.
0
∞
2.
0
3.523
,
√
dx
1 +
√
π
−
2
ln
=
2
+
1
πx
(1 + x2 ) cosh 4
2
BI (97)(3)
2β − 1
xβ−1
dx = β−1 β Γ(β) ζ(β)
sinh ax
2
a
[Re β > 1,
22n − 1
x2n−1
dx =
sinh ax
2n
[a > 0,
π
a
2n
|B2n |
a > 0]
WH
n = 1, 2, . . .]
WH, GW(352)(2a)
∞
3.
0
2
xβ−1
1
dx =
Γ(β) Φ −1, β,
cosh ax
(2a)β
2
∞
2
2
Γ(β)
(−1)k
=
(2a)β
2k + 1
β
k=0
[Re β > 0,
∞
4.
0
∞
5.
0
∞
6.
0
∞
7.
0
∞
8.
0
∞
9.
0
∞
10.
0
∞
11.
0
∞
0
∞
1.
2.11
0
|E2n |
[a > 0]
BI(84)(12)a, GW(352)(1a)
π3
x2 dx
=
cosh x
8
(cf. 4.261 6)
BI (84)(3)
π4
x3 dx
=
sinh x
8
(cf. 4.262 1 and 2)
BI (84)(5)
5 5
x4 dx
=
π
cosh x
32
BI (84)(7)
x5
π6
dx =
sinh x
4
BI (84)(8)
61 7
x6
dx =
π
cosh x
128
BI (84)(9)
17 8
x7
dx =
π
sinh x
16
BI (84)(10)
∞
x1/2 dx √ 1
= π
(−1)k
cosh x
(2k + 1)3/2
BI (98)(7)a
∞
√ dx
(−1)k
π
=
2
x1/2 cosh x
(2k + 1)1/2
k=0
xμ−1
0
2n+1
EH I 35, ET I 322(1)
k=0
12.
3.524
π
x2n
dx =
cosh ax
2a
a > 0]
sinh βx
Γ(μ)
dx =
sinh γx
(2γ)μ
1
ζ μ,
2
BI (98)(25)a
1−
β
γ
1
β
− ζ μ,
1+
2
γ
[Re γ > |Re β|, Re μ > −1]
ET I 323(10)
∞
x2m
2m
aπ
π d
sinh ax
tan
dx =
2m
sinh bx
2b da
2b
[b > |a|]
BI (112)(20)a
3.524
Hyperbolic and algebraic functions
∞
3.
0
4.11
5.
∞
sinh ax dx
= Γ(1 − p)
p
sinh bx x
1
1
−
[b(2k + 1) − a]1−p
[b(2k + 1) + a]1−p
k=0
[b > |a|,
∞
377
p < 1]
BI (131)(2)a
2m+1
aπ
sinh ax
π d
sec
dx =
[b > |a|]
BI (112)(18)a
2m+1
cosh bx
2b da
2b
0
∞
Γ(μ)
cosh βx
1
β
1
β
dx =
xμ−1
ζ
μ,
1
−
+
ζ
μ,
1
+
sinh γx
(2γ)μ
2
γ
2
γ
0
[Re γ > |Re β|, Re μ > 1]
x2m+1
ET I 323(12)
∞
6.
x2m
0
∞
7.
0
∞
cosh ax dx
· p = Γ(1 − p)
(−1)k
cosh bx x
∞
∞
9.8
x2m+1
π3
sinh ax
aπ
aπ
dx = 3 sin
sec3
sinh bx
4b
2b
2b
∞
10.
x4
sinh ax
aπ
π
dx = 8
sec
sinh bx
2b
2b
0
x6
0
∞
12.
x
0
∞
13.
∞
14.
x3
x5
0
x7
0
16.
∞
x
0
sin
π
aπ
sinh ax
dx =
sec
cosh bx
2b
2b
π
aπ
sinh ax
dx =
sec
cosh bx
2b
2b
4
sin
6
sin
sinh ax
π
aπ
dx =
sec
cosh bx
2b
2b
π
aπ
cosh ax
dx =
sec
sinh bx
2b
2b
8
sin
2
BI(131)(1)a
[b > |a|]
BI (112)(19)a
[b > |a|]
BI (84)(18)
BI (82)(17)a
aπ
aπ
aπ
+ 2 cos4
45 − 30 cos2
2b
2b
2b
[b > |a|]
BI (82)(21)a
[b > |a|]
BI (84)(15)a
[b > |a|]
BI (82)(14)a
aπ
aπ
· 6 − cos2
2b
2b
aπ
aπ
aπ
120 − 60 cos2
+ cos4
2b
2b
2b
[b > |a|]
∞
15.
π
aπ
sinh ax
dx = 16
sec
sinh bx
2b
2b
7
aπ
aπ
· 2 + sin2
2b
2b
π2
aπ
sinh ax
aπ
dx = 2 sin
sec2
cosh bx
4b
2b
2b
0
· sin
p < 1]
[b > |a|]
∞
11.
5
BI(112)(17)
1
1
+
[b(2k + 1) − a]1−p
[b(2k + 1) + a]1−p
aπ
π d2m+1
cosh ax
tan
dx =
sinh bx
2b da2m+1
2b
x2
0
[b > |a|]
[b > |a|,
0
π d
cosh ax
aπ
dx =
sec
2m
cosh bx
2b da
2b
k=0
8.
2m
BI (82)(18)a
aπ
aπ
aπ
aπ
5040 − 4200 cos2
+ 546 cos4
− cos6
2b
2b
2b
2b
[b > |a|]
BI (82)(22)a
[b > |a|]
BI (84)(16)a
378
Hyperbolic Functions
∞
17.
x3
π
aπ
cosh ax
dx = 2
sec
sinh bx
2b
2b
4
x5
π
aπ
cosh ax
dx = 8
sec
sinh bx
2b
2b
6
0
∞
18.
0
x7
0
∞
20.
∞
21.
0
∞
23.
0
3.525
∞
1.
0
∞
2.
0
∞
3.
0
[b > |a|]
BI (82)(15)a
aπ
aπ
+ 2 cos4
2b
2b
315 − 420 cos2
x4
π
aπ
cosh ax
dx =
sec
cosh bx
2b
2b
5
24 − 20 cos2
BI (82)(19)a
aπ
aπ
aπ
+ 126 cos4
− 4 cos6
2b
2b
2b
[b > |a|]
BI(82)(23)a
[b > |a|]
BI (84)(17)a
aπ
aπ
+ cos4
2b
2b
[b > |a|]
∞
22.
π
aπ
cosh ax
dx = 16
sec
sinh bx
2b
2b
8
cosh ax
aπ
π3
aπ
dx = 3 2 sec3
− sec
cosh bx
8b
2b
2b
0
15 − 15 cos2
x2
0
aπ
2b
[b > |a|]
∞
19.
1 + 2 sin2
3.525
x6
aπ
π
cosh ax
dx =
sec
cosh bx
2b
2b
7
720 − 840 cos2
BI (82)(16)a
aπ
aπ
aπ
+ 182 cos4
− cos6
2b
2b
2b
sinh ax dx
aπ π
·
= ln tan
+
cosh bx x
4b
4
[b > |a|]
BI (82)(20)a
[b > |a|]
BI (95)(3)a
dx
1
sinh ax
a
·
= − cos a + sin a ln [2 (1 + cos a)]
sinh πx 1 + x2
2
2
dx
1
1 − sin a
sinh ax
π
·
= sin a + cos a ln
π
2
sinh 2 x 1 + x
2
2
1 + sin a
[π ≥ |a|]
BI (97)(10)a
[π ≥ 2|a|]
BI (97)(11)a
1
cosh ax x dx
1
·
= (a sin a − 1) + cos a ln [2 (1 + cos a)]
sinh πx 1 + x2
2
2
[π > |a|]
∞
4.
0
1
1 + sin a
cosh ax x dx
π
·
= cos a − 1 + sin a ln
sinh π2 x 1 + x2
2
2
1 − sin a
+π
> |a|
2
∞
sinh ax
x dx
a+π
a π
·
= −2 sin + sin a − cos a ln tan
2
2
2
4
0 cosh πx 1 + x
BI (97)(12)a
,
BI (97)(13)a
5.
6.
0
[π > |a|]
∞
GW (352)(12)
dx
a+π
cosh ax
a π
·
= 2 cos − cos a − sin a ln tan
cosh πx 1 + x2
2
2
4
[π > |a|]
GW (352)(11)
3.527
Hyperbolic and algebraic functions
∞
7.
0
∞
0
∞
0
∞
cos
π
x dx
cosh ax
π
b
· 2
+
π
=
sinh bx c + x2
2bc
bc + kπ
[b > |a|]
BI (97)(19)
k(b−a)
sinh ax cosh bx dx
1
(a + b + c)π
(b + c − a)π
·
= ln tan
cot
cosh cx
x
2
4c
4c
[c > |a| + |b|]
∞
2.
0
BI (97)(18)
k=1
1.
[b ≥ |a|]
k=1
8.
3.526
∞
k(b−a)
dx
sinh ax
π sin b π
· 2
=
sinh bx c + x2
c
bc + kπ
379
∞
3.
0
1
a
sinh2 ax dx
·
= ln sec π
sinh bx
x
2
b
Γ(μ)
xμ−1
dx =
sinh βx cosh γx
(2γ)μ
BI (93)(10)a
[b > |2a|]
1
Φ −1, μ,
2
1+
β
γ
BI (95)(5)a
1
β
+ Φ −1, μ,
1−
2
γ
[Re γ > |Re β|, Re μ > 0]
ET I 323(11)
3.527
∞
1.
0
2.
3.6
5.
6.
7.
8.
9.
[Re a > 0,
Re μ > 2]
BI (86)(7)a
m = 1, 2, . . .]
BI(86)(5)a
∞
x2m
π 2m
|B2m |
dx
=
2
a2m+1
0 sinh ax
∞ μ−1
x
4 1 − 22−μ Γ(μ) ζ(μ − 1)
dx =
2
μ
(2a)
0 cosh ax
1
= 2 ln 2
a
4.
xμ−1
4
Γ(μ) ζ(μ − 1)
dx =
(2a)μ
sinh2 ax
[a > 0,
[Re a > 0,
Re μ > 0,
[Re a > 0,
μ = 2]
μ = 2]
BI (86)(6)a
∞
x dx
ln 2
[a = 0]
= 2
2
a
cosh
ax
0
2m
∞
2 − 2 π 2m
x2m
dx =
|B2m |
[a > 0, m = 1, 2, . . .]
2
(2a)2m a
0 cosh ax
∞
∞
2 Γ(μ) sinh ax
(−1)k
dx
=
xμ−1
[Re μ > 1, a > 0]
aμ
(2k + 1)μ−1
cosh2 ax
0
k=0
∞
x sinh ax
π
[a > 0]
dx = 2
2a
cosh2 ax
0
∞
sinh ax
2m + 1 π 2m+1
x2m+1
|E2m |
[a > 0, m = 0, 1, . . .]
dx =
2
a
2a
cosh ax
0
∞
cosh ax
22m+1 − 1
x2m+1
(2m + 1)! ζ(2m + 1)
dx
=
a2 (2a)2m
sinh2 ax
0
[a = 0,
m = 1, 2, . . .]
LO III 396
BI(86)(2)a
BI (86)(15)a
BI (86)(8)a
BI (86)(12)a
BI (86)(13)a
380
Hyperbolic Functions
10.
11
11.8
12.
13.
14.11
15.10
16.∗
3.528
∞
cosh ax
22m − 1 π 2m
|B2m |
dx
=
a
a
sinh2 ax
0
√
∞
x sinh ax
π
Γ(μ)
dx
=
2μ+1
2 Γ μ+ 1
4μa
cosh
ax
0
2
∞ 2
x dx
π2
=
2
3
−∞ sinh x
∞
cosh ax
π2
x2
dx
=
2a3
sinh2 ax
0
∞
sinh x
dx = 4G
x2
cosh2 x
0
∞
tanh x2 dx
= ln 2
cosh x
0
∞
cosh ax
2 Γ (μ) ζ (μ − 1) 1 − 21−μ
xμ−1
=
2
μ
a
sinh ax
0
∞
1.
0
∞
2.
0
3.529
∞
1.
0
∞
2.
0
∞
3.
0
3.531
∞
1.7
0
∞
2.10
0
3.
4.
x2m
3.528
[a > 0,
m = 1, 2, . . .]
[μ > 0,
a > 0]
1
1
−
sinh x x
[a > 0]
BI (86)(11)a
[a = 0]
BI (86)(10)a
BI (93)(17)a
BI (87)(8)
[n = 0, 1, . . .]
dx
= − ln 2
x
aπ
cosh ax − 1 dx
·
= − ln cos
sinh bx
x
2b
a
b
−
sinh ax sinh bx
LI (86)(9)
BI (102)(2)a
(1 + xi)2n−1 − (1 − xi)2n−1
dx = 2
i sinh πx
2
(1 + xi)2n − (1 − xi)2n
dx = (−1)n+1 2|E2n | + 2
i sinh πx
2
BI (86)(14)a
BI (87)(7)
BI (94)(10)a
[b > |a|]
dx
= (b − a) ln 2
x
GW (352)(66)
BI (94)(11)a
4 +π
π ,
x dx
=√
ln 2 − L
= 1.1719536193 . . .
2 cosh x − 1
3
3 3
[see 8.26 for L(x)]
t ln 2 − L(t)
x dx
=
cosh 2x + cos 2t
sin 2t
LI (88)(1)
LO III 402
∞
t π 2 − t2
x2 dx
= ·
3
sin t
0 cosh x + cos t
2
∞
4
t π − t2 7π 2 − 3t2
x dx
=
15
sin t
0 cosh x + cos t
[0 < t < π]
BI (88)(3)a
[0 < t < π]
BI (88)(4)a
3.533
Hyperbolic and algebraic functions
∞
sin 2kaπ
x2m dx
= 2(2m)! cosec 2aπ
cosh x − cos 2aπ
k 2m+1
k=1
= 2 22m−1 − 1 π 2m |B2m |
∞
5.3
0
1
2
a=
a=
1
2
xμ−1 dx
cosh x − cos t
0
i Γ(μ) −it −it
e Φ e , μ, 1 − eit Φ eit , μ, 1
sin t
= 2 − 23−μ Γ(μ) ζ(μ − 1)
[μ = 2,
t = π]
= 2 ln 2
[μ = 2,
t = π]
[Re μ > 0,
=
∞
7.
0
0 < a < 1,
BI (88)(5)a
∞
6.3
381
0 < t < 2π,
t = π] ET I 323(5)
∞
xμ dx
sin kt
2 Γ(μ + 1) =
(−1)k−1 μ+1
cosh x + cos t
sin t
k
[μ > −1,
0 < t < π]
BII (96)(14)a
k=1
u
8.
0
1
x dx
= cosec 2t [L(θ + t) − L(θ − t) − 2 L(t)]
cosh 2x − cos 2t
2
[θ = arctan (tanh u cot t) ,
t = nπ]
LO III 402
3.532
∞
1.11
0
2.
0
3.533
k=0
∞
π
x cosh x dx
= cosec t
ln 2 − L
cosh 2x − cos 2t
2
x
0
0
n > −1]
b > 0,
t
2
−L
(π − t)
2
GW (352)(5)
t = nπ
∞
sinh ax dx
2
(cosh ax − cos t)
x3
=
π−t
cosec t
a2
sinh x dx
2
(cosh x + cos t)
=
t π2 − t
sin t
LO III 288a
[t = mπ]
∞
2.6
3.
k
θ+t
1
θ−t
x cosh x dx
ψ+t
= cosec t L
−L
+L π−
cosh 2x − cos 2t 2
2
2
2
ψ−t
t
π−t
+L
− 2L
− 2L
2
2
2
u
t
ψ
u
t
θ
tan = tanh cot , tan = coth cot ;
2
2
2
2
2
2
0
b−a
b+a
[a > 0,
u
1.
∞
2n! xn dx
1
=
a cosh x + b sinh x
a+b
(2k + 1)n+1
[a > 0,
LO III 403
0 < t < π]
(cf. 3.5141)
BI (88)(11)a
2
[0 < t < π]
(cf. 3.531 3)
BI (88)(13)
382
Hyperbolic Functions
∞
4.11
x2m+1
0
sinh x dx
2
(cosh x − cos 2aπ)
= 2(2m + 1)! cosec 2aπ
= 2(2m + 1) 2
3.534
∞
sin 2kaπ
k=1
2m−1
k 2m+1
− 1 π 2m |B2m |
0 < a < 1,
a=
1
2
a=
1
2
BI (88)(14)
3.534
1
π
I 1 (a)
1 − x2 cosh ax dx =
1.
2a
0
1
cosh ax
π
√
2.
dx = I 0 (a)
2
2
1−x
0
1
π
x
arcsin (tanh a)
dx
√
= √
[a > 0]
3.535
·
·
2
sinh
ax
sinh a
cosh
2a
−
cosh
2ax
2
2a
0
3.536
∞
π2
x2
11
dx =
1.
2
12
0 cosh x
√ ∞ 2
∞
x tanh x2 dx
π
(−1)k
√
2.
=
2
2
2k + 1
cosh x
0
k=0
∞
νπ
sin μπ
1−μ−ν
xμ−1
2 sin 2
Γ(μ) Γ
3.
sinh (ν arcsinh x) √
dx =
×Γ
μ
2
2 π
2
1+x
0
WA 94(9)
WA 94(9)
BI (80)(11)
BI (98)(7)
BI (98)(8)
1−μ+ν
2
[−1 < Re μ < 1 − |Re ν|]
∞
ET I 324(14)
νπ
cos μπ
1−μ−ν
1−μ+ν
xμ−1
2 cos 2
Γ(μ) Γ
cosh (ν arccosh x) √
dx =
×Γ
μπ
2
2
2
2
1+x
ET I 324(15)
[0 < Re μ < 1 − |Re ν|]
4.
0
3.54 Combinations of hyperbolic functions and exponentials
3.541
∞
1.
e−μx sinhν βx dx =
0
∞
2.
0
1
sinh βx
dx =
e−μx
ψ
sinh bx
2b
e−μx
−∞
∞
e−x
4.
0
5.
0
μ
ν
− ,ν + 1
2β
2
1 μ+β
+
2
2b
−ψ
[Re β > 0,
1 μ−β
+
2
2b
Re ν > −1, Re μ > Re βν]
EH I 11(25), ET I 163(5)
[Re (μ + b ± β) > 0]
∞
3.
1
B
2ν+1 β
∞
π
μπ
sinh μx
dx =
tan
sinh βx
2β
β
1 π
aπ
sinh ax
dx = − cot
sinh x
a
2
2
e−px dx
(cosh px)
2q+1
=
1
22q−2
B(q, q) −
p
2qp
[Re β > 2|Re μ|]
[0 < a < 2]
[p > 0,
q > 0]
EH I 16(14)a
BI (18)(6)
BI (4)(3)
LI (27)(19)
3.545
Hyperbolic functions and exponentials
∞
6.
e−μx
0
∞
7.
μ+1
2
dx
=β
cosh x
e−μx tanh x dx = β
0
8.
9.
10.
3.542
μ
1
−
2
μ
∞
1.
e−μx (cosh βx − 1) dx =
ν
0
[Re μ > −1]
ET I 163(7)
[Re μ > 0]
ET I 163(9)
[Re μ > 0]
ET I 163(8)
[Re μ > 0]
LI (27)(15)
[0 0,
∞
2.
1
B
2ν β
e
−μx
ν−1
(cosh x − cosh u)
dx = −i
0
1
Re ν > − ,
2
Re μ > Re βν
ET I 163(6)
1
1
2 iπν
2 −ν
e Γ(ν) sinhν− 2 u Q μ−
1 (cosh u)
2
π
[Re ν > 0, Re μ > Re ν − 1]
EH I 155(4), ET I 164(23)
3.543
1.
∞
iπeitb
e−ibx dx
=−
cosh πb − e−2itb
sinh πb cosh t
−∞ sinh x + sinh t
[t > 0]
∞
2.
0
∞
0
∞
sin kt
e−μx
dx = 2 cosec t
cosh x − cos t
μ+k
[Re μ > −1,
BI (6)(10)a
∞
cos kt
1 − e−x cos t −(μ−1)x
e
dx = 2
cosh x − cos t
μ+k
[Re μ > 0,
t = 2nπ]
BI (6)(9)a
k=0
∞
epx − cos t
2
dx =
1
p
t cosec t +
1
1 + cos t
(cosh px + cos t)
∞
exp − n + 12 x
dx = Q n (cosh u) ,
3.544
2 (cosh x − cosh u)
u
3.545
∞
π
aπ
1
sinh ax
dx =
cosec
−
1.
px + 1
e
2p
p
2a
0
∞
1
π
aπ
sinh ax
dx =
−
cot
2.
px
−1
2a 2p
p
0 e
0
t = 2nπ]
k=1
3.
4.
ET I 121(30)
[p > 0]
BI (27)(26)a
[u > 0]
EH II 181(33)
[p > a,
p > 0]
BI (27)(3)
[p > a,
p > 0]
BI (27)(9)
384
3.546
1.
2.
3.
4.
Hyperbolic Functions
3.546
√
a
1 π
a2
√ exp
√
Φ
2
4β
β
2
β
0
∞
2
2
a
1 π
exp
e−βx cosh ax dx =
2 β
4β
0
∞
2
1 π
a2
−1
e−βx sinh2 ax dx =
exp
4 β
β
0
∞
2
1 π
a2
+1
e−βx cosh2 ax dx =
exp
4 β
β
0
3.547 1.
∞
e−βx sinh ax dx =
2
∞
exp (−β sinh x) sinh γx dx =
0
[Re β > 0]
FI II 720a
[Re β > 0]
ET I 166(40)
[Re β > 0]
ET I 166(41)
[Re β > 0]
exp (−β cosh x) sinh γx sinh x dx =
0
∞
exp (−β sinh x) cosh γx dx =
3.
0
ET I166(38)a
γπ
π
π
cot
[J γ (β) − Jγ (β)] − [Eγ (β) + Y γ (β)] = γ S −1,γ (β)
2
2
2
∞
2.
[Re β > 0]
WA 341(5), ET I 168(14)a
γ
K γ (β)
β
πγ
π
π
tan
[Jγ (β) − J γ (β)] − [Eγ (β) + Y γ (β)] = S 0,γ (β)
2
2
2
[Re β > 0, γ not an integer]
ET I 168(16)a, WA 341(4), EH II 84(50)
∞
4.
exp (−β cosh x) cosh γx dx = K γ (β)
[Re β > 0]
ET I 168(16)a, WA 201(5)
[Re β > 0]
ET I 168(7), EH II 85(51)
0
∞
5.
exp (−β sinh x) sinh γx cosh x dx =
0
γ
S 0,γ (β)
β
∞
6.
exp (−β sinh x) sinh[(2n + 1)x] cosh x dx = O 2n+1 (β)
0
[Re β > 0]
∞
7.
exp (−β sinh x) cosh γx cosh x dx =
0
1
S 1,γ (β)
β
ET I 167(5)
[Re β > 0]
∞
exp (−β sinh x) cosh 2nx cosh x dx = O 2n (β)
8.
[Re β > 0]
ET I 168(6)
0
∞
9.
0
∞
10.11
1
exp (−β cosh x) sinh2ν x dx = √
π
exp [−2 (β coth x + μx)] sinh2ν x dx =
0
11.
0
∞
2
β
exp −
ν
Γ ν+
1
2
K ν (β)
Re β > 0,
Re ν > − 12
EH II 82(20)
1 ν
β Γ (μ − ν) W −μ,ν− 12 (4β)
2
[Re β > 0, Re μ > Re ν]
β2
sinh x sinhν−1 x coshν x dx = −π D ν βeiπ/4 D ν βe−iπ/4
2
+
π,
Re ν > 0, |arg β| ≤
4
EH II 120(10)
3.548
Hyperbolic functions and exponentials
12.
13.
14.
,
1 3 +
exp (2νx − 2β sinh x)
√
dx =
π β J ν+ 14 (β) J ν− 14 (β) + Y ν+ 14 (β) Y ν− 14 (β)
2
sinh x
0
[Re β > 0]
EH I 169(20)
∞
+
,
exp (−2νx − 2β sinh x)
1 3
√
dx =
π β J ν+ 14 (β) Y ν− 14 (β) − J ν− 14 (β) Y ν+ 14 (β)
2
sinh x
0
ET I 169(21)
[Re β > 0]
∞
exp (−2β sinh x) sinh 2νx
1 π 3 β νπi (1)
(2)
√
e H 1 +ν (β) H 1 −ν (β)
dx =
2
4i
2
sinh x
0
2
(1)
(2)
−νπi
−e
H 1 −ν (β) H 1 +ν (β)
∞
2
∞
15.
0
2
[Re β > 0]
17.8
18.
19.
20.
∞
ET I 170(24)
exp (−2β sinh x) cosh 2νx
1 π 3 β νπi (1)
(2)
√
dx =
e H 1 +ν (β) H 1 −ν (β)
2
2
4
2
sinh x
(1)
(2)
−νπi
+e
H 1 −ν (β) H 1 +ν (β)
2
16.
385
exp (−2β cosh x) cosh 2νx
√
dx =
cosh x
2
[Re β > 0]
ET I 170(25)
β
K 1 (β) K ν− 14 (β)
π ν+ 4
0
ET I 170(26)
[Re β > 0]
∞
exp [−2β (cosh x − 1)] cosh 2νx
β 2β
√
· e K ν+ 14 (β) K ν− 14 (β)
dx =
π
cosh x
0
[Re β > 0]
ET I 170(27)
∞
cos ν + 14 π exp (−2νx − 2β sinh x) + sin ν + 14 π exp (2νx − 2β sinh x)
√
dx
sinh x
0
+
,
1
=
π 3 β J 14 +ν (β) J 14 −ν (β) + Y 14 +ν (β) Y 14 −ν (β)
2
ET I 169(22)
[Re β > 0]
∞
1
1
sin ν + 4 π exp (−2νx − 2β sinh x) − cos ν + 4 π exp (2νx − 2β sinh x)
√
dx
sinh x
0
+
,
1
=
π 3 β J 14 +ν (β) Y 14 −ν (β) − J 14 −ν (β) Y 14 +ν (β)
2
ET I 169(23)
[Re β > 0]
∞
exp [−β(cosh x − 1)] cosh νx sinh x
dx = eβ K ν (β)
cosh
x
(cosh
x
−
1)
0
[Re β > 0]
3.548 1.
∞
e−μx sinh ax2 dx =
π
4
e−μx cosh ax2 dx =
π
4
4
0
2.
0
∞
4
a
exp
2μ
a2
8μ
I 14
a
exp
2μ
a2
8μ
I − 14
a2
8μ
ET I 169(19)
[Re μ > 0,
a ≥ 0]
ET I 166(42)
[Re μ > 0,
a > 0]
ET I 166(43)
a2
8μ
386
3.549
Hyperbolic Functions
∞
1.
3.549
e−βx sinh [(2n + 1) arcsinh x] dx = O 2n+1 (β)
[Re β > 0]
(cf. 3.547 6)
0
ET I 167(5)
∞
2.
e−βx cosh (2n arcsinh x) dx = O 2n (β)
[Re β > 0]
(cf. 3.547 8)
0
ET I 168(6)
∞
3.
e−βx sinh (ν arcsinh x) dx =
ν
S 0,ν (β)
β
[Re β > 0]
(cf. 3.5475)
e−βx cosh (ν arcsinh x) dx =
1
S 1,ν (β)
β
[Re β > 0]
(cf. 3.547 7)
0
∞
4.
0
ET I 168(7)
A number of other integrals containing hyperbolic functions and exponentials, depending on arcsinh x
or arccosh x, can be found by ﬁrst making the substitution x = sinh t or x = cosh t.
3.55–3.56 Combinations of hyperbolic functions, exponentials, and powers
3.551 1.
∞
xμ−1 e−βx sinh γx dx =
0
xμ−1 e−βx cosh γx dx =
0
∞
3.
1
Γ(μ) (β − γ)−μ + (β + γ)−μ
2
[Re μ > 0,
xμ−1 e−βx coth x dx = Γ(μ) 21−μ ζ μ,
0
β
2
− β −μ
∞
4.
xn e−(p+mq)x sinhm qx dx = 2−m n!
0
5.11
1 −βx
e
x
0
ET I 164(19)
x
e
x
∞
ET I 164(21)
m |Re γ|]
ET I 163(12)
cosh γx dx =
1
[− Ei(γ − β) − Ei(−γ − β)]
2
[Re β > |Re γ|]
ET I 164(15)
∞ −βx
1
0
1
β+γ
sinh γx dx =
+ Ei(γ − β) − Ei(−γ − β)
ln
2
β−γ
e
7.
q > 0,
[β > γ]
0
8.6
m
m
(−1)k
k (p + 2kq)n+1
k=0
[p > 0,
Re β > 0]
∞ −βx
6.
Re β > |Re γ|]
[Re μ > 1,
Re β > |Re γ|]
ET I 164(18)
∞
2.
1
Γ(μ) (β − γ)−μ − (β + γ)−μ
2
[Re β > −1,
xe−x coth x dx =
π2
−1
4
BI (82)(6)
3.554
Hyperbolic functions, exponentials, and powers
∞
9.
e
−βx
0
∞
10.6
β
dx
= ln + 2 ln
tanh x
x
4
xe−x coth(x/2) dx =
0
3.552
∞
1.
0
∞
2.
0
∞
3.
0
Γ
β
4
[Re β > 0]
β
1
+
4
2
ET I 164(16)
π2
−1
3
xμ−1 e−βx
1
1−μ
dx = 2
Γ(μ) ζ μ, (β + 1)
sinh x
2
[Re μ > 1,
1
x2m−1 e−ax
π
dx =
|B2m |
sinh ax
2m
a
[a > 0,
Re β > −1]
ET I 164(20)
2m
xμ−1 e−x
dx = 21−μ 1 − 21−μ Γ(μ) ζ(μ)
cosh x
= ln 2
m = 1, 2, . . .]
[Re μ > 0,
EH I 38(24)a
μ = 1]
[if μ = 1]
EH I 32(5)
∞
4.
0
Γ
387
∞
5.
0
x2m−1 e−ax
1 − 21−2m
π
dx =
|B2m |
cosh ax
2m
a
2m
[a > 0,
m = 1, 2, . . .]
EH I 39(25)a
∞
x2 e−2nx
1
dx = 4
sinh x
(2k + 1)3
[n = 0, 1, 2, . . .]
(cf. 4.261 13)
k=n
BI(84)(4)
∞
6.11
0
π4
x3 e−2nx
dx =
− 12
sinh x
8
n
k=1
1
(2k − 1)4
[n = 0, 1, . . .]
(cf. 4.262 6)
BI (84)(6)
3.553
∞
1.
0
∞
2.11
0
sinh2 ax e−x dx
1
= ln (aπ cosec aπ)
sinh x
x
2
[a < 1]
BI (95)(7)
sinh2 x2 e−x dx
1 4
·
= ln
cosh x
x
2 π
(cf. 4.267 2)
BI (95)(4)
3.554
1.
∞
11
e
−βx
0
2.
3.
∞
Γ β+3
4
dx
= 2 ln
(1 − sech x)
β+1
x
Γ
4
− ln
β
4
[Re β > 0]
β+1
1
β
− cosech x dx = ψ
[Re β > 0]
− ln
x
2
2
0
&
∞%
sinh 12 − β x
dx
= 2 ln Γ(β) − ln π + ln (sin πβ)
− (1 − 2β)e−x
x
sinh
x
0
2
e−βx
[0 < Re β < 1]
ET I 164(17)
ET I 163(10)
EH I 21(7)
388
Hyperbolic Functions
∞
4.
e−βx
0
∞
−
5.
0
∞
6.
1
− coth x
x
sinh qx
+ 2qe−x
sinh x2
dx = ψ
β
2
− ln
3.555
1
β
+
2
β
dx
1
= 2 ln Γ q +
x
2
ET I 163(11)
+ ln cos πq − ln π
xμ−1 e−βx (coth x − 1) dx = 21−μ Γ(μ) ζ μ,
0
[Re β > 0]
q2 <
1
2
WH
β
+1
2
[Re β > 0;
3.555
∞
1.
0
1
sinh2 ax dx
= ln
·
1 − epx
x
4
p
2aπ
sin
2aπ
p
Re μ > 1]
[0 < 2|a| < p]
ET I 164(22)
(cf. 3.545 2)
BI (93)(15)
∞
2.
0
3.556
∞
1.
x
−∞
∞
2.
0
3.557
∞
1.
0
π2
pπ
1 − epx
dx = − tan2
sinh x
2
2
1 − e−px 1 − e−(p+1)x
·
dx = 2p ln 2
sinh x
x
1
2
n−1
m
π
= 2 cosec
n
(cf. 4.255 3)
[p > −1]
2
k=1
(−1)k−1 sin
Γ
km
π ln
n
Γ
BI (93)(9)
BI (101)(4)
BI (95)(8)
Γ
km
π ln
n
Γ
n+q+k
2n
Γ
p+k
2n
n+p+k
2n
Γ
q+k
2n
n+q−k
n
Γ
p+k
n
n+p−k
n
Γ
q+k
n
[p > −1,
∞
(cf. 3.545 1)
dx
e−px − e−qx
·
cosh x − cos m
π
x
n
k=1
2.
a<
[p < 1]
n−1
m
= 2 cosec
π
(−1)k−1 sin
n
1
sinh2 ax dx
·
= − ln (aπ cot aπ)
x
e +1
x
4
q > −1]
[m + n odd]
[m + n even]
BI (96)(1)
2
dx
(1 − e−x )
·
m
cosh
x
+
cos
π
x
0
n
n−1
m
π
(−1)k−1 sin
= 2 cosec
n
n+k+1 2 k+2 k Γ
Γ 2n Γ 2n
km
π × ln 2n2 k+1
n+k
n
Γ 2n
Γ 2n Γ n+k+2
k=1
2n
n−1
n−k+1 2 k+2 k 2
Γ
Γ n Γ n
m
km
k−1
n
π
π × ln = 2 cosec
(−1)
sin
n−k n−k+2 k+1 2
n
n
Γ
Γ
Γ
k=1
n
n
[m + n odd]
[m + n even]
n
BI (96)(2)
3.558
Hyperbolic functions, exponentials, and powers
∞
3.
0
e−px sin m
m
dx
−x
nπ
π−
e tan
·
m
2n
cosh x + cos n π
x
m
π ln(2n) + 2
(−1)k−1 sin
2n
Γ
km
π ln
n
Γ
n−1
= tan
k=1
n−1
2
m
π ln n + 2
(−1)k−1 sin
= tan
2n
k=1
∞
4.
0
∞
5.
0
dx
1 + e−x
a
· 1−p = 2 sec Γ(p)
cosh x + cos a x
2
−x
2
0
∞
7.
∞
∞
5.
∞
6.
0
LI (96)(5)a
BI (88)(6)
x
n−1
n−k
1 − (−1)n e−nx
nπ 2
+
4
dx
=
(−1)k 2
2 x
3
k
cosh 2
k=1
LI (85)(1)
x2
∞
BI (85)(5)
LI (85)(6)
k=1
x2
n−1
n−k
1 + (−1)n e−nx
dx
=
6n
ζ(3)
−
8
k3
cosh2 x2
k=1
LI (85)(4)
x3
n−1
n−k
1 − e−nx
4
4
nπ
dx
=
−
24
2 x
15
k4
sinh 2
k=1
BI (85)(9)
x3
n−1
1 + (−1)n e−nx
n−k
7
4
nπ
dx
=
+
24
(−1)k 4
2 x
30
k
cosh 2
k=1
BI (85)(8)
0
LI (96)(5)
BI (85)(3)
0
7.
BI (96)(3)
BI (88)(8)
cos kaπ
e−x − cos aπ
dx = 2 · (2m + 1)!
cosh x − cos aπ
k 2m+2
k=1
[m + n even]
2
n−1
n−k
1 − e−nx
dx
=
8n
ζ(3)
−
8
x
k3
0
sinh2
k=1
2
∞
∞
n−1
−2nx
n−k
1
2 x1 − e
x e
−
8
dx = 8n
2
(2k − 1)3
(2k − 1)3
sinh x
0
p+k
n
n−1
n−k
1 − e−nx
2nπ 2
−
4
dx
=
2 x
3
k2
sinh 2
k=1
∞
0
4.
p+n−k
n
[m + n odd]
x
∞
2.
2
p+k
2n
∞
x2m+1
0
3.
[p > 0]
1
cos k − 2 λ
(−1)k−1
k q+1
p+n+k
2n
k=1
1.
k=1
1
k−1 cos k − 2 a
(−1)
kp
a
π
e − cos a
dx = |a|π −
−
cosh x − cos a
2
3
0
3.558
∞
Γ
km
π ln
n
Γ
[q > −1]
−x
x
∞
x
2
xq e cosh
Γ(q + 1)
dx =
cosh x + cos λ
cos λ2 k=1
∞
6.
389
390
Trigonometric Functions
%
∞
3.559
e
−x
0
∞
1.
tanh x2
π
dx = 2 ln √
x cosh x
2 2
∞
2.
1
Γ(2μ)(2β)−μ exp
2
γ2
8β
x2μ−1 e−βx cosh γx dx =
1
Γ(2μ)(2β)−μ exp
2
γ2
8β
2
2
0
∞
3.
xe−βx sinh γx dx =
2
0
∞
4.
−βx2
xe
0
γ
4β
γ
cosh γx dx =
4β
[Re μ > 0,
π
exp
β
γ
4β
π
exp
β
γ2
4β
[Re β > 0]
Φ
√ ∞
π 2β + γ 2
2 −βx2
√
x e
sinh γx dx =
exp
8β 2 β
0
∞
6.
0
γ
γ
− D −2μ √
D −2μ − √
2β
2β
1
Re μ > − 2 , Re β > 0
ET I 166(44)
γ
γ
+ D −2μ √
D −2μ − √
2β
2β
Re β > 0]
ET I 166(45)
2
5.
[a > 0]
BI (93)(18)
x2μ−1 e−βx sinh γx dx =
0
1
1
dx
= a − + ln Γ(a) − ln(2π)
x
2
2
BI (96)(6)
e
0
&
∞ −2x
3.561
3.562
1 (1 − e−x ) (1 − ax) − xe−x (2−a)x
e
a− +
2
4 sinh2 x2
3.559
√ π 2β + γ 2
2 −βx2
√
x e
cosh γx dx =
exp
8β 2 β
γ
√
2 β
+
BI(81)(12)a,ET I 165(34)
1
2β
[Re β > 0]
γ2
4β
Φ
γ
√
2 β
+
ET I 166(35)
γ
4β 2
[Re β > 0]
ET I 166(36)
[Re β > 0]
ET I 166(37)
2
γ
4β
3.6–4.1 Trigonometric Functions
3.61 Rational functions of sines and cosines and trigonometric functions of multiple
angles
3.611
2π
n
(1 − cos x) sin nx dx = 0
1.
BI (68)(10)
0
2π
n
(1 − cos x) cos nx dx = (−1)n
2.
0
3.
π
n
(cos t + i sin t cos x) dx =
0
π
2n−1
π
(cos t + i sin t cos x)
0
BI (68)(11)
−n−1
dx = π P n (cos t)
EH I 158(23)a
3.613
3.612
Rational functions of sines and cosines
π
1.6
0
391
sin nx cos mx
dx = 0
sin x
=π
for n > m,
if m + n is odd and positive
=0
for n > m,
if m + n is even
for n ≤ m;
LI (64)(3)
π
2.
0
sin nx
dx = 0
sin x
=π
for n even
for n odd
BI (64)(1, 2)
π/2
3.
0
FI II 145
π/2
1 1
(−1)k−1
sin 2nx
dx = 2 1 − + − · · · +
sin x
3 5
2n − 1
0
π/2
π
(−1)n−1
sin 2nx
sin 2nx
1 1
dx = 2
dx = (−1)n−1 4 1 − + − · · · +
cos x
cos x
3 5
2n − 1
0
0
π2
π
cos(2n + 1)x
cos(2n + 1)x
dx = 2
dx = (−1)n π
cos
x
cos
x
0
0
π/2
π
sin 2nx cos x
dx =
sin x
2
0
4.
5.
6.
7.
3.613
π
1.6
0
2.
π
sin(2n − 1)x
dx =
sin x
2
π
6
0
3.
0
π
cos nx dx
=√
1 + a cos x
1 − a2
√
1 − a2 − 1
a
cos nx dx
πan
=
1 − 2a cos x + a2
1 − a2
π
= 2
(a − 1) an
n
a2 < 1,
n≥0
a2 < 1,
n≥0
a2 > 1,
n≥0
GW (332)(21b)
GW (332)(22a)
GW (332)(22b)
LI (45)(17)
BI (64)(12)
BI (65)(3)
π
sin nx sin x dx
π
= an−1
2
1 − 2a cos x + a
2
π
= n+1
2a
a2 < 1,
n≥1
a2 > 1,
n≥1
BI(65)(4), GW(332)(34a)
392
Trigonometric Functions
4.
π
10
0
π
5.
0
6.
7.
8.
cos nx cos x dx
π 1 + a2 n−1
·
=
a
1 − 2a cos x + a2
2 1 − a2
2
π
a +1
= n+1 · 2
2a
a −1
πa
=
1 − a2
π
=
a (a2 − 1)
cos(2n − 1)x dx
=
1 − 2a cos 2x + a2
3.614
a2 < 1,
n≥1
a2 > 1,
n≥1
n = 0,
a2 < 1
n = 0,
a2 > 1
π
0
cos 2nx cos x dx
=0
1 − 2a cos 2x + a2
a2 = 1
π
0
0
π
11.6
0
π
12.
0
13.
0
=1
=1
<1
>1
BI (65)(9, 10)
BI (65)(12)
BI (65)(6, 7)
cos(2n − 1)x cos x dx π a
= ·
1 − 2a cos 2x + a2
2 1−a
1
π
= ·
2 (a − 1)an
n−1
a2 < 1
a2 > 1
BI (65)(11)
π
10.
BI (65)(8)
π
9.
BI(65)(5), GW(332)(34b)
2
cos(2n − 1)x cos 2x dx
=0
a
2
1 − 2a cos 2x + a
0
π
π
sin 2nx sin x dx
sin(2n − 1)x sin 2x dx
=
=0
2
1
−
2a
cos
2x
+
a
1 − 2a cos 2x + a2
0
0
2
a
π
2
sin(2n − 1)x sin x dx π an−1
a
= ·
2
1
−
2a
cos
2x
+
a
2
1
+
a
0
2
1
π
a
= ·
2 (1 + a)an
π
sin nx − a sin(n − 1)x
sin mx dx = 0
1 − 2a cos x + a2
π
= am−n
2
cos nx − a cos(n − 1)x
π |m|−n
a
cos mx dx =
−1
2
1 − 2a cos x + a
2
for m < n
for m ≥ n
2
a <1
sin nx − a sin[(n + 1)x]
dx = 0
1 − 2a cos x + a2
cos nx − a cos[(n + 1)x]
dx = πan
1 − 2a cos x + a2
a2 < 1
a2 < 1
a2 < 1
LI (65)(13)
BI (65)(14)
BI (68)(13)
BI (68)(14)
3.616
Rational functions of sines and cosines
π
7
3.614
0
sin x
sin px · dx
·
a2 − 2ab cos x + b2 1 − 2ap cos px + a2p
πbp−1
= p+1
2a
(1 − bp )
πap−1
=
2b (bp − a2p )
393
[0 < b ≤ a ≤ 1,
0 < a ≤ 1,
p = 1, 2, 3, . . .]
p = 1, 2, 3, . . .
a2 < b,
BI (66)(9)
3.615
1.
2.
3.
3.616
2n
√
2
1 − a2
a <1
BI (47)(27)
a
0
π
1
cos x sin 2nx dx
s
s
π
2n
sin
2n
arctan
arccos
=
−
tan
2
b
2
2
2a2
0 1 + (a + b sin x)
π
1
cos x cos(2n + 1)x dx
s
s
π
2n+1
cos
arccos
(2n
+
1)
arctan
=
tan
2
b
2
2
2a2
1 + (a + b sin x)
0
<
2
2
2
where s = − 1 + b − a + (1 + b2 − a2 ) + 4a2 BI (65)(21, 22)
π/2
π
1.
cos 2nx dx
(−1)n π
√
=
1 − a2 sin2 x
2 1 − a2
1 − 2a cos x + a2
n
dx = π
0
π
2.10
0
dx
1
n =
2
(1 − 2a cos x + a2 )
0
1−
n
n
k
a2k
BI (63)(1)
k=0
2π
dx
n
(1 − 2a cos x + a2 )
n−1
(n + k − 1)!
π
=
n
2
2
(1 − a )
(k!) (n − k − 1)!
=
2
(a2
π
n
− 1)
k=0
n−1
(n + k − 1)!
a2
1 − a2
1
2
k=0
k
k
(k!) (n − k − 1)! (a2 − 1)
a2 < 1
a2 > 1
BI (331)(63)
π
3.
1 − 2a cos x + a2
n
cos nx dx = (−1)n πan
BI (63)(2)
0
4.
π
0
1 − 2a cos x + a2
n
cos mx dx
n
1 2π =
1 − 2a cos x + a2 cos mx dx
2 0
=0
n−m
= π(−a)m 1 + a2
[n < m]
[(n−m)/2]
k=0
n
k
n−k
m+k
a
1 + a2
2k
[n ≥ m]
GW (332)(35a)
5.
0
2π
sin nx dx
m =0
(1 − 2a cos 2x + a2 )
GW (332)(32a)
394
Trigonometric Functions
π
6.
0
π
7.
0
0
3.61710
0
[a = 0,
cos nx dx
cos nx dx
1 2π
=
m
2 0 (1 − 2a cos x + a2 )m
(1 − 2a cos x + a2 )
m−1
k
2m − k − 2
1 − a2
a2m+n−2 π m + n − 1
=
2m−1
k
m−1
a2
(1 − a2 )
k=0
m−1
m+n−1
k
2m − k − 2 2
π
a −1
=
2m−1
n
2
k
m
−
1
a (a − 1)
k=0
cos 2nx dx
n+1 =
2
2
a cos x + b2 sin2 x
π
dx
n+1/2
(1 − 2a cos x + a2 )
2n
n
2 n
π/2
dx
n+1/2
1 − k 2 sin2 x
where the Fn (k) satisﬁes the recurrence relation
k dFn (k)
,
n = 0, 1, 2, . . .
Fn+1 (k) = Fn (k) +
2n + 1 dk
and
π/2
dx
F0 (k) = K(k) ≡
1/2
0
1 − k 2 sin2 x
is the complete elliptic integral of the ﬁrst kind.
Introducing the complete elliptic integral of the second kind
π/2
1/2
1 − k 2 sin2 x
dx
E(k) =
Fn (k) =
0
the derivatives
0
d K(k)
E(k)
K(k)
=
−
,
dk
k (1 − k 2 )
k
combined with the recurrence relation lead to
d E(k)
E(k) − K(k)
=
dk
k
d F 0 (k)
dk
E(k)
E(k)
= K(k) +
− K(k) =
,
1 − k2
1 − k2
E(k)
E(k)
k d
+
F2 (k) =
2
1−k
3 dk 1 − k 2
4 − 2k 2
1
=
E(k)
−
K(k)
3 (1 − k 2 )
1 − k2
F1 (k) = F 0 (k) + k
a2 < 1
a2 > 1
GW (332)(31)
b2 − a
π
(2ab)2n+1
[a > 0, b > 0]
2 |a|
2
=
,
|a| = 1
2n+1 Fn
|1
+ a|
|1 + a|
with
±1]
GW (332)(32c)
π/2
8.
1
sin x dx
1
1
−
m =
2(m − 1)a (1 − a)2m−2
(1 + a)2m−2
(1 − 2a cos 2x + a2 )
3.617
GW (332)(30b)
3.623
Powers of trigonometric functions
395
3.62 Powers of trigonometric functions
3.621
π/2
π/2
sinμ−1 x dx =
1.
0
cosμ−1 x dx = 2μ−2 B
π/2
0
1
Γ
cos3/2 x dx = √
6 2π
π/2
sin3/2 x dx =
2.
0
π/2
0
π/2
sin2m x dx =
3.
cos2m x dx =
0
0
π/2
0
cos2m+1 x dx =
0
π/2
0
6.∗
7.∗
3.622
π/2 √
π/2
1.
sin x dx =
tan±μ x dx =
0
π/4
tanμ x dx =
2.
0
0
ln 2 (−1)k
+
2
2n − 2k
BI (34)(3)
n−1
π/2
0
μ
μ
1
B
,ν −
2
2
2
[0 < Re μ < 2 Re ν]
0
π/4
2.6
BI(42)(6), BI(45)(22)
tanμ x sin2 x dx =
1+μ
β
4
μ+1
2
−
1
4
[Re μ > −1]
BI (34)(4)
tanμ x cos2 x dx =
1−μ
β
4
μ+1
2
+
1
4
[Re μ > −1]
BI (34)(5)
0
0
π/2
cotμ−1 x sin2ν−2 x dx =
tanμ−1 x cos2ν−2 x dx =
BI (34)(1)
k=0
1.
3.6
[Re μ > −1]
BI (34)(2)
π/4
0
BI (42)(1)
π (−1)k
+
4
2n − 2k − 1
tan2n+1 x dx = (−1)n
[|Re μ| < 1]
k=0
4.11
3.623
3
4
n−1
tan2n x dx = (−1)n
Re ν > 0]
2
μ+1
2
π/4
3.
FI II 151
[Re μ > 0,
μπ
π
sec
2
2
1
β
2
FI II 151
LO V 113(50), LO V 122, FI II 788
2
Γ
π
0
1 2
π/2
Γ 4
dx
√
√
=
2 2π
sin
x
0
2
(2m)!!
(2m + 1)!!
μ ν
1
B
,
2
2 2
sinμ−1 x cosν−1 x dx =
5.
1
4
FI II 789
(2m − 1)!! π
(2m)!! 2
π/2
sin2m+1 x dx =
4.
μ μ
,
2 2
π/4
396
3.624
1.
2.3
3.11
4.8
5.
6.6
3.625
Trigonometric Functions
π/4
1
sinp x
dx =
[p > −1]
p+2
cos
x
p+1
0
1
π/2
π/2
1
μ
1 Γ 2 + 14 Γ(1 − μ)
sinμ− 2 x
cosμ− 2 x
dx =
dx =
cos2μ−1 x
2
Γ 54 − μ2
sin2μ−1 x
0
0
1
− 2 < Re μ < 1
π/4
1
(2n)!!
cosn− 2 (2x)
dx = π
2
2n+1
2n+1
cos
(x)
2
(n!)
0
π/4
cosμ 2x
[Re μ > −1]
dx = 22μ B(μ + 1, μ + 1)
2(μ+1) x
cos
0
π/4 2μ−2
Γ μ − 12 Γ(1 − μ)
sin
x
1−2μ
√
dx = 2
B(2μ − 1, 1 − μ) =
cosμ 2x
2 π
0
1
2 < Re μ < 1
π/2
2
sin ax
aπ 1
− sin πa [2a β(a) − 1] ,
[a > 0]
dx =
sin
x
2
2
0
π/4
1.
0
π/4
2.
0
π/4
3.
0
π/4
4.8
0
3.626
3.624
π/4
1.
0
π/4
2.
0
0
3.62811
0
sin2n x cosp 2x
dx =
cos2p+2n+2 x
π
2
1
2
B n + 12 , p + 1
[p > −1]
(cf. 3.251 1)
LI (55)(12)
BI (38)(3)
BI (35)(1)
BI (35)(4)
BI (35)(2)
BI (35)(3)
1
(2n − 2)!!(2m − 1)!!
sin2n−1 x cosm− 2 2x
dx =
cos2n+2m x
(2n + 2m − 1)!!
BI (38)(6)
1
(2n − 1)!!(2m − 1)!! π
sin2n x cosm− 2 2x
dx =
·
cos2n+2m+1 x
(2n + 2m)!!
2
BI (38)(7)
sin2n−1 x √
(2n − 2)!!
cos 2x dx =
2n+2
cos
x
(2n + 1)!!
(cf. 3.251 1)
BI (38)(4)
sin2n x √
(2n − 1)!! π
·
cos 2x dx =
2n+3
cos
x
(2n + 2)!! 2
(cf. 3.251 1)
BI (38)(5)
Γ(μ) Γ 12 − μ
cotμ x
μπ
√
dx =
sin
μ
μ
sin x
2 π
0
2
−1 < Re μ < 12
1
sec2p x sin2p−1 x dx = √ Γ(p) Γ 12 − p
0 −1]
(cf. 3.251 1)
GW (331)(34b)
tanμ x
dx =
cosμ x
π/2
BI (55)(12)a
WA 691
3.631
Powers of trigonometric functions
397
3.63 Powers of trigonometric functions and trigonometric functions of linear
functions
3.631 1.
sinν−1 x sin ax dx =
0
2ν−1 ν B
π/2
2 sinν−2 x sin νx dx =
2.7
0
π
3.6
νπ
1
cos
1−ν
2
sinν x sin νx dx = 2−ν π sin
0
π sin aπ
2
ν+a+1 ν−a+1
,
2
2
π
νπ
2
[Re ν > 0]
LO V 121(67a), WA 337a
[Re ν > 1]
GW(332)(16d), FI I 152
[Re ν > −1]
LO V 121(69)
π
sinn x sin 2mx dx = 0
4.
GW (332)(11a)
0
π
π/2
sin2n x sin(2m + 1)x dx =
5.
sin2n x sin(2m + 1)x dx
0
0
(−1)m 2n+1 n!(2n − 1)!!
(2n − 2m − 1)!!(2m + 2n + 1)!!
(−1)n 2n+1 n!(2m − 2n − 1)!!(2n − 1)!!
=
(2m + 2n + 1)!!
=
[m ≤ n] ∗
[m ≥ n] ∗
GW (332)(11b)
π
π/2
sin2n+1 x sin(2m + 1)x dx = 2
6.
0
sin2n+1 x sin(2m + 1)x dx
0
=
(−1)m π
22n+1
2n + 1
n−m
[n ≥ m]
=0
[n < m]
BI(40)(12), GW(332)(11c)
π
sinn x cos(2m + 1)x dx = 0
7.
GW (332)(12a)
0
π
sinν−1 x cos ax dx =
8.
0
2ν−1 ν B
π/2
cosν−1 x cos ax dx =
9.
0
π/2
sinν−2 x cos νx dx =
10.
0
∗ In
2ν ν B
π cos aπ
2
ν+a+1 ν−a+1
,
2
2
[Re ν > 0]
LO V 121(68)a, WA 337a
[Re ν > 0]
GW (332)(9c)
[Re ν > 1]
GW(332)(16b), FI II 15 2
π
ν+a+1 ν−a+1
,
2
2
νπ
1
sin
ν−1
2
3.631.5, for m = n we should set (2n − 2m − 1)!! = 1
398
Trigonometric Functions
11.
12.
3.632
π
π
νπ
[Re ν > −1]
cos
ν
2
2
0
π
π/2
2n
(−1)m
sin2n x cos 2mx dx = 2
sin2n x cos 2mx dx =
π
2n
n
−m
2
0
0
sinν x cos νx dx =
LO V 121(70)a
[n ≥ m]
=0
[n < m]
BI(40)(16), GW(332)(12b)
π
sin2n+1 x cos 2mx dx
π/2
=2
sin2n+1 x cos 2mx dx =
13.7
0
(−1)m 2n+1 n!(2n + 1)!!
(2m − 2n − 3)!!(2m + 2n + 1)!!
0
(−1)n+1 2n+1 n!(2m − 2n + 3)!!(2n + 1)!!
=
(2m + 2n + 1)!!
[n ≥ m − 1]
[n < m − 1]
GW (332)(12c)
π/2
cosν−2 x sin νx dx =
14.
0
1
ν−1
π
[Re ν > 1]
cos x sin nx dx = 1 − (−1)
m
15.
π/2
0
π
cosn x sin mx dx =
17.11
0
cosm x sin nx dx
FI II 153
π
2n+1
n
k
if m ≤ n and n − m = 2k
otherwise
GW (332)(15a)
π
cosm x cos ax dx =
18.6
0
π/2
0r−1
2n+1
k
k=1
1 + (−1)m+n
0
n
2k
1
cosn x sin nx dx =
16.
(m
+
n
−
2k
−
2)!!
m!(n
−
m
−
2)!!
m!
+s
= 1 − (−1)m+n
(m − k)!
(m + n)!!
(m + n)!!
k=0
⎧
⎡
⎤
⎪
⎨2 if n − m = 4l + 2 > 0
m if m ≤ n
⎢
⎥
s = 1 if n − m = 2l + 1 > 0
⎣r =
⎦ GW (332)(13a)
⎪
n if m ≥ n
⎩
0 if n − m = 4l or n − m < 0
0
m+n
GW(332)(16c), FI II 152
m
a+m
(−1) sin aπ
a+m
;1 −
; −1
2 F 1 −m, −
2m (m + a)
2
2
[a = 0, ±1, ±2, . . .]
WA 313
π/2
cosν−2 x cos νx dx = 0
19.
[Re ν > 1]
GW(332)(16a), FI II 152
[Re n > −1]
LO V 122(78), FI II 153
0
π/2
cosn x cos nx dx =
20.10
0
3.632
1.
0
π
π
2n+1
p+a ,
+ π
Γ 2
Γ p−a
2
− x dx = 2p−1
Γ(p)
sinp−1 x cos a
2
Γ(p − a) Γ(p + a)
2
p < a2
BI (62)(11)
3.634
Powers of trigonometric functions
π
2
2.
−π
2
+
π ,
dx =
cosν−1 x sin a x +
2
399
π sin aπ
2
ν+a+1 ν−a+1
,
2
2
2ν−1 ν B
[Re ν > 0]
π/2
cosp x sin[(p + 2n)x] dx = (−1)n−1
3.10
0
n−1
k=0
WA 337a
(−1)k 2k n − 1
p+k+1
k
[n > 0]
π
4.
−π
cosn−1 x cos[m(x − a)] dx = 1 − (−1)n+m =
=
π/2
cosp+q−2 x cos[(p − q)x] dx =
5.
0
π
2
−π
2
LI (41)(12)
cosn−1 x cos[m(x − a)] dx
[1 − (−1)n+m ] π cos ma
n+m+1 n−m+1
,
2n−1 n B
2
2
[n ≥ m]
LO V 123(80), LO V 139(94a)
π
2p+q−1 (p + q − 1) B(p, q)
[p + q > 1]
3.633
π/2
cosp−1 x sin ax sin x dx =
1.
0
2p+1 p(p + 1) B
π/2
cosn x sin nx sin 2mx dx =
2.
0
π/2
cosn x cos nx cos 2mx dx =
π/2
cosn−1 x cos[(n + 1)x] cos 2mx dx =
0
π/2
cosp+q x cos px cos qx dx =
4.
0
aπ
p+a
p−a
+ 1,
+1
2
2
0
3.
π/2
cosp+q x sin px sin qx dx =
5.6
0
π
2p+q+2
π
2p+q+2
π
2n+1
1+
∞
k=1
WH
LO V 150(110)
π
2n+2
BI (42)(19, 20)
n−1
m−1
[n > m − 1]
BI (42)(21)
1
(p + q + 1) B(p + 1, q + 1)
p
k
n
m
q
k
[p + q > −1]
GW (332)(10c)
Γ(p + q + 1)
π
= p+q+2
−1
2
Γ(p + 1) Γ(q + 1)
[p + q > −1]
3.634
π/2
sinμ−1 x cosν−1 x sin(μ + ν)x dx = sin
1.
0
BI (42)(16)
μπ
B(μ, ν)
2
[Re μ > 0,
Re ν > 0]
BI(42)(23), FI II 814a
400
Trigonometric Functions
π/2
sinμ−1 x cosν−1 x cos(μ + ν)x dx = cos
2.
0
3.635
μπ
B(μ, ν)
2
[Re μ > 0,
Re ν > 0]
BI(42)(24), FI II 814a
π/2
cosp+n−1 x sin px cos[(n + 1)x] sin x dx =
3.
0
π
2p+n+1
Γ(p + n)
n! Γ(p)
[p > −n]
3.635
π/4
1.
cos
0
2.
μ−1
1
2x tan x dx =
ψ
4
π/2
cosp+2n x sin px tan x dx =
7
0
μ+1
2
0
π/2
1.
tan±μ x sin 2x dx =
0
π/2
2.
0
3.
π/2
11
0
μπ
μπ
cosec
2
2
tan±μ x cos 2x dx = ∓
tan2μ x
dx =
cos x
π/2
0
[Re μ > 0]
BI (34)(7)
∞
n Γ(p + n − k)
k
2p+2n+1 Γ(p)
(n − k)!
k=0
Γ(p + 2n)
pπ
= p+2+n+1
2
Γ(n + 1) Γ(p + n + 1)
[p > −2n]
BI (42)(22)
cosn−1 x sin[(n + 1)x] cot x dx =
3.636
π
π/2
3.
μ
−ψ
2
BI (42)(15)
μπ
μπ
sec
2
2
π
2
BI (45)(18)
[0 < Re μ < 2]
BI (45)(20)a
[|Re μ| < 1]
Γ μ + 12 Γ(−μ)
cot2μ x
√
dx =
sin x
2 π
− 12 < Re μ < 1
BI (45)(21)
(cf. 3.251 1)
BI (45)(13, 14)
3.637
π/2
tanp x sinq−2 x sin qx dx = − cos
1.
0
(p + q)π
B(p + q − 1, 1 − p)
2
[p + q > 1 > p]
π/2
tanp x sinq−2 x cos qx dx = sin
2.
0
(p + q)π
B(p + q − 1, 1 − p)
2
[p + q > 1 > p]
π/2
cotp x cosq−2 x sin qx dx = cos
3.
0
GW (332)(15d)
GW (332)(15b)
pπ
B(p + q − 1, 1 − p)
2
[p + q > 1 > p]
GW (332)(15c)
3.642
Trigonometric functions: powers and rational functions
π/2
cotp x cosq−2 x cos qx dx = sin
4.
0
401
pπ
B(p + q − 1, 1 − p)
2
[p + q > 1 > p]
3.638
π/4
1.
1
cosμ+ 2
0
π/4
2.
0
sin2μ x dx
π/2
3.
0
2x cos x
=
π
sec μπ
2
μ− 12
|Re μ| <
1
2
GW (332)(15a)
(cf. 3.192 2)
BI (38)(8)
1
Γ μ + 2 Γ(1 − μ)
sin
2x dx
2
√
=
·
sin
cosμ 2x cos x
2μ − 1
π
π
cosp−1 x sin px
dx =
sin x
2
2μ − 1
π
4
1
− 2 < Re μ < 1
[p > 0]
BI (38)(17)
GW(332)(17), BI(45)(5)
3.64–3.65 Powers and rational functions of trigonometric functions
3.641
π/2
1.
0
sinp−1 x cos−p x
dx =
a cos x + b sin x
π/2
0
π cosec pπ
sin−p x cosp−1 x
dx =
a sin x + b cos x
a1−p bp
[ab > 0,
π/2
2.
sin
1−p
x cos x
3
(sin x + cos x)
0
p
dx =
0
π/2
p
sin x cos
1−p
x
3
(sin x + cos x)
dx =
0 0, Re ν > 0]
BI (48)(28)
μ+ν = 2μ 2ν B(μ, ν)
2
2a b
0
a2 sin x + b2 cos2 x
π/2
B n2 , n2
sinn−1 x cosn−1 x dx
[ab > 0]
GW (331)(59a)
n =
2(ab)n
a2 cos2 x + b2 sin2 x
0
π/2
sin2n x dx
sin2n x dx
1 π
=
n+1
2 0 a2 cos2 x + b2 sin2 x n+1
0
a2 cos2 x + b2 sin2 x
π/2
cos2n x dx
cos2n x dx
1 π
(2n − 1)!!π
=
=
n+1 = n+1
n+1
2
2
2
2
2
2
2
2
2
2
n!ab2n+1
0
0
a sin x + b cos x
a sin x + b cos x
[ab > 0]
4.
0
π/2
n
cosp+2n x cos px dx
=
π
n+1
a2 cos2 x + b2 sin2 x
k=0
2n − k
n
GW (331)(58)
p+k−1
bp−1
k
(2a)2n−k+1 (a + b)p+k
[a > 0, b > 0, p > −2n − 1]
GW (332)(30)
402
3.643
Trigonometric Functions
π/2
1.
0
π/2
2.
0
3.644
π
1.
0
π
2.
0
3.643
cosp x cos px dx
π
(1 + a)p−1
=
·
1 − 2a cos 2x + a2
2p+1
1−a
a2 < 1,
p > −1
GW (332)(33c)
m−1 m−k−1
β
2n
sin2n x cosμ x cos βx
(−1)n π(1 − a)2n−2m+1 dx
=
m
k
l
22m−β−1 (1 + a)2m+β+1
(1 − 2a cos 2x + a2 )
k=0
l=0
2m − k − l − 2
×
(a − 1)k
l
m
−
1(−2)
2
a < 1, β = 2m − 2n − μ − 2, μ > −1
k
ν−1
m + 1 − 2ν m + 1 − 2ν
sinm x
p p2 − q 2
dx = 2m−2 2
,
B
+
p + q cos x
q ν=1 −4q 2
2
2
.
⎧
⎪
πp
q2
⎪
⎪
if m = 2k + 2
⎪
⎨ q 2 1 − 1 − p2
k ≥ 1, q = 0,
where A =
⎪
⎪
1 p+q
⎪
⎪
if m = 2k + 1
⎩ ln
q p−q
sinm x
dx = 2m−1 B
1 + cos x
m−1 m+1
,
2
2
GW (332)(33)
p2 − q 2
−q 2
k
A
p2 − q 2 ≥ 0
[m ≥ 2]
m−1 m+1
sinm x
dx = 2m−1 B
,
[m ≥ 2]
1
−
cos
x
2
2
0
.
π
pπ
sin2 x
q2
dx = 2 1 − 1 − 2
4.
q
p
0 p + q cos x
π
p
sin3 x
1
p+q
p2
dx = 2 2 +
5.
1 − 2 ln
q
q
q
p−q
0 p + q cos x
π
n
cosn x dx
(2n − 2k − 1)!!(2k − 1)!!
π
√
(−1)k
3.645
n+1 = n
n
2
2
(n − k)!k!
2 (a + b) a − b k=0
0 (a + b cos x)
2
a > b2
π
3.
3.646
π/2
1.
0
π/2
2.
0
3.647
0
π/2
cosn x sin nx sin 2x
π
dx =
2
1 − 2a cos 2x + a
4a
1+a
2
n
1
− n
2
a2 < 1
π
cosp x cos px dx
ap−1
=
·
2
2
2
2b (a + b)p
sin x + b cos x
[p > −1,
a > 0,
k
LI (64)(16)
BI (50)(6)
∞
1 − a cos 2nx
π m k
π
m
a + m+1
cos
x
cos
mx
dx
=
2
m+2
kn
1 − 2a cos 2nx + a
2
2
k=1
2
a <1
a2
a+b
a−b
b > 0]
LI (50)(7)
BI (47)(20)
3.652
3.648
Trigonometric functions: powers and rational functions
π/4
1.
0
403
tanl x dx
1 + cos m
n π sin 2x
n−1
n+l+k
m l+k
km
1
cosec π
π ψ
(−1)k−1 sin
−ψ
2n
n
n
2n
2n
k=0
n−1
2
n+l−k
m l+k
km
1
π ψ
(−1)k−1 sin
= cosec π
−ψ
n
n
n
n
n
=
[m + n is odd]
[m + n is even]
k=0
[l is a natural number]
π/2
2.
0
3.649
π/2
1.
0
μπ
tan±μ x sin 2x dx
π
cosec
=
1−
1 ∓ 2a cos 2x + a2
4a
2 μπ
π
cosec
=
1+
4a
2
1−a
1+a
a−1
a+1
π/2
μπ
tan±μ x (1 ∓ a cos 2x)
π
dx = sec
1+
1 ∓ 2a cos 2x + a2
4
2 0
μπ
π
= sec
1−
4
2
2.
3.651
1.
2.
3.652
1
μ+2
tanμ x dx
=
ψ
1 + sin x cos x
3
3
0
π/4
μ
1
μ+2
tan x dx
=
β
1 − sin x cos x
3
3
0
π/4
π/2
1.
0
tan±μ x dx
= π cosec t sin μt cosec(μπ)
1 + cos t sin 2x
tanμ x dx
=
(sin x + cos x) sin x
π/2
0
−ψ
μ+1
3
+β
μ+1
3
μ
μ
|Re μ| < 1,
a2 < 1
a2 > 1
π/2
2.
0
tanμ x dx
=
(sin x − cos x) sin x
π/2
0
3.
0
π/2
μ+ 12 x
cot
dx
=
(sin x + cos x) cos x
0
π/2
BI (47)(4)
[−2 < Re μ < 1]
μ
2
1−a
a <1
1+a μ
2
a−1
a >1
a+1
[|Re μ| < 1]
BI (50)(3)
BI (50)(4)
[Re μ > −1]
BI (36)(3)
[Re μ > −1]
BI (36)(4)a
cotμ x dx
= π cosec μπ
(sin x + cos x) cos x
BI (49)(1)
cotμ x dx
= −π cot μπ
(cos x − sin x) cos x
[0 < Re μ < 1]
[0 < Re μ < 1]
t2 < π 2
BI (36)(5)
BI (49)(2)
μ− 12 x
tan
dx
= π sec μπ
(sin x + cos x) cos x
|Re μ| < 12
BI (61)(1, 2)
404
3.653
Trigonometric Functions
π/2
1.
0
π/2
2.11
0
tan1−2μ x dx
=
a2 cos2 x + b2 sin2 x
μ
tan x dx
=
1 − a sin2 x
π/2
0
π/2
0
3.653
cot1−2μ x dx
π
= 2μ 2−2μ
2
2
2
2
2a b
sin μπ
a sin x + b cos x
[0 < Re μ < 1]
μ
π sec
cot x dx
= 2
1 − a cos x
2 (1 −
GW (331)(59b)
μπ
2
a)μ+1
[|Re μ| < 1,
π/2
+ π
,
μπ
tan±μ x dx
π
cosec
t
sec
cos
−
t
μ
=
2
2
2
1 − cos2 t sin2 2x
0
|Re μ| < 1,
a < 1]
BI (49)(6)
3.
t2 < π 2
BI(49)(7), BI(47)(21)
4.
5.
π/2
π/2
+ π
,
tan±μ x sin 2x
μπ
sin
−
t
μ
dx
=
π
cosec
2t
cosec
2
2
1 − cos2 t sin2 2x
0
|Re μ| < 1, t2 < π 2
BI (47)(22)a
π/2
π/2
+ μπ
,
π
μπ
tanμ x sin2 x dx
cotμ x cos2 x dx
=
=
cosec
2t
sec
cos
−
(μ
+
1)t
2
1 − cos2 t sin2 2x
1 − cos2 t sin2 2x 2
0
0
2
|Re μ| < 1, t2 < π 2
BI(47)(23)a, BI(49)(10)
6.
0
3.654
π/2
1.
tanμ x cos2 x dx
=
1 − cos2 t sin2 2x
tanμ+1 x cos2 x dx
(1 + cos t sin 2x)
0
2
π/2
0
+ μπ
,
μπ
cotμ x sin2 x dx
π
cosec
2t
sec
cos
−
(μ
−
1)t
=
2
2
1 − cos2 t sin2 2x
2
|Re μ| < 1, t2 < π 2
BI(47)(24)a, BI(49)(9)
π/2
=
0
cotμ+1 x sin2 x dx
2
(1 + cos t sin 2x)
=
π (μ sin t cos μt − cos t sin μt)
2 sin μπ sin3 t
|Re μ| < 1, t2 < π 2
BI(48)(3), BI(49)(22)
π/2
2.
2
(sin x + cos x)
0
π/2
3.
0
3.655
0
tan±μ x dx
π/2
=
μπ
sin μπ
μπ
tan±(μ−1)x dx
π
= ± cot
2
2
2
2
cos x − sin x
tan2μ−1 x dx
=
1 − 2a cos t1 sin2 x + cos t2 cos2 x + a2
π/2
[0 < Re μ < 1]
BI (56)(9)a
[0 < Re μ < 2]
BI (45)(27, 29)
cot2μ−1 x dx
1 − 2a cos t1 cos2 x + cos t2 sin2 x + a2
0
π cosec μπ
=
2 )μ (1 − 2a cos t + a2 ) 1 − μ
(1
−
2a
cos
t
+
a
2
1 0 < Re μ < 1, t21 < π 2 , t22 < π 2
BI (50)(18)
3.662
3.656
Trigonometric functions: powers of linear functions
π/4
1.
0
tanμ x dx
1
=
2
2
12
1 − sin x cos x
μ+1
μ+2
−ψ
6
6
μ+4
μ+5
μ+2
μ+1
+ψ
+ 2ψ
− 2ψ
6
6
3
3
[Re μ > −1]
(cf. 3.651 1 and 2) LI (36)(10)
−ψ
+ψ
π/2
2.
0
tanμ−1 x cos2 x dx
=
1 − sin2 x cos2 x
405
π/2
0
π
cotμ−1 x sin2 x dx
μπ
= √ cosec
cosec
2
2
6
1 − sin x cos x
4 3
[0 < Re μ < 4]
2+μ
π
6
LI (47)(26)
3.66 Forms containing powers of linear functions of trigonometric functions
3.661
2π
1.
(a sin x + b cos x)
2n+1
(a sin x + b cos x)
2n
dx = 0
BI (68)(9)
0
2π
2.
0
π
3.
n
(a + b cos x) dx =
0
4.
0
π
dx
n+1
(a + b cos x)
n
(2n − 1)!!
· 2π a2 + b2
(2n)!!
dx =
1
2
n
n
(a + b cos x) dx = π a2 − b2 2 P n
2π
0
π/2
π/2
μ
(sec x − 1) sin x dx =
1.
√
a
− b2
a2
n/2
k
π (−1)k (2n − 2k)! n−2k 2
= n
a
a − b2
2
k!(n − k)!(n − 2k)!
k=0
2
a > b2
a
dx
1 2π
π
√
=
=
n+1 P n
2 − b2
2 0 (a + b cos x)n+1
2
2
2
a
(a − b )
n
a+b
π
(2n − 2k − 1)!!(2k − 1)!!
√
=
·
n
n
2
2
(n − k)!k!
a−b
2 (a + b) a − b
k=0
3.662
BI (68)(8)
0
[a > |b|]
π/2
μ
μ
(cosec x − 1) sin 2x dx = (1 − μ)μπ cosec μπ
0
π/2
μ
(sec x − 1) tan x dx =
3.
0
0
π/2
[−1 < Re μ < 2]
BI (55)(13)
BI (48)(7)
μ
(cosec x − 1) cot x dx = −π cosec μπ
0
π/4
μ
(cot x − 1)
4.
GW(332)(38), LI(64)(14)
0
2.
k
(cosec x − 1) cos x dx = μπ cosec μπ
[|Re μ| < 1]
GW (332)(37a)
π
dx
= − cosec μπ
sin 2x
2
[−1 < Re μ < 0]
BI (46)(4,6)
[−1 < Re μ < 0]
BI (38)(22)a
406
Trigonometric Functions
π/4
dx
= μπ cosec μπ
cos2 x
μ
(cot x − 1)
5.
0
3.663
u
ν− 12
(cos x − cos u)
1.
cos ax dx =
0
ν−1
(cos x − cos u)
2.
[|Re μ| < 1]
BI (38)(11)a
π
1
sinν u Γ ν +
P −ν 1 (cos u)
2
2 a− 2
Re ν > − 12 ; a > 0,
√
u
3.663
cos[(ν + β)x] dx =
0
0<u<π
EH I 159(27), ET I 22(28)
2ν−1
π Γ(β + 1) Γ(ν) Γ(2ν) sin
u ν
C β (cos u)
1
ν
2 Γ(β + 2ν) Γ ν + 2
[Re ν > 0, Re β > −1, 0 0,
0
π
2.
0
z+
√
arg z +
z 2 − 1 cos x
q
z + z 2 − 1 cos x
μ
π
z+
cos nx dx =
0
4.
z 2 − 1 cos x = arg z for x =
dx
q = π P q−1 (z)
z 2 − 1 cos x
+
Re z > 0,
3.
arg z +
π,
2
z 2 − 1 cos x = arg z for x =
SM 482
π,
2
WH
π
P nq (z)
(q +
1)(q
+
2)
·
·
·
(q
+
n)
π
Re z > 0, arg z + z 2 − 1 cos x = arg z for x = ,
2
z lies outside the interval (−1, 1) of the real axis
WH, SM 483(15)
π
0
sin2ν−1 x dx
2
22ν−1 Γ(μ + 1) [Γ(ν)]
C νμ (z)
Γ(2ν
+
μ)
√
1
1−ν
π Γ(ν) Γ(2ν) Γ(μ + 1) ν
π 2
ν
2 −ν
=
z − 1 4 2 Γ(ν) P μ+ν−
C μ (z) = 2
1 (z)
1
2
2
Γ(2ν + μ) Γ ν + 2
EH I 155(6)a, EH I 178(22)
[Re ν > 0]
2π +
,
ν
ν−1
dx
β + β 2 − 1 cos(a − x)
γ + γ 2 − 1 cos x
0
= 2π P ν βγ − β 2 − 1 γ 2 − 1 cos a
=
5.
[Re β > 0,
Re γ > 0]
EH I 157(18)
3.667
3.665
Trigonometric functions: powers of linear functions
π
1.
0
π
2.
0
sinμ−1 x dx
2μ−1
μ μ
μ = μ B 2, 2
2
2
(a + b cos x)
(a − b )
[Re μ > 0,
3.666
π
(β + cos x)
1.
μ−ν− 12
0
π
2.6
0
4.
5.10
3.667
FI II 790a
EH I 81(9)
Re ν > − 12
√
π
μ+ν
(cosh β + sinh β cos x)
sin−2ν x dx = ν sinhν (β) Γ 12 − ν P νμ (cosh β)
2
Re ν < 12
π
1
μ
0
|a| < 1]
μ 1
2ν+ 2 e−iμπ β 2 − 1 2 Γ ν + 12 Q μν− 1 (β)
2
sin2ν x dx =
Γ ν + μ+ 12
Re ν + μ + 12 > 0,
(cos t + i sin t cos x) sin2ν−1 x dx = 2ν− 2
3.
0 0,
407
2π
√
1
EH I 155(5)a
EH I 156(7)
−ν
2
π sin 2 −ν t Γ(ν) P μ+ν−
1 (cos t)
2
Re ν > 0, t2 < π 2
1
i3m 2π Γ(ν + 1)
cos ma P m
ν (cos t)
Γ(ν + m + 1)
0
+
,
π
0<t<
2
2π
3m
2π
Γ(ν
+
1)
i
ν
sin ma P m
[cos t + i sin t cos(a − x)] sin mx dx =
ν (cos t)
Γ(ν + m + 1)
0
+
π,
0<t<
2
EH I 158(23)
ν
[cos t + i sin t cos(a − x)] cos mx dx =
π/4
1.
2μ
(cos x + sin x)
0
sinμ−1 2x dx
π/4
2.
EH I 159(26)
√
=
π
Γ(μ)
2μ+1 Γ μ + 12
sinμ x dx
μ+1
(cos x − sin x)
0
EH I 159(25)
cos x
= −π cosec μπ
[Re μ > 0]
[−1 < Re μ < 0]
BI (37)(1)
(cf. 3.192 2)
BI (37)(16)
π/4
3.
0
π/4
4.
0
5.
0
π/4
μ
π
(cos x − sin x)
dx = − cosec μπ
sinμ x sin 2x
2
[−1 < Re μ < 0]
sinμ x dx
π
= cosec μπ
μ
2
(cos x − sin x) sin 2x
[0 < Re μ < 1]
sinμ x dx
= μπ cosec μπ
μ
(cos x − sin x) cos2 x
[|Re μ| < 1]
BI (35)(27)
LI (37)(20)a
BI (37)(17)
408
Trigonometric Functions
π/4
sinμ x dx
6.
μ−1
(cos x − sin x)
0
π/2
sinμ−1 x cosν−1 x
7.
μ+ν
(sin x + cos x)
0
3.668
π
4
−π
4
v
2.
u
μ−1
π/2
0
3.670
2.∗
π
√
0
dx
=
a ± b cos x
[Re μ > 0,
BI(35)(24), BI(37)(18)
Re ν > 0]
BI (48)(8)
π
2 sin (π cos2 t)
μ−1
1 − 2a cos u + a2
sin x dx
π
·
=
μ ·
2
2
1 − 2a cos x + a
sin μπ
(1 − 2a cos v + a )
0 < Re μ < 1, a2 < 1
π√
0
[|Re μ| < 1]
dx = B(μ, ν)
sinp−1 x cosq−p−1 x dx
=
q
(a cos x + b sin x)
a ± b cos x dx =
1.
1−μ
μπ cosec μπ
2
dx =
(cos u − cos x)
μ
(cos x − cos v)
3.669
=
cos 2t
cos x + sin x
cos x − sin x
1.
cos3 x
3.668
π/2
√
−π/2
π/2
0
FI II 788
BI (73)(2)
sinq−p−1 x cosp−1 x
B(p, q − p)
q dx =
aq−p bp
(a sin x + b cos x)
[q > p > 0, ab > 0]
√
a ± b cos x dx = 2 a + b K
π/2
dx
2
√
=√
E
a+b
−π/2 a ± b sin x
2b
a+b
BI (331)(9)
[a > b > 0]
2b
a+b
[a > b > 0]
3.67 Square roots of expressions containing trigonometric functions
3.671
π/2
1.
0
1
sinα x cosβ x 1 − k 2 sin2 x dx = B
2
α+1 β+1
α+1 1 α+β+2 2
,
,− ;
;k
F
2
2
2
2
2
[α > −1, β > −1, |k| < 1]
GW (331)(93)
π/2
2.
0
sinα x cosβ x
1
dx = B
2
2
2
1 − k sin x
α+1 β+1
,
2
2
F
α+1 1 α+β+2 2
, ;
;k
2
2
2
[α > −1, β > −1,
|k| < 1]
GW (331)(92)
3.
0
π
2n
sin
x dx
1 − k 2 sin x
2
=
π
2n
∞
j=0
(2j − 1)!! (2n + 2j − 1)!! 2j
k
22j j!(n + j)!
∞
2
(2n − 1)!!π [(2j − 1)!!]
√
=
2n 1 − k 2 j=0 22j j!(n + j)!
k2
2
k −1
j
k2 < 1
k2 <
1
2
LI (67)(2)
3.676
4.∗
Trigonometric functions and square roots
π√
a + b cos x dx =
√
√
a + b sin x dx = 2 a + bE
−π/2
0
5.∗
π/2
π
√
0
dx
=
a ± b cos x
π/2
dx
2
√
K
=
a
+
b
a
±
b
sin
x
−π/2
409
2b
a+b
[a > b]
2b
a+b
[a > b]
3.672
π/4
1.
0
π/4
2.
0
π
2
3.673
dx
sinn x
(2n)!!
·
=2·
cosn+1 x
(2n
+ 1)!!
cos x (cos x − sin x)
BI (39)(5)
dx
sinn x
(2n − 1)!!
·
π
=
n+1
cos
x
(2n)!!
sin x (cos x − sin x)
BI (39)(6)
√
u
3.674
1.8
π
2
0
π
2.
0
π
3.8
0
3.675
1.
2.
3.676
1.
2.
√
dx
π − 2u
= 2K sin
4
sin x − sin u
dx
1 − (p2 /2) (1 − cos 2x)
= K(p),
=2
1 − 2p cos x + p2
2
=
p
cos x dx
1 − 2p cos x +
p2
=
2
1 1+p
K
p 1+p
√
2 p
1+p
p2 ≤ 1
p2 ≥ 1
BI (67)(5)
√
2 p
− (1 + p)E
1+p
2
p <1
BI (67)(6)
BI (67)(7)
π
WH
FI II 684, WH
π/2
sin x dx
1
= arctan p
2
2
p
0
1 + p sin x
π/2
<
tan2 x 1 − p2 sin2 x dx = ∞
0
[1 > p > 0]
sin x dx
sin n + 12 x dx
π
= P n (cos u)
2
2 (cos u − cos x)
u
u
cos n + 12 x dx
π
= P n (cos u)
2
2 (cos x − cos u)
0
BI (74)(11)
BI (60)(5)
BI (53)(8)
410
Trigonometric Functions
π/2
3.
0
3.677
1.
2.
3.678
dx
1
= K
2
2
2
2
p
p cos x + q sin x
p2 − q 2
p
3.677
[0 < q < p]
√ √ √
1
sin2 x dx
2
2
−√ K
= 2E
2
2
2
2
0
1 + sin x
% √ √ &
π/2
√
cos2 x dx
2
2
= 2 K
−E
2
2
2
0
1 + sin x
π/2
dx
= ln 2
tan
x
0
π/4
tan2 x dx
1
= 1 − k 2 − E(k) + K(k)
2
2
2
0
1 − k sin 2x
u
π
cos 2x − cos 2u
π2
2
dx = (1 − cos u)
u <
cos 2x + 1
2
4
0
π/4
n− 12
√
(cos x − sin x)
(2n − 1)!!
π
cosec x dx =
n+1
cos
x
(2n)!!
0
π/4
n− 1
√
(cos x − sin x) 2
(2n − 1)!!(2m − 1)!!
tanm x cosec x dx =
π
n+1 x
cos
(2n + 2m)!!
0
2.
3.
4.
5.
3.679
BI (60)(2)
BI (60)(3)
π/4
sec1/2 2x − 1
1.
FI II 165
π/2
1.
0
BI (38)(23)
BI (39)(2)
LI (74)(6)
BI (38)(24)
BI (38)(25)
dx
cos2 x
·
1 − cos2 β cos2 x
1 − k 2 sin2 x
π
1
∗
−
K
E
(β,
k
=
)
−
E
F
(β,
k
)
+
K
F
(β,
k
)
sin β cos β 1 − k 2 sin2 β 2
MO 138
π/2
2.
0
2
sin x
dx
·
1 − 1 − k 2 sin2 β sin2 x
1 − k 2 sin2 x π
1
∗
−
K
E
(β,
k
=
)
−
E
F
(β,
k
)
+
K
F
(β,
k
)
k 2 sin β cos β 1 − k 2 sin2 β 2
MO 138
3.
0
∗ In
π/2
sin x
K E (β, k) − E F (β, k)
dx
=
·
2
2
sin β sin x
k 2 sin β cos β 1 − k 2 sin2 β
1 − k 2 sin2 x
2
1−
3.679, k =
√
k2
1 − k2 .
MO 138
3.683
Trigonometric functions: various powers
411
3.68 Various forms of powers of trigonometric functions
3.681
π/2
1.
0
π/2
2.
0
π/2
3.
0
π/2
4.8
0
sin2μ−1 x cos2ν−1 x dx
1
= B(μ, ν) F , μ; μ + ν; k 2
2
2
1 − k 2 sin x
2μ−1
[Re μ > 0,
Re ν > 0]
EH I 115(7)
[Re μ > 0,
Re ν > 0]
EH I 10(20)
2ν−1
sin
x cos
x dx
B(μ, ν)
μ+ν =
μ
2
2 (1 − k 2 )
1 − k 2 sin x
sinμ x dx
μ −1
cosμ−3 x 1 − k 2 sin2 x 2
Γ μ+1
Γ 2 − μ2
2
= k 3 π(μ − 1)(μ − 3)(μ − 5)
1 + (μ − 3)k + k 2
1 − (μ − 3)k + k 2
−
(1 + k)μ−3
(1 − k)μ−3
[−1 < Re μ < 4]
BI (54)(10)
sinμ+1 x dx
μ
(1 − k)−μ − (1 + k)−μ
√
Γ
Γ 1+
μ+1 =
2
2kμ
π
2
2
μ
2
cos x 1 − k sin x
1−μ
2
[−2 < Re μ < 1]
π/2
3.682
0
sinμ x cosν x
1
B
dx =
2a
(a − b cos2 x)
μ+1 ν+1
,
2
2
F
BI (61)(5)
μ+ν
b
ν+1
, ;
+ 1;
2
2
a
[Re μ > −1, Re ν > −1,
a > |b| ≥ 0]
GW (331)(64)
3.683
π/4
(sinn 2x − 1) tan
1.
0
2.
3.
π
+ x dx =
4
11
2
k
n
π/4
(cosn 2x − 1) cot x dx = −
0
k=1
1
= − [C + ψ(n + 1)]
2
[n ≥ 0]
BI(34)(8), BI(35)(11)
π/4
π/4
π
+ x dx =
(sinμ 2x − 1) cosecμ 2x tan
(cosμ 2x − 1) secμ 2x cot x dx
4
0
0
1
= [C + ψ(1 − μ)]
2
[Re μ < 1]
BI (35)(20)
π/4
π4
2μ
2μ
π
+ x dx =
sin 2x − 1 cosecμ 2x tan
cos 2x − 1 secμ 2x cot x dx
4
0
0
1
π
=−
+ cot μπ
2μ
2
BI (35)(21)
π/4
π/4
(1 − secμ 2x) cot x dx =
4.
0
(1 − cosecμ 2x) tan
0
1
π
+ x dx = [C + ψ(1 − μ)]
4
2
[Re μ < 1]
BI (35)(13)
412
Trigonometric Functions
π/4
3.684
0
(cotμ x − 1) dx
=
(cos x − sin x) sin x
π/2
0
3.684
(tanμ x − 1) dx
= −C − ψ(1 − μ)
(sin x − cos x) cos x
[Re μ < 1]
BI (37)(9)
3.685
π/4
1.
0
μ−1
π
+ x dx =
sin
2x − sinν−1 2x tan
4
=
π/2
2.
0
μ−1
dx
sin
=
x − sinν−1 x
cos x
π/2
0
π/4
cosμ−1 2x − cosν−1 2x cot x dx
0
1
[ψ(ν) − ψ(μ)]
2
[Re μ > 0, Re ν > 0]
dx
1+
ν
μ ,
cosμ−1 x − cosν−1 x
=
ψ
−ψ
sin x
2
2
2
[Re μ > 0,
π/2
(sinμ x − cosecμ x)
3.
0
dx
=
cos x
BI(34)(9), BI(35)(12)
π/2
(cosμ x − secμ x)
0
Re ν > 0]
π
μπ
dx
= − tan
sin x
2
2
[|Re μ| < 1]
π/4
(sinμ 2x − cosecμ 2x) cot
4.
0
π
+ x dx =
4
BI (46)(2)
BI (46)(1, 3)
π/4
(cosμ 2x − secμ 2x) tan x dx
0
π
1
− cosec μπ
2μ
2
[|Re μ| < 1]
BI (35)(19, 22)
π/4
π/4
π
+ x dx =
5.
(sinμ 2x − cosecμ 2x) tan
(cosμ 2x − secμ 2x) cot x dx
4
0
0
π
1
+ cot μπ
=−
2μ
2
[|Re μ| < 1]
BI (35)(14)
π/4
μ−1
π
+ x dx
sin
6.
2x + cosecμ 2x cot
4
0
π/4
μ−1
π
=
cos
2x + secμ 2x tan x dx = cosec μπ
4
0
[0 < Re μ < 1]
BI (35)(18, 8)
π/4
π/4
μ−1
μ−1
π
π
μ
+ x dx =
sin
cos
7.
2x − cosec 2x tan
2x − secμ 2x cot x dx = cot μπ
4
2
0
0
[0 < Re μ < 1]
BI(35)(7), LI(34)(10)
π/2
π/2
π
tan x dx
cot x dx
=
=
3.686
BI(47)(28), BI(49)(14)
μ
μ
μ
μ
cos x + sec x
sin x + cosec x
4μ
0
0
3.687
π/2 μ−1
π/2
cos ν−μ
sin
x + sinν−1 x
cosμ−1 x + cosν−1 x
μ ν
4 π
1.
dx =
,
B
dx =
ν+μ
cosμ+ν−1 x
2 2
sinμ+ν−1 x
0
0
π
2 cos
4
[Re μ > 0, Re ν > 0, Re(μ + ν) < 2]
=
BI (46)(7)
3.688
Trigonometric functions: various powers
π/2
2.
0
sinμ−1 x − sinν−1 x
dx =
cosμ+ν−1 x
π/2
0
413
sin ν−μ
cosμ−1 x − cosν−1 x
μ ν
4 π
B
dx =
,
2 2
sinμ+ν−1 x
2 sin ν+μ
π
4
[Re μ > 0, Re ν > 0, Re(μ + ν) < 4]
BI(46)(8)
π/2
3.
0
sinμ x + sinν x
cot x dx =
sinμ+ν x + 1
π
2
0
μ−ν π
π
cosμ x + cosν x
tan x dx =
sec
·
cosμ+ν x + 1
μ+ν
μ+ν 2
[Re μ > 0, Re ν > 0]
BI (49)(15)a, BI (47)(29)
π/2
4.
0
sinμ x − sinν x
cot x dx =
sinμ+ν x − 1
π
2
0
μ−ν π
π
cosμ x − cosν x
tan x dx =
tan
·
cosμ+ν x − 1
μ+ν
μ+ν 2
[Re μ > 0, Re ν > 0]
BI(149)(16)a, BI(47)(30)
π/2
5.
0
π/2
6.
0
3.688
π/4
1.
0
π/4
2.
0
μ π
π
cosμ x + secμ x
tan x dx =
sec
·
cosν x + secν x
2ν
ν 2
[|Re ν| > |Re μ|]
BI (49)(12)
π
μ π
cosμ x − secμ x
tan x dx =
tan
·
cosν x − secν x
2ν
ν 2
[|Re ν| > |Re μ|]
BI (49)(13)
tanν x − tanμ x dx
·
= ψ(μ) − ψ(ν)
cos x − sin x
sin x
[Re μ > 0,
BI (37)(10)
tanμ x − tan1−μ x dx
·
= π cot μπ
cos x − sin x
sin x
[0 < Re μ < 1]
π/4
(tanμ x + cotμ x) dx =
3.
0
μπ
π
sec
2
2
π/4
(tanμ x − cotμ x) tan x dx =
4.
0
π/4
5.
0
π/4
6.
0
π/4
7.
0
π/4
8.
0
π/4
9.
0
10.
0
π/4
π
μπ
1
− cosec
μ
2
2
Re ν > 0]
[|Re μ| < 1]
BI (37)(11)
BI (35)(9)
[0 < Re μ < 2]
BI (35)(15)
π
μπ
tanμ−1 x − cotμ−1 x
dx = cot
cos 2x
2
2
[|Re μ| < 2]
BI (35)(10)
1
π
μπ
tanμ x − cotμ x
tan x dx = − + cot
cos 2x
μ
2
2
[−2 < Re μ < 0]
BI (35)(23)
tanμ x + cotμ x
dx = π cosec t cosec μπ sin μt
1 + cos t sin 2x
[t = nπ,
tanμ−1 x + cotμ x
dx = π cosec μπ
(sin x + cos x) cos x
[0 < Re μ < 1]
BI (37)(3)
tanμ x − cotμ x
1
dx = −π cosec μπ +
(sin x + cos x) cos x
μ
[0 < Re μ < 1]
BI (37)(4)
tanν x − cotμ x
dx = ψ(1 − μ) − ψ(1 + ν)
(cos x − sin x) cos x
[Re μ < 1,
|Re μ| < 1]
Re ν > −1]
BI (36)(6)
BI (37)(5)
414
Trigonometric Functions
π/4
11.
0
π/4
12.
0
13.
14.
3.689
tanμ−1 x − cotμ x
dx = π cot μπ
(cos x − sin x) cos x
[0 < Re μ < 1]
BI (37)(7)
1
tanμ x − cotμ x
dx = π cot μπ −
(cos x − sin x) cos x
μ
[0 < Re μ < 1]
BI (37)(8)
π/4
dx
π
1
·
=
[Re μ = 0]
BI (37)(12)
μ
tan x + cot x sin 2x
8μ
0
√
π/2
π/2
1
1
π
Γ(ν)
dx
dx
=
= 2ν+1
ν ·
ν ·
μ
μ
μ
μ
2
μ Γ ν + 12
(tan x + cot x) tan x
(tan x + cot x) sin 2x
0
0
μ
[ν > 0]
π/4
(tanμ x − cotμ x) (tanν x − cotν x) dx =
15.
0
BI(49)(25), BI(49)(26)
μπ
2
sin νπ
2π sin
2
cos μπ + cos νπ
[|Re μ| < 1,
π/4
(tanμ x + cotμ x) (tanν x + cotν x) dx =
16.
0
π/4
17.
0
π/4
18.
0
π/4
19.
0
π/2
20.
3.689
π/2
1.
0
2.
|Re ν| < 1]
BI (35)(25)
[0 < Re ν < 1]
BI (37)(14)
dx
π
νπ
tanν x + cotν x
·
=
sec
tanμ x + cotμ x sin 2x
4μ
2μ
[0 < Re ν < 1]
BI (37)(13)
[μ > 0,
BI (49)(29)
ν
(1 + tan x) − 1
μ+ν
dx
= ψ(μ + ν) − ψ(μ)
sin x cos x
ν > 0]
π
μπ
μt
(sinμ x + cosecμ x) cot x dx
= cosec t cosec
sin
sinν x − 2 cos t + cosecν x
ν
ν
ν
[μ < ν]
BI (35)(16)
dx
π
νπ
tanν x − cotν x
·
=
tan
tanμ x − cotμ x sin 2x
4μ
2μ
(1 + tan x)
0
|Re ν| < 1]
sin μπ
(tanμ x − cotμ x) (tanν x + cotν x)
dx = −π
cos 2x
cos μπ + cos νπ
[|Re μ| < 1,
BI (35)(17)
νπ
2π cos μπ
2 cos 2
cos μπ + cos νπ
[|Re μ| < 1,
|Re ν| < 1]
LI (50)(14)
π
μt2
t2
sinμ x − 2 cos t1 + cosecμ x
μπ
· cot x dx = cosec t2 cosec
sin
− cosec t2 cos t1
ν
νx
sin
x
+
2
cos
t
+
cosec
ν
ν
ν
ν
2
0
[(ν > μ > 0) or (ν < μ < 0) or (μ > 0, ν < 0, and μ + ν < 0) or (μ < 0, ν > 0, and μ + ν > 0)]
π/2
BI (50)(15)
3.691
Trigonometric functions: complicated arguments
415
3.69–3.71 Trigonometric functions of more complicated arguments
3.691
∞
1.
2
sin ax
dx =
0
0
1
2.
sin ax2 dx =
0
3.
4.
∞
1
cos ax dx =
2
2
π
2a
π √ S
a
2a
cos ax2 dx =
∞
5.
0
∞
6.
1
sin ax2 cos 2bx dx =
2
2
cos ax sin 2bx dx =
0
∞
7.
0
1
2
[a > 0]
2
π
2a
cos
b2
+ sin S
a
b
√
a
2
b
b
− sin
a
a
0
∞
9.
0
π
2a
π
2a
b2
sin C
a
11.
0
[a > 0, b > 0]
b2
π
1 π
cos
+
=
2 a
a
4
[a > 0, b > 0]
b2
− cos S
a
b
√
a
2
cos
b
b2
+ sin
a
a
(cos ax + sin ax) sin b2 x2 dx
1 π
=
2b 2
1 + 2C
(cos ax + sin ax) cos b2 x2 dx
1 π
=
2b 2
1 + 2C
a
2b
a
2b
√
∞
2bc
π
sin 2 cos
sin a2 x2 sin 2bx sin 2cx dx =
2a
a
0
b
√
a
ET I 82(1)a
ET I 82(18), BI(70)(13) GW(334)(5a)
10.
ET I 8(5)a
b
√
a
[a > 0,
b > 0]
[a > 0,
b > 0]
ET I 83(3)a
GW(334)(5a), BI(70)(14), ET I 24(7)
∞
8.
cos ax2 cos 2bx dx =
FI II 743a, ET I 64(7)a
[a > 0]
√ π
C
a
2a
0
∞
2
π
b2
sin ax sin 2bx dx =
cos C
2a
a
0
1
[a > 0]
∞
sin a2 x2 cos 2bx cos 2cx dx =
√
2bc
π
cos 2 cos
2a
a
a2
cos
− 1 − 2S
4b2
[a > 0, b > 0]
a
2b
a2
sin
+ 1 − 2S
4b2
[a > 0, b > 0]
a
2b
sin
a2
4b2
ET I 85(22)
cos
a2
4b2
ET I 25(21)
b 2 + c2
π
−
a2
4
[a > 0,
b > 0,
c > 0]
ET I 84(15)
b > 0,
c > 0]
ET I 84(21)
b 2 + c2
π
+
2
a
4
[a > 0,
416
Trigonometric Functions
∞
12.
2 2
cos a x
0
∞
13.
0
∞
14.
0
∞
15.
0
∞
16.
0
∞
17.
0
∞
18.
0
∞
19.
0
∞
20.
√
π
2bc
sin 2 sin
sin 2bx sin 2cx dx =
2a
a
1 π
sin ax2 cos bx2 dx =
4 2
1 π
=
4 2
2 2
1
cos ax − sin2 bx2 dx =
8
π
−
b
2 2
1
cos ax − cos2 bx2 dx =
8
π
+
b
4 2
1
cos ax − sin4 bx2 dx =
8
π
+
a
sin2n ax2 dx =
21.
2n+1
sin
2
ax dx =
0
[a > b > 0]
[b > a > 0]
b > 0]
BI (178)(1)
π
a
[a > 0,
b > 0]
BI (178)(3)
[a > 0,
b > 0]
BI (178)(5)
[a > 0, b > 0]
π
π
1
−
+
32
2a
2b
BI (178)(2)
π
−
b
π
b
π
a
[a > 0,
π
π
−
a
b
b > 0]
BI (178)(4)
[a > 0,
b > 0]
BI (178)(6)
cos2n ax2 dx = ∞
BI (177)(5, 6)
0
∞
1
22n+1
n
n+k
(−1)
k=0
2n + 1
k
π
2(2n − 2k + 1)a
[a > 0]
∞
22.
cos2n+1 ax2 dx =
0
1
22n+1
n
k=0
2n + 1
k
∞
1.
sin a − x2 + cos a − x2 dx =
0
2.
∞
cos
0
x2
π
−
2
8
cos ax dx =
π
cos
2
BI (70)(9)
π
2(2n − 2k + 1)a
[a > 0]
3.692
ET I 25(19)
[a > 0,
π
b
√
4 2
1
8+ 2
cos ax − cos4 bx2 dx =
64
c > 0]
π
a
π
−
a
b > 0,
BI (177)(21)
√
4 2
1
8− 2
sin ax − sin4 bx2 x =
64
0
1
1
√
+√
a+b
a−b
1
1
√
−√
b+a
b−a
2 2
1
sin ax − sin2 bx2 dx =
8
∞
b 2 + c2
π
−
2
a
4
[a > 0,
3.692
π
sin a
a
a2
π
−
2
8
BI(177)(7)a, BI(70)(10)
GW(333)(30c), BI(178)(7)a
[a > 0]
ET I 24(8)
3.695
Trigonometric functions: complicated arguments
∞
3.
0
∞
4.
0
∞
5.
1
sin a 1 − x2 cos bx dx = −
2
1
cos a 1 − x2 cos bx dx =
2
sin ax2 +
0
b2
a
417
b2
π
π
cos a +
+
a
4a
4
[a > 0]
ET I 23(2)
2
b
π
π
sin a +
+
a
4a
4
[a > 0]
∞
cos 2bx dx =
cos ax2 +
0
b2
a
cos 2bx dx =
1
2
ET I 24(10)
π
2a
[a > 0]
6.8
3.693
∞
∞
BI (70)(19, 20)
∞
+
,
π
cos x2 − 1 − cos x2 + 1 dx =
4n+1 [(2n)!]2 n + 1
−∞
n=0 2
2
1.
sin ax2 + 2bx dx =
0
∞
2.
cos ax2 + 2bx dx =
0
π
2a
cos
b2
a
b2
a
1
− S2
2
− sin
b2
a
1
− C2
2
[a > 0]
π
2a
cos
b2
a
b2
a
1
− C2
2
+ sin
b2
a
1
− S2
2
[a > 0]
3.694
∞
1.
sin ax2 + 2bx + c dx =
0
∞
2.
2
3 3
cos ax + 2bx + c dx =
0
3.695
∞
1.
sin a x
0
2.
0
∞
b2
a
b2
π
cos
2a
a
b2
π
sin
+
2a
a
π
cos a3 x3 cos(bx) dx =
6a
b
3a
b
3a
J 13
J 13
BI (70)(3)
BI (70)(4)
b2
b2
1
1
− C2
− S2
sin c +
cos c
2
a
2
a
b2
b2
1
1
− S2
− C2
sin c −
cos c
2
a
2
a
[a > 0]
GW (334)(4a)
b2
b2
1
1
− C2
− S2
cos c −
sin c
2
a
2
a
b2
b2
1
1
− S2
− C2
cos c +
sin c
2
a
2
a
[a > 0]
GW (334)(4b)
b2
π
cos
2a
a
b2
π
sin
+
2a
a
π
sin(bx) dx =
6a
b2
a
2b
3a
2b
3a
b
3a
b
3a
+ J − 13
2b
3a
b
3a
√
3
b
2b
−
K 13
π
3a 3a
[a > 0, b > 0]
ET I 83(5)
√
3
b
b
2b
2b
+ J − 13
+
K 13
3a 3a
π
3a 3a
[a > 0,
b > 0]
ET I 24(11)
418
3.696
Trigonometric Functions
∞
4
sin ax
1.
2
sin bx
0
∞
2.
0
∞
3.
0
∞
4.
0
π
dx = −
4
π
sin ax4 cos bx2 dx = −
4
π
cos ax4 sin bx2 dx =
4
π
cos ax4 cos bx2 dx =
4
b
sin
2a
3.696
b2
3
− π J 14
8a 8
b2
8a
[a > 0,
b
sin
2a
b2
π
−
8a
8
J − 14
b
cos
2a
2
b
3
− π J 14
8a 8
b2
π
−
8a
8
J − 14
sin
a2
x
√
aπ
sin(bx) dx = √ J 1 2a b
2 b
sin
a2
x2
1
sin b2 x2 dx =
4b
∞
0
3.698
∞
1.
0
∞
2.8
2
sin
a
x2
cos
a2
x2
0
∞
3.
0
∞
4.
cos
0
a2
x2
1
cos b2 x2 dx =
4b
2 2
sin b x
1
dx =
4b
1
cos b2 x2 dx =
4b
[a > 0,
1.
2.
3.
b > 0]
ET I 83(4), ET I 25(24)
b > 0]
ET I 25(25)
b2
8a
b > 0]
ET I 83(6)
π
sin 2ab − cos 2ab + e−2ab
2
[a > 0,
b > 0]
π
sin 2ab + cos 2ab − e−2ab
2
ET I 24(13)
[a > 0,
b > 0]
ET I 84(12)
b > 0]
ET I 24(14)
π
cos 2ab − sin 2ab + e−2ab
2
√
2π
(cos 2ab + sin 2ab)
[a > 0, b > 0]
dx =
4a
0
√
∞
2π
b2
2 2
(cos 2ab − sin 2ab)
[a > 0, b > 0]
cos a x + 2 dx =
x
4a
0
√
∞
∞
2π
b2
b2
2 2
2 2
sin a x − 2ab + 2 dx =
cos a x − 2ab + 2 dx =
x
x
4a
0
0
[a > 0, b > 0]
∞
b2
sin a x + 2
x
2 2
ET I 83(9)
π
sin 2ab + cos 2ab + e−2ab
2
[a > 0,
3.699
ET I 84(19)
b
8a
[a > 0,
3.697
b > 0]
2
[a > 0,
b
cos
2a
ET I 83(2)
b2
8a
[a > 0,
b > 0]
BI (70)(27)
BI (70)(28)
BI(179)(11, 12)a, ET I 83(6)
3.715
Trigonometric functions: complicated arguments
∞
4.
0
∞
5.
cos a2 x2 −
0
u
3.711
0
√
2π −2ab
e
dx =
4a
√
2π −2ab
e
dx =
4a
b2
sin a x − 2
x
2 2
b2
x2
419
[a > 0,
b > 0]
GW (334)(9b)a
[a > 0,
b > 0]
GW (334)(9b)a
πau
sin a u2 − x2 cos bx dx = √
J 1 u a2 + b 2
2
2
2 a +b
[a > 0,
b > 0,
u > 0]
ET I 27(37)
3.712
∞
1.
sin (axp ) dx =
Γ
∞
2.
1
cos (axp ) dx =
Γ
[a > 0,
p > 1]
EH I 13(40)
1
p
π
cos 2p
1
[a > 0,
p > 1]
EH I 13(39)
pa p
0
3.713
π
sin 2p
pa p
0
1
p
∞
1.
0
∞
1 (−b)k − kq+1
a p Γ
sin (ax + bx ) dx =
p
k!
p
kq + 1
p
q
k=0
k(q − p) + 1
π
sin
2p
[a > 0, b > 0, p > 0,
q > 0]
BI (70)(7)
∞
2.
∞
cos (axp + bxq ) dx =
0
1 (−b)k −(kq+1)/p
a
Γ
p
k!
k=0
kq + 1
k(q − p) + 1
π
cos
p
2p
[a > 0, b > 0, p > 0,
q > 0]
BI (70)(8)
3.714 1.
∞
cos (z sinh x) dx = K 0 (z)
[Re z > 0]
WA 202(14)
[Re z > 0]
MO 36
[Re z > 0]
MO 37
0
∞
2.
sin (z cosh x) dx =
0
∞
3.
0
π
J 0 (z)
2
cos (z cosh x) dx = −
π
Y 0 (z)
2
∞
4.
cos (z sinh x) cosh μx dx = cos
0
π
cos (z cosh x) sin2μ x dx =
5.
0
3.715
1.
√
π
μπ
K μ (z)
2
2
z
μ
Γ μ+
1
2
π
sin (z sin x) sin ax dx = sin aπ s 0,a (z) = sin aπ
0
[Re z > 0,
|Re μ| < 1]
I μ (z)
Re z > 0,
Re μ > − 12
∞
k=1
(12
−
WA 202(13)
WH
(−1)k−1 z 2k−1
− a2 ) . . . [(2k − 1)2 − a2 ]
a2 ) (32
[a > 0]
WA 338(13)
420
Trigonometric Functions
π
2.
π/2
3.
sin (z sin x) sin 2x dx =
0
π
sin (z sin x) sin nx dx
π/2
π
= [1 − (−1)n ]
sin (z sin x) sin nx dx = [1 − (−1)n ] J n (z)
2
0
[n = 0, ±1, ±2, . . .] WA 30(6), GW(334)(153a)
0
1
sin (z sin x) sin nx dx =
2
3.715
−π
2
(sin z − z cos z)
z2
LI (43)(14)
π
4.
sin (z sin x) cos ax dx = (1 + cos aπ) s 0,a (z)
0
= (1 + cos aπ)
∞
k=1
(−1)k−1 z 2k−1
(12 − a2 ) (32 − a2 ) . . . [(2k − 1)2 − a2 ]
[a > 0]
WA 338(14)
π
5.
sin (z sin x) cos[(2n + 1)x] dx = 0
GW (334)(53b)
0
π
cos (z sin x) sin ax dx = −a (1 − cos aπ) s −1,a (z)
∞
1
(−1)k−1 z 2k
= −a (1 − cos aπ) − 2 +
a
a2 (22 − a2 ) (42 − a2 ) . . . [(2k)2 − a2 ]
6.
0
k=1
[a > 0]
WA 338(12)
π
7.
cos (z sin x) sin 2nx dx = 0
GW (334)(54a)
0
π
cos (z sin x) cos ax dx = −a sin aπ s −1,a (z)
∞
1
(−1)k−1 z 2k
= −a sin aπ − 2 +
a
a2 (22 − a2 ) (42 − a2 ) . . . [(2k)2 − a2 ]
8.
0
k=1
9.
π
cos (z sin x) cos nx dx =
1
2
[a > 0]
WA 338(11)
π
cos (z sin x) cos nx dx
π/2
π
n
= [1 + (−1) ]
cos (z sin x) cos nx dx = [1 + (−1)n ] J n (z)
2
0
0
−π
GW (334)(54b)
π/2
cos (z sin x) cos2n x dx =
10.8
0
11.
π/2
sin (z cos x) sin 2x dx =
0
π (2n − 1)!!
J n (z)
2
zn
2
(sin z − z cos z)
z2
[n = 0, 1, 2, . . .]
FI II 486, WA 35a
LI (43)(15)
3.716
Trigonometric functions: complicated arguments
π/2
12.8
0
421
aπ
aπ
π
s 0,a (z) = cosec
[Ja (z) − J−a (z)]
sin (z cos x) cos ax dx = cos
2
4
2
π
aπ
= − sec
[Ea (z) + E−a (z)]
4
4
∞
(−1)k−1 z 2k−1
aπ = cos
2
(12 − a2 ) (32 − a2 ) . . . [(2k − 1)2 − a2 ]
k=1
π
13.
sin (z cos x) cos nx dx =
0
1
2
[a > 0]
π
sin (z cos x) cos nx dx = π sin
−π
π/2
sin (z cos x) cos[(2n + 1)x] dx = (−1)n
14.
0
π/2
15.11
sin (a cos x) tan x dx = si(a) +
0
π/2
2ν
16.
sin (z cos x) sin
0
π/2
17.7
0
√
π
x dx =
2
nπ
J n (z)
2
π
J 2n+1 (z)
2
π
2
2
z
WA 339
WA 30(8)
[a > 0]
ν
Γ ν+
1
2
GW (334)(55b)
BI (43)(17)
Hν (z)
Re ν > − 12
WA 358(1)
aπ
s −1,a (z)
cos (z cos x) cos ax dx = −a sin
2
aπ
aπ
π
π
[Ja (z) + J−a (z)] = cosec
[Ea (z) − E−a (z)]
= sec
4
2 4
2
∞
aπ
(−1)k−1 z 2k
1
= −a sin
− 2+
2
a
a2 (22 − a2 ) (42 − a2 ) . . . [(2k)2 − a2 ]
k=1
π
18.
cos (z cos x) cos nx dx =
0
1
2
π
cos (z cos x) cos nx dx = π cos
−π
π/2
cos (z cos x) cos 2nx dx = (−1)n ·
19.
0
√
π/2
20.
cos (z cos x) sin
2ν
x dx =
0
π
cos (z cos x) sin2μ x dx =
21.
0
ν
Γ ν+
μ
Γ μ+
1
2
1
2
J ν (z)
Re ν > − 12
J μ (z)
Re μ > − 12
[a > 0]
cos (a tan x) dx =
π −a
e
2
[a ≥ 0]
π/2
nπ
J n (z)
2
GW (334)(56b)
WA 30(9)
1 −a
e Ei(a) − ea Ei(−a)
2
0
2.
2
z
WA 339
π
J 2n (z)
2
sin (a tan x) dx =
π/2
1.
2
z
π
2
√
π
0
3.716
[a > 0]
(cf. 3.723 1)
WA 35, WH
WH
BI (43)(1)
BI (43)(2)
422
Trigonometric Functions
π/2
3.
sin (a tan x) sin 2x dx =
aπ −a
e
2
[a ≥ 0]
BI (43)(7)
cos (a tan x) sin2 x dx =
1 − a −a
πe
4
[a ≥ 0]
BI (43)(8)
cos (a tan x) cos2 x dx =
1 + a −a
πe
4
[a ≥ 0]
BI (43)(9)
[a > 0]
BI (43)(5)
0
π/2
4.
0
π/2
5.
0
π/2
6.
sin (a tan x) tan x dx =
0
π −a
e
2
π/2
cos (a tan x) tan x dx = −
7.
3.717
0
1 −a
e Ei(a) + ea Ei(−a)
2
[a > 0]
π/2
sin (a tan x) sin2 x tan x dx =
8.
0
[a ≥ 0]
(cf. 3.742 1)
BI (43)(3)
cos2 (a tan x) dx =
π
1 + e−2a
4
[a ≥ 0]
(cf. 3.742 3)
BI (43)(4)
π/2
0
π/2
sin2 (a tan x) cot2 x dx =
11.
0
π/2
12.
0
π −2a
e
+ 2a − 1
4
[a ≥ 0]
BI (43)(19)
dx
=C
1 − sec2 x cos (tan x)
tan x
π/2
13.
BI (43)(11)
π
1 − e−2a
4
0
10.
[a > 0]
BI (43)(6)
sin2 (a tan x) dx =
π/2
9.
2 − a −a
πe
4
(cf. 3.723 5)
sin (a cot x) sin 2x dx =
0
BI (51)(14)
aπ −a
e
2
[a ≥ 0]
(cf. 3.716 3)
In general, formulas 3.716 remain valid if we replace tan x in the argument of the sine or cosine with
cot x if we also replace sin x with cos x, cos x with sin x, hence tan x with cot x, cot x with tan x, sec x
with cosec x, and cosec x with sec x in the factors. Analogously,
π/2
π/2
π
dx
dx
=
= sin a
[a ≥ 0]
sin (a cosec x) sin (a cot x)
sin (a sec x) sin (a tan x)
3.717
cos x
sin x
2
0
0
BI (52)(11, 12)
3.718
π/2
1.
sin
0
π
p − a tan x tanp−1 x dx =
2
π/2
cos
0
π
π
p − a tan x tanp x dx = e−a
2
2
2
p < 1, p = 0, a ≥ 0
BI (44)(5, 6)
π/2
sin (a tan x − νx) sinν−2 x dx = 0
2.
[Re ν > 0,
a > 0]
NH 157(15)
0
3.
π/2
sin (n tan x + νx)
0
π
cosν−1 x
dx =
sin x
2
[Re ν > 0]
BI (51)(15)
3.722
Trigonometric and rational functions
π/2
cos (a tan x − νx) cosν−2 x dx =
4.
0
πe−a aν−1
Γ(ν)
423
[Re ν > 1,
a > 0]
LO V 153(112), NT 157(14)
π/2
5.
cos (a tan x + νx) cosν x dx = 2−ν−1 πe−a
[Re ν > −1,
a ≥ 0]
BI (44)(4)
0
π/2
cos (a tan x − γx) cosν x dx =
6.
0
π/2
7.
0
ν
(2a)
πa 2 W γ2 ,− ν+1
2
·
ν
γ+ν
+1
22
Γ 1+ 2
a > 0, Re ν > −1,
sin nx − sin (nx − a tan x)
cosn−1 x dx =
sin x
ν+γ
= −1, −2, . . .
2
π/2
[n = 0,
−a
π (1 − e ) [n = 1,
EH I 274(13)a
a > 0] ,
a ≥ 0]
LO V 153(114)
3.719
1.
π
6
sin (νx − z sin x) dx = π Eν (z)
WA 336(2)
cos (nx − z sin x) dx = π J n (z)
WH
cos (νx − z sin x) dx = π Jν (z)
WA 336(1)
0
π
2.
0
3.
π
0
3.72–3.74 Combinations of trigonometric and rational functions
3.721
∞
1.
0
∞
2.
1
3.
∞
8
1
3.722
∞
1.
0
2.11
3.
π
sin(ax)
dx = sign a
x
2
FI II 645
sin(ax)
dx = − si(a)
x
BI 203(1)
cos(ax)
dx = − ci(a)
x
BI 203(5)
sin(ax)
dx = ci(aβ) sin(aβ) − cos(aβ) si(aβ)
x+β
[|arg β| < π,
a > 0]
BI(16)(1), FI II 646a
∞
sin(ax)
dx = πeiaβ
−∞ x + β
∞
cos(ax)
dx = − sin(aβ) si(aβ) − cos(aβ) ci(aβ)
x+β
0
[a > 0,
Im β > 0]
[|arg β| < π,
a > 0]
ET I 8(7), BI(160)(2)
424
Trigonometric Functions
4.
8
5.10
7.10
∞
∞
0
0
4.
5.11
6.
7.
8.
9.
10.
11.
[a > 0,
Im β > 0]
sin(ax)
1 −aβ
e
dx =
Ei(aβ) − eaβ Ei(−aβ)
2
2
β +x
2β
[a > 0,
β > 0]
cos(ax)
π −aβ
e
dx =
2
2
β +x
2β
[a ≥ 0,
Re β > 0]
ET I 65(14), BI(160)(3)
FI II 741, 750, ET I 8(11), WH
∞
3.
β not real and positive]
ET I 8(8), BI(161)(2)a
2.
Im β > 0]
∞
0
β not real and positive]
FI II 646, BI(161)(1)
cos(ax)
dx = −iπeiaβ
−∞ β − x
3.723 1.11
Im β > 0]
∞
sin(ax)
dx = −πeiaβ
[a > 0,
−∞ β − x
∞
cos(ax)
dx = − cos(aβ) ci(aβ) + sin(aβ) [si(aβ) + π]
β−x
0
[a > 0,
8.11
∞
cos(ax)
dx = −iπeiaβ
[a > 0,
−∞ x + β
∞
sin(ax)
dx = sin(βa) ci(βa) − cos(βa) [si(βa) + π]
β−x
0
[a > 0,
6.8
3.723
x sin(ax)
π
dx = e−aβ
β 2 + x2
2
[a > 0,
Re β > 0]
FI II 741, 750, ET I 65(15), WH
∞
x sin(ax)
dx = πe−aβ
[a > 0, Re β > 0]
BI (202)(10)
2
2
−∞ β + x
∞
x cos(ax)
1
dx = − e−aβ Ei(aβ) + eaβ Ei(−aβ)
[a > 0, β > 0]
BI (160)(6)
2
2
β +x
2
0
∞
sin[a(b − x)]
π
dx = e−ac sin(ab)
[a > 0, b > 0, c > 0]
LI (202)(9)
2 + x2
c
c
−∞
∞
cos[a(b − x)]
π
dx = e−ac cos(ab)
[a > 0, b > 0, c > 0]
LI (202)(11)a
2 + x2
c
c
−∞
∞
π ,
sin(ax)
1+
sin(aβ)
ci(aβ)
−
cos(aβ)
si(aβ)
+
dx
=
2
2
β
2
0 β −x
BI (161)(3)
[|arg β| < π, a > 0]
∞
cos(ax)
π
sin(ab)
[a > 0, b > 0]
dx =
BI(161)(5), ET I 9(15)
2 − x2
b
2b
0
∞
x sin(ax)
π
[a > 0]
dx = − cos(ab)
FI II 647, ET II 252(45)
2 − x2
b
2
0
∞
+
x cos(ax)
π,
dx
=
cos(aβ)
ci(aβ)
+
sin(aβ)
si(aβ)
+
β 2 + x2
2
0
BI (161)(6)
[|arg β| < π, a > 0]
3.726
Trigonometric and rational functions
12.
3.724
1.
2.
3.
3.725
425
∞
cos(ab) − 1
sin(ax)
dx = π
x(x
−
b)
b
−∞
[a > 0,
b > 0]
√ 2
cq − b
sin(aq) + c cos(aq) πe−a p−q
p − q2
a > 0, p > q 2
∞
√
b + cx
b − cq
−a p−q 2
cos(ax)
dx
=
cos(aq)
+
c
sin(aq)
πe
2
p − q2
−∞ p + 2qx + x
a > 0, p > q 2
∞
cos[(b − 1)t] − x cos(bt)
cos(ax) dx = πe−a sin t sin (bt + a cos t)
1 − 2x cos t + x2
−∞
a > 0, t2 < π 2
∞
b + cx
sin(ax) dx =
2
−∞ p + 2qx + x
∞
1.
0
∞
2.
0
∞
3.
0
ET II 252(44)
π sin(ax) dx
=
1 − e−aβ
2
2
2
x (β + x )
2β
[Re β > 0,
π
sin(ax) dx
= 2 (1 − cos(ab))
2
2
x (b − x )
2b
[a > 0]
π −βb
sin(ax) cos(bx)
dx =
e
sinh(aβ)
x (x2 + β 2 )
2β 2
π
π
= − 2 e−aβ cosh(bβ) + 2
2β
2β
a > 0]
BI (202)(12)
BI (202)(13)
BI (202)(14)
BI (172)(1)
BI (172)(4)
[0 < a < b]
[a > b > 0]
ET I 19(4)
3.726
∞
1.11
0
2.7
0
x sin(ax) dx
± b2 x + bx2 ± x3
1 + −ab
π ,
=±
e Ei(ab) − eab Ei(−ab) − 2 ci(ab) sin(ab) + 2 cos(ab) si(ab) +
4b
2
πe−ab − π cos(ab)
+
4b
[a > 0, b > 0; if the lower sign is taken, then the integral is a principal value integral]
b3
ET I 65(21)a, BI(176)(10, 13)
∞
x2 sin(ax) dx
b3 ± b2 x + bx2 ± x3
=
[a > 0,
b > 0;
1 + ab
π ,
e Ei(−ab) − e−ab Ei(ab) + 2 ci(ab) sin(ab) − 2 cos(ab) si(ab) +
4
2
−ab
+ cos(ab)
±π e
if the lower sign is taken, then the integral is a principal value integral]
ET I 66(22), BI(176)(11, 14)
426
3.727
Trigonometric Functions
∞
1.
0
∞
2.8
0
∞
3.
0
4.
5.
6.11
√
cos(ax)
π 2
ab
dx =
exp − √
4
4
3
b +x
4b
2
ab
ab
cos √ + sin √
2
2
[a > 0,
+
π
sin(ax)
1
dx = 3 2 sin(ab) ci(ab) − 2 cos(ab) si(ab) +
b4 − x4
4b
2
,
−ab
ab
Ei(ab) − e Ei(−ab)
+e
b > 0]
BI(160)(25)a, ET I 9(19)
[a > 0,
b > 0]
BI (161)(12)
cos(ax)
π dx = 3 e−ab + sin(ab)
b4 − x4
4b
[a > 0,
∞
7.
0
b > 0]
BI (161)(13)
(cf. 3.723 5 and 3.723 11)
ab
ab
cos √ − sin √
2
2
[a > 0,
b > 0]
∞
x cos(ax)
π sin(ab) − e−ab
dx =
4
4
b −x
4b
9.
0
√
ab
x2 cos(ax)
π 2
exp − √
dx =
b4 + x4
4b
2
b > 0]
BI (160)(23)a
x2 sin(ax) dx
1+
2 sin(ab) ci(ab)
=
4
4
b −x
4b
,
π
−2 cos(ab) si(ab) +
− e−ab Ei(ab) + eab Ei(−ab)
2
[a > 0, b > 0]
0
BI (161)(16)
b > 0]
∞
8.11
12.7
(cf. 3.723 2 and 3.723 9)
x sin(ax)
ab
π
ab
sin √
[a > 0,
dx = 2 exp − √
4 + x4
b
2b
2
2
0
∞
x sin(ax)
π dx = 2 e−ab − cos(ab)
[a > 0,
4
4
b −x
4b
0
∞
x cos(ax)
1
π
dx
=
2 cos(ab) ci(ab) + 2 sin(ab) si(ab) +
b4 − x4
4b2
2
0
−ab
ab
− e Ei(ab) − e Ei(−ab)
11.
b > 0]
∞
[a > 0,
10.
3.727
BI (161)(17)
BI (160)(26)a
BI (161)(14)
2
[a > 0,
b > 0]
BI (161)(18)
b > 0]
BI (160)(24)
b > 0]
BI (161)(15)
b > 0]
BI(161)(19)
∞
ab
x3 sin(ax)
ab
π
cos √
[a > 0,
dx = exp − √
4 + x4
b
2
2
2
0
∞ 3
x sin(ax)
−π −ab
e
dx =
− cos(ab)
[a > 0,
4
4
b −x
4
0
∞ 3
π
x cos(ax) dx 1 +
= 2 cos(ab) ci(ab) + 2 sin(ab) si(ab) +
4 − x4
b
4
2
0
,
−ab
ab
+e Ei(ab) + e Ei(−ab)
[a > 0,
3.729
Trigonometric and rational functions
∞
13.
3
(x2 + b2 )
0
∞
14.
x3 sin ax
4
(x2 + b2 )
0
3.728
x3 sin ax
∞
1.
0
dx =
πe−ab 3a − ba2
16b
dx =
πe−ab a 3 + 3ab − a2 b2
3
96b
π βe−aγ − γe−aβ
cos(ax) dx
=
(β 2 + x2 ) (γ 2 + x2 )
2βγ (β 2 − γ 2 )
∞
π e−aβ − e−aγ
x sin(ax) dx
=
2
2
2
2
2 (γ 2 − β 2 )
0 (β + x ) (γ + x )
[a > 0,
427
Re β > 0,
Re γ > 0]
BI (175)(1)
2.
∞
π βe−aβ − γe−aγ
x2 cos(ax) dx
=
2
2
2
2
2 (β 2 − γ 2 )
0 (β + x ) (γ + x )
[a > 0,
Re β > 0,
Re γ > 0]
BI (174)(1)
3.
∞
π β 2 e−aβ − γ 2 e−aγ
x3 sin(ax) dx
=
2
2
2
2
2 (β 2 − γ 2 )
0 (β + x ) (γ + x )
[a > 0,
Re β > 0,
Re γ > 0]
BI (175)(2)
4.
(b2
0
∞
6.
0
7.
8.
9.
3.729
π (b sin(ac) − c sin(ab))
cos(ax) dx
=
2
2
2
− x ) (c − x )
2bc (b2 − c2 )
π (cos(ab) − cos(ac))
x sin(ax) dx
=
(b2 − x2 ) (c2 − x2 )
2 (b2 − c2 )
∞
1.
∞
2.
+
x sin(ax) dx
2
∞
3.
cos(px)
0
4.
0
=
2
x2 )
(b2 + x2 )
0
cos(ax) dx
(b2
0
Re γ > 0]
[a > 0,
b > 0,
c > 0]
[a > 0]
BI (175)(3)
BI (174)(3)
∞
x2 cos(ax) dx
π (c sin(ac) − b sin(ab))
=
2
2
2
2
2 (b2 − c2 )
0 (b − x ) (c − x )
2
∞
π b cos(ab) − c2 cos(ac)
x3 sin(ax) dx
=
2
2
2
2
2 (b2 − c2 )
0 (b − x ) (c − x )
∞
π e−ac − cos ba
x sin ax
dx =
2
2
2
2
2
a2 + c 2
0 (b − x ) (c + x )
Re β > 0,
BI (174)(2)
∞
5.
[a > 0,
∞
π
(1 + ab)e−ab
4b3
=
π −ab
ae
4b
1 − x2
(1 + x2 )
x3 sin(ax) dx
(b2 + x2 )
2
=
2
dx =
[a > 0,
b > 0,
c > 0]
BI (175)(4)
[a > 0,
b > 0,
c > 0]
BI (174)(4)
[a > 0,
c > 0,
b real]
[a > 0,
b > 0]
BI (170)(7)
[a > 0,
b > 0]
BI (170)(3)
πp −p
e
2
π
(2 − ab)e−ab
4
BI (43)(10)a
[a > 0,
b > 0]
BI (170)(4)
428
3.731
1.
Trigonometric Functions
√
√
Notation: 2A2 = b4 + c2 + b2 , 2B 2 = b4 + c2 − b2 ,
∞
cos(ax) dx
π e−aA (B cos(aB) + A sin(aB))
√
=
2
2
2
2
2c
b 4 + c2
0 (x + b ) + c
2.
∞
x sin(ax) dx
=
2
b2 )
π −aA
e
sin(aB)
2c
(x2 +
+ c2
∞ 2
2
x + b cos(ax) dx
0
3.
3.731
(x2 +
0
∞
4.
x x2 + b
(x2
0
3.732 1.
∞
0
∞
∞
3.
0
∞
4.
+
c2
BI (176)(3)
[a > 0,
b > 0,
c > 0]
BI (176)(1)
[a > 0,
b > 0,
c > 0]
BI (176)(4)
[a > 0,
b > 0,
c > 0]
BI (176)(2)
π e−aA (A cos(aB) − B sin(aB))
√
2
b 4 + c2
=
π −aA
e
cos(aB)
2
ET I 65(16)
[a > 0,
∞
1.
0
∞
2.
0
3.
0
|Im γ| < Re β]
γ+x
γ−x
− 2
sin(ax) dx = πe−aβ cos(aγ)
β 2 + (γ + x)2
β + (γ − x)2
[a > 0, Re β > 0,
ET I 8(13)
γ + iβ is not real]
LI (175)(17)
γ+x
γ−x
+ 2
cos(ax) dx = πe−aβ sin(aγ)
2
+ (γ + x)
β + (γ − x)2
[a > 0,
3.733
γ + iβ is not real]
1
1
π
+ 2
cos(ax) dx = e−aβ cos(aγ)
2
2
+ (γ − x)
β + (γ + x)
β
β2
0
sin(ax) dx
2
b2 )
c > 0]
β2
0
+
2
+ c2
=
b > 0,
1
1
π
−
sin(ax) dx = e−aβ sin(aγ)
β 2 + (γ − x)2
β 2 + (γ + x)2
β
[a > 0, Re β > 0,
2.
2
b2 )
[a > 0,
|Im a| < Re β]
LI (176)(21)
cos(ax) dx
π
sin (t + ab sin t)
= 3 exp (−ab cos t)
x4 + 2b2 x2 cos 2t + b4
2b
sin 2t
+
a > 0, b > 0,
|t| <
π,
2
x sin(ax) dx
π
sin (ab sin t)
= 2 exp (−ab cos t)
x4 + 2b2 x2 cos 2t + b4
2b
sin 2t+
a > 0,
|t| <
π,
2
b > 0,
BI (176)(7)
BI(176)(5), ET I 66(23)
∞
x4
sin (t − ab sin t)
x2 cos(ax) dx
π
exp (−ab cos t)
=
+ 2b2 x2 cos 2t + b4
2b
sin 2t
+
a > 0,
b > 0,
|t| <
π,
2
BI (176)(8)
3.736
Trigonometric and rational functions
4.
5.
3.734
429
∞
x3 sin(ax) dx
sin (2t − ab sin t)
π
= exp (−ab cos t)
4
2
2
4
2
sin 2t
0 x + 2b x cos 2t + b
+
π,
a > 0, b > 0, |t| <
2
∞
π
sin(ax) dx
sin (2t + ab sin t)
= 4 1 − exp (−ab cos t)
4
2 2
4
2b
sin 2t
0 x (x + 2b x cos 2t + b )
+
π,
a > 0, b > 0, |t| <
2
π
sin(ax) dx
ab
ab
1.
= 4 1 − exp − √
cos √
[a > 0, b > 0]
4
4
2b
2
2
0 x (b + x )
∞
π sin(ax) dx
= 4 2 − e−ab − cos(ab)
2.
[a > 0, b > 0]
4
4
4b
0 x (b − x )
∞
sin(ax) dx
π
1 −ab
3.735
= 4 1 − e (2 + ab)
[a > 0, b > 0]
2
2 2
2b
2
0 x (b + x )
3.736
∞
π cos(ax) dx
1.
= 5 sin(ab) + (2 + ab)e−ab
2
2
4
4
8b
0 (b + x ) (b − x )
∞
(b2
0
∞
3.
0
∞
4.
∞
5.
6.
0
BI (172)(10)
WH, BI (172)(22)
[a > 0,
b > 0]
BI (176)(5)
[a > 0,
b > 0]
BI (174)(5)
[a > 0,
b > 0]
BI (175)(6)
[a > 0,
b > 0]
BI (174)(6)
[a > 0,
b > 0]
BI (175)(7)
[a > 0,
b > 0]
BI (174)(7)
π x sin(ax) dx
= 4 (1 + ab)e−ab − cos(ab)
2
4
4
+ x ) (b − x )
8b
π x cos(ax) dx
= 3 sin(ab) − abe−ab
(b2 + x2 ) (b4 − x4 )
8b
2
π x3 sin(ax) dx
= 2 (1 − ab)e−ab − cos(ab)
2
4
4
+ x ) (b − x )
8b
π x cos(ax) dx
=
sin(ab) + (ab − 2)e−ab
2
4
4
+ x ) (b − x )
8b
4
(b2
0
BI (172)(7)
(b2
0
BI (176)(22)
∞
2.
BI (176)(6)
∞
π
x sin(ax) dx
=
(ab − 3)e−ab − cos(ab)
2
4
4
+ x ) (b − x )
8
5
(b2
430
3.737
Trigonometric Functions
∞
1.8
0
2.
3.
3.737
(2n − k − 2)!(2ab)k
cos(ax) dx
πe−ab
=
n
(2b)2n−1 (n − 1)!
k!(n − k − 1)!
(b2 + x2 )
√
k=0
n−1
n−1
e−ab p
d
(−1)
π
= 2n−1
√
2b
(n − 1)! dpn−1
p
p=1
n−1
n−1
−abp
e
d
π
(−1)
= 2n−1
2b
(n − 1)! dpn−1 (1 + p)n p=1
[a > 0, b > 0]
n−1
GW(333)(67b), WA 209, WA 192
n−1
πae−aβ (2n − k − 2)!(2aβ)k
2
2 n+1
22n n!β 2n−1
k!(n − k − 1)!
0 (x + β )
k=0
π −aβ
= e
[n = 0, β ≥ 0]
2
[a > 0, Re β > 0]
GW (333)(66c)
∞
sin(ax) dx
π
e−aβ
=
1 − n Fn (aβ)
n+1
2n+2
2
2
2β
2 n!
0 x (β + x )
a > 0, Re β > 0, F0 (z) = 1, F1 (z) = z + 2, . . . , Fn (z) = (z + 2n)Fn−1 (z) − zFn−1
(z)
∞
x sin(ax) dx
(b2
0
∞
5.
∞
6.
+
4
x2 )
=
x sin ax
n+1
+ β2)
πa
(1 + ab)e−ab
16b3
[a > 0,
b > 0]
BI(170)(5), ET I 67(35)a
πa 3 + 3ab + a2 b2 e−ab
[a > 0, b > 0] BI(170)(6), ET I 67(35)a
96b5
πe−aβ
dx = 2n
2n−1 (2n − 3)!!(2 − βa)
2 n!β 2n−2
&
n−1
(2n − k − 2)!2k (βa)k−1 k(k + 1) − 2(k + 1)βa + β 2 a2
−
k!(n − k − 1)!
=
3
(x2
0
+
3
x2 )
x sin(ax) dx
(b2
0
=
GW (333)(66e)
∞
4.
x sin(ax) dx
k=1
3.738
∞
1.
0
n
xm−1 sin(ax)
πβ m−2n (2k − 1)π
dx
=
−
exp
−aβ
sin
x2n + β 2n
2n
2n
k=1
(2k − 1)mπ
(2k − 1)π
+ aβ cos
× cos
2n
2n
+
[m is even] ,
2.
0
∞
a > 0,
|arg β| <
π
,
2n
n
xm−1 cos(ax)
πβ m−2n (2k − 1)π
dx
=
exp
−aβ
sin
x2n + β 2n
2n
2n
k=1
(2k − 1)mπ
(2k − 1)π
+ aβ cos
× sin
2n
2n
,
+
π
, 0 < m < 2n + 1
[m is odd] ,
a > 0, |arg β| <
2n
,
0 < m ≤ 2n
ET I 67(38)
BI(160)(29)a, ET I 10(29)
3.741
3.739
Trigonometric and rational functions
∞
1.
sin(ax) dx
x (x2 + 22 ) (x2 + 42 ) . . . (x2 + 4n2 )
% n−1
&
π(−1)n
k 2n
2(k−n)a
n 2n
=
(−1)
+ (−1)
2
e
k
n
(2n)!22n+1
0
k=0
[a > 0,
∞
2.
(x2
0
+
12 ) (x2
(x2
0
π2−2n−1
2
(2n + 1) (n!)
+
12 ) (x2
∞
4.
0
∞
0
∞
2.
0
3.
0
[a = 0,
n ≥ 0]
n ≥ 0]
LI (174)(9)
n
2n
π21−2n cos ax dx
=
(−1)k k
e−2ak
n−k
(x2 + 22 ) (x2 + 42 ) . . . (x2 + 4n2 )
(2n)!
k=1
1.
n ≥ 0]
x sin(ax) dx
+ 32 ) . . . [x2 + (2n + 1)2 ]
n
2n + 1
π(−1)n
(−1)k
=
(2n − 2k + 1)e(2k−2n−1)a
k
(2n + 1)!22n+1
[a > 0,
[a ≥ 0,
k=0
k=0
3.741
LI(174)(8)
BI(175)(8)
∞
3.
n ≥ 0]
cos(ax) dx
+ 32 ) . . . [x2 + (2n + 1)2 ]
n
2n + 1 (2k−2n−1)a
π (−1)n
(−1)k
=
e
k
(2n + 1)! 22n+1
=
431
sin(ax) sin(bx)
1
dx = ln
x
4
sin(ax) cos(bx)
π
dx =
x
2
π
=
4
=0
a+b
a−b
[n ≥ 1,
a ≥ 0]
[a > 0,
b > 0,
2
a = b]
FI II 647
[a > b ≥ 0]
[a = b > 0]
[b > a ≥ 0]
FI II 645
∞
sin(ax) sin(bx)
aπ
dx =
x2
2
bπ
=
2
[0 < a ≤ b]
[0 0,
b > 0,
Re β > 0]
[β > 0,
a ≥ b ≥ 0]
[β > 0,
b ≥ a ≥ 0]
BI(163)(1)a, GW(333)(71c)
∞
x cos(ax) cos(bx)
1
dx = − eaβ ebβ Ei [− β(a + b)] + e−bβ Ei [β(b − a)]
2
2
β +x
4
1
− e−aβ ebβ Ei [β(a − b)] + e−bβ Ei [β(a + b)]
4
=∞
[a = b]
[a = b]
BI (163)(2)
∞
5.
0
x sin(ax) cos(bx)
π
dx = e−aβ cosh(bβ)
x2 + β 2
2
π −2aβ
= e
4
π
= − e−bβ sinh(aβ)
2
[0 0,
BI (162)(3)
0
7.
a ≥ b ≥ 0]
,
cos(ax) cos(bx)
π + −|a−b|β
e
dx =
+ e−(a+b)β
2
2
β +x
4β
π −aβ
=
e
cosh bβ
2β
π −bβ
e
=
cosh aβ
2β
4.
[β > 0,
∞
0
Re β > 0]
BI(162)(1)a, GW(333)(71a)
3.
b > 0,
sin(ax) cos(bx)
1 −aβ bβ
e
e Ei [β(a − b)] + e−bβ Ei [β(a + b)]
dx =
2
2
β +x
4β
1
− eaβ ebβ Ei [− β(a + b)] + e−bβ Ei [β(b − a)]
4β
0
[a > 0,
∞
2.
sin(ax) sin(bx)
π
dx =
e−|a−b|β − e−(a+b)β
β 2 + x2
4β
π −aβ
e
sinh bβ
=
2β
π −bβ
e
=
sinh aβ
2β
3.742
sin(ax) sin(bx)
π
dx = − cos(ap) sin(bp)
2
2
p −x
2p
π
= − sin(2ap)
4p
π
= − sin(ap) cos(bp)
2p
[a > b > 0]
[a = b > 0]
[b > a > 0]
BI (166)(1)
∞
sin(ax) cos(bx)
π
x dx = − cos(ap) cos(bp)
p2 − x2
2
π
= − cos(2ap)
4
π
= sin(ap) sin(bp)
2
[a > b > 0]
[a = b > 0]
[b > a > 0]
BI (166)(2)
3.747
Trigonometric and rational functions
∞
8.
0
cos(ax) cos(bx)
π
sin(ap) cos(bp)
dx =
p2 − x2
2p
π
=
sin(2ap)
4p
π
=
cos(ap) sin(bp)
2p
433
[a > b > 0]
[a = b > 0]
[b > a > 0]
BI (166)(3)
3.743
∞
1.
0
∞
2.
0
∞
3.
0
4.
5.6
Re β > 0]
ET I 80(21)
x dx
sin(ax)
π sinh(aβ)
· 2
=− ·
2
cos(bx) x + β
2 cosh(bβ)
[0 < a < b,
Re β > 0]
ET I 81(30)
x dx
cos(ax)
π cosh(aβ)
·
= ·
sin(bx) x2 + β 2
2 sinh(bβ)
[0 < a < b,
Re β > 0]
ET I 23(37)
[0 < a < b,
Re β > 0]
ET I 23(36)
dx
cos(ax)
π cosh(aβ)
· 2
·
=
2
cos(bx)
x
+
β
2β
cosh(bβ)
0
∞
dx
sin(ax)
· 2
PV
=0
sin x
b − x2
0
π
= sin(a − 1)b
b
∞
sin(ax)
cos(bx)
0 ∞
sin(ax)
cos(bx)
0
3.7443
3.7453
xn+1
0
∞
2.
0
n
dx -
∞
1.
3.747
[0 < a < b,
∞
3.746
dx
sin(ax)
π sinh(aβ)
·
·
=
sin(bx) x2 + β 2
2β sinh(bβ)
π/2
0
π/2
0
∞
3.
0
dx
π sinh(aβ)
=
·
x (x2 + β 2 )
2β 2 cosh(bβ)
dx
·
=0
x (c2 − x2 )
x dx
=
sin x
k=1
m
2
0
0
ET I 82(32)
[0 < a < b,
c > 0]
ET I 82(31)
⎡
⎣a >
n
&
ak ,
ak > 0
k=1
n
k=1
|ak | +
m
FI II 646
⎤
|bj |⎦
WH
j=1
&
∞
7
22k−1 − 1
1
+
ζ(2k) = 2πG − ζ(3)
m
42k−1 (m + 2k)
2
k=1
π/2 π
%
Re β > 0]
a0 >
k=1
π
xm
dx =
sin x
2
[0 < a < b,
%
n
π sin (ak x) =
ak
2
− x dx
= 2G
cos x
π
x dx
=
(x2 + b2 ) sin(ax)
2 sinh(ab)
[m = 2]
LI (206)(2)
BI(204)(18), BI(206)(1), GW(333)(32)
[b > 0]
GW (333)(79c)
π
x tan x dx = −π ln 2
4.
[b real, b/π ∈ Z]
·
k=1
2.
if 1 ≤ a ≤ 2
n
m
n
sin(ax)
π dx
sin
(a
x)
cos
(b
x)
=
ak
k
j
xn+1
2
j=1
1.7
k=0
if 0 ≤ a ≤ 1
BI (218)(4)
434
Trigonometric Functions
3.748
π/2
x tan x dx = ∞
5.
BI (205)(2)
0
π/4
x tan x dx = −
6.
0
π/2
7.
x cot x dx =
0
π/4
8.
π/2
9.
0
π
ln 2
2
tan ax
0
π/2
11.
0
π
dx
=
x
2
xm tan x dx =
0
π/2
0
xm cot x dx =
0
∞
1.
0
3.
1
2
π
x cot x dx =
2
π/4
3.
GW(333)(33b), BI(218)(12)
[a > 0]
p
2.
BI (204)(2)
LO V 279(5)
π
x cot x
dx = ln 2
cos 2x
4
π/4
1.
2.
FI II 623
1
π
ln 2 + G = 0.7301810584 . . .
8
2
π
1 π π
π
− x tan x dx =
− x tan x dx = ln 2
2
2 0
2
2
∞
10.
3.749
BI (204)(1)
x cot x dx =
0
3.748
1
π
ln 2 + G = 0.1857845358 . . .
8
2
1
2
π
4
BI (206)(12)
∞
m
k=1
4k − 1 ζ(2k)
42k−1 (m + 2k)
LI (204)(5)
∞
1
1
−2
ζ(2k)
p
4k (p + 2k)
k=1
∞
π m 2
ζ(2k)
−
4
m
42k−1 (m + 2k)
p
x tan(ax) dx
π
= 2ab
x2 + b2
e +1
LI (205)(7)
LI (204)(6)
k=1
[a > 0,
b > 0]
GW (333)(79a)
∞
x cot(ax) dx
π
[a > 0, b > 0]
= 2ab
2 + b2
x
e
−1
0
∞
∞
∞
x tan(ax) dx
x cot(ax) dx
x cosec(ax) dx
=
=
=∞
2
2
2
2
b −x
b −x
b2 − x2
0
0
0
GW (333)(79b)
BI (161)(7, 8, 9)
3.75 Combinations of trigonometric and algebraic functions
3.751
∞
1.
0
2.9
0
∞
sin(ax) dx
√
=
x+β
cos(ax) dx
√
=
x+β
,
π +
cos(aβ) − sin(aβ) + 2 C
aβ sin(aβ) − 2 S
aβ cos(aβ)
2a
[a > 0,
|arg β| < π]
ET I 65(12)a
,
π +
aβ cos(aβ) − 2 S
aβ sin(aβ)
cos(aβ) + sin(aβ) − 2 C
2a
[a > 0,
|arg β| < π]
ET I 8(9)a
3.755
Trigonometric and algebraic functions
∞
3.
u
∞
4.
u
3.752
1.
1
8
sin(ax)
√
dx =
x−u
cos(ax)
√
dx =
x−u
1
1
1.8
0
2.
3.
4.
5.
3.754
u > 0]
ET I 65(13)
π
[cos(au) − sin(au)]
2a
[a > 0,
u > 0]
ET I 8(10)
π
(−1)k a2k+1
=
H1 (a)
(2k − 1)!!(2k + 3)!!
2a
[a > 0]
0
[a > 0,
k=0
2.
3.753
π
[sin(au) + cos(au)]
2a
∞
sin(ax) 1 − x2 dx =
0
KU 65(6)a
∞
sin(ax) dx (−1)k a2k+1
π
√
=
2 = 2 H0 (a)
2
1−x
[(2k + 1)!!]
k=0
∞
0
∞
2.
0
0
3.755
∞
0
2.
0
WA 30(7)a
[a > 0]
WA 200(14)
WA 200(15)
[a > 0]
WA 30(6)
sin(ax) dx
π
= [I 0 (aβ) − L0 (aβ)]
2
2
2
β +x
[a > 0,
Re β > 0]
cos(ax) dx
= K 0 (aβ)
β 2 + x2
[a > 0,
Re β > 0]
x sin(ax)
<
dx = a K 0 (aβ)
3
2
2
(β + x )
<
1.
BI (149)(9)
ET I 66(26)
WA 191(1), GW(333)(78a)
∞
3.
[a > 0]
1
1.
BI (149)(6)
π
J 1 (a)
cos(ax) 1 − x2 dx =
2a
cos(ax) dx
π
√
= J 0 (a)
2
2
1−x
0
∞
sin(ax) dx
π
√
= J 0 (a)
2
2
x −1
1
∞
cos(ax)
π
√
dx = − Y 0 (a)
2−1
2
x
1
1
x sin(ax)
π
√
dx = J 1 (a)
2
2
1−x
0
435
∞
<
x2 + β 2 − β sin(ax) dx
=
x2 + β 2
x2 + β 2 + β cos(ax) dx
=
x2 + β 2
[a > 0,
π −aβ
e
2a
[a > 0]
π −aβ
e
2a
[a > 0,
Re β > 0]
ET I 66(27)
ET I 66(31)
Re β > 0]
ET I 10(25)
436
3.756
Trigonometric Functions
∞
1.
0
2.
x
n
2 −1
cos(ax)
0
3.757
∞
0
2.
n
-
ak > 0,
a>
%
cos (ak x) dx = 0
∞
11
0
sin(ax)
√
dx =
x
cos(ax)
√
dx =
x
n
&
ak
k=2
ak > 0,
a>
k=1
1.11
%
n
sin(ax) sin (ak x) dx = 0
n
x 2 −1 k=2
∞
3.756
n
ET I 80(22)
&
ak
ET I 22(26)
k=1
π
2a
[a > 0]
BI (177)(1)
π
2a
[a > 0]
BI (177)(2)
3.76–3.77 Combinations of trigonometric functions and powers
3.761
1
xμ−1 sin(ax) dx =
1.
0
−i
[ 1 F 1 (μ; μ + 1; ia) −
2μ
1F 1
(μ; μ + 1; −ia)]
[a > 0,
u
∞
3.
1
μ = 0]
ET I 68(2)a
∞
2.8
Re μ > −1,
π
i − π iμ
e 2 Γ(μ, iu) − e 2 iμ Γ(μ, −iu)
xμ−1 sin x dx =
2
sin(ax)
a2n−1
dx
=
x2n
(2n − 1)!
%2n−1
(2n − k − 1)!
k=1
a2n−k
[Re μ < 1]
sin a + (k − 1)
&
π
+ (−1)n ci(a)
2
[a > 0]
∞
4.
xμ−1 sin(ax) dx =
0
μπ
2
π sec
Γ(μ)
μπ
= μ
sin
aμ
2
2a Γ(1 − μ)
[a > 0;
EH II 149(2)
LI (203)(15)
0 < |Re μ| < 1]
FI II 809a, BI(150)(1)
π
m/2
m!
(−1)n+1 (nπ)m−2k
(−1)k
nm+1
(m − 2k)!
k=0=
>
= >
m!
m−2 m
m/2
2 −1
−(−1)
nm+1
xm sin(nx) dx =
5.10
0
GW(333)(6)
1
xμ−1 cos(ax) dx =
6.8
0
1
[ 1 F 1 (μ; μ + 1; ia) +
2μ
1F 1
(μ; μ + 1; −ia)]
[a > 0,
∞
7.
u
Re μ > 0]
ET I 11(2)
π
1 − π iμ
e 2 Γ(μ, iu)s + e 2 iμ Γ(μ, −iu)
xμ−1 cos x dx =
2
[Re μ < 1]
EH II 149(1)
3.763
Trigonometric functions and powers
∞
8.
1
437
% 2n
&
cos(ax)
a2n (2n − k)!
π
+ (−1)n+1 ci(a)
dx =
cos a + (k − 1)
x2n+1
(2n)!
a2n−k+1
2
k=1
[a > 0]
∞
9.8
xμ−1 cos(ax) dx =
0
μπ
2
π cosec
Γ(μ)
μπ
= μ
cos
μ
a
2
2a Γ(1 − μ)
[a > 0,
LI (203)(16)
0 < Re μ < 1]
FI II 809a, BI(150)(2)
π
(−1)n
nm+1
(m−1)/2
m!
(nπ)m−2k−1
(m − 2k − 1)!
k=0
(m+1)/2 2[(m + 1)/2] − m
+(−1)
· m!
nm+1
xm cos(nx) dx =
10.
0
(−1)k
GW (333)(7)
m/2
π/2
m
11.
x cos x dx =
0
(−1)k
k=0
m!
(m − 2k)!
m−2k
π
2
+ (−1)
m/2
2
:m;
2
− m m!
GW (333)(9c)
2nπ
xm cos kx dx = −
12.
0
3.762
m−1
j=0
∞
1.
j!
k j+1
xμ−1 sin(ax) sin(bx) dx =
0
m
j+1
π
(2nπ)m−j cos
j
2
BI (226)(2)
,
+
μπ
1
−μ
cos
Γ(μ) |b − a| − (b + a)−μ
2
2
[a > 0, b > 0,
a = b,
−2 < Re μ < 1]
(for μ = 0, see 3.741 1, for μ = −1, see 3.741 3)
BI(149)(7), ET I 321(40)
∞
2.
[a > 0,
∞
3.
,
μπ
1
−μ
sin
Γ(μ) (a + b)−μ + |a − b| sign(a − b)
2
2
b > 0, |Re μ| < 1]
(for μ = 0 see 3.741 2) BI(159)(8)a, ET I 321(41)
xμ−1 sin(ax) cos(bx) dx =
0
+
xμ−1 cos(ax) cos(bx) dx =
0
+
,
μπ
1
−μ
cos
Γ(μ) (a + b)−μ + |a − b|
2
2
[a > 0, b > 0,
0 < Re μ < 1]
ET I 20(17)
3.763
1.
0
∞
νπ
sin(ax) sin(bx) sin(cx)
1
ν−1
cos
Γ(1
−
ν)
(c + a − b)
dx
=
− (c + a + b)ν−1
xν
4
2
ν−1
ν−1
− |c − a + b|
sign(a − b − c) + |c − a − b|
sign(a + b − c)
[c > 0,
0 < Re ν < 4,
ν = 1, 2, 3,
a ≥ b > 0]
GW(333)(26a)a, ET I 79(13)
438
Trigonometric Functions
∞
2.
0
∞
3.
0
∞
4.
0
3.764 1.
∞
sin(ax) sin(bx) sin(cx)
dx = 0
x
π
=
8
π
=
4
∞
2.
∞
0
[c = a − b and c = a + b]
[a − b < c < a + b]
[a ≥ b > 0,
sin(ax) sin(bx) sin(cx)
πbc
dx =
x3
2
πbc π(a − b − c)2
−
=
2
8
xp sin(ax + b) dx =
1
pπ
Γ(1 + p) cos b +
ap+1
2
xp cos(ax + b) dx = −
0
3.765 1.10
[c < a − b and c > a + b]
c > 0]
FI II 645
sin(ax) sin(bx) sin(cx)
1
dx = (c + a + b) ln(c + a + b)
2
x
4
1
1
− (c + a − b) ln(c + a − b) − |c − a − b| ln |c − a − b|
4
4
1
× sign(a + b − c) + |c − a + b| ln |c − a + b| sign(a − b − c)
4
[a ≥ b > 0, c > 0] BI(157)(8)a, ET I 79(11)
0
3.764
[0 < c < a − b and c > a + b]
[a − b < c < a + b]
[a ≥ b > 0,
c > 0]
BI(157)(20), ET I 79(12)
[a > 0,
−1 0,
−1 < p < 0]
GW (333)(30b)
1
πp
Γ(1 + p) sin b +
ap+1
2
sin ax
dx
xν (x + b)
1
πν
ν 3 ν
Γ(−1 − ν) 1 F 2 1; 1 + , + ; − a2 b2 sign(a)
2
2 2 2
4
ν
1
π cosec(πν) sin(ab)
ν
πν
−
− aν Γ(−ν) 1 F 2 1; 1 + , 1 + ; − a2 b2 sign(a) sin
bν
2
2
4
2
[Im a = 0, −1 < Re b < 2, arg b = π] MC
= a1+ν b cos
∞
2.
0
Γ(1 − ν) iaβ
cos(ax)
dx =
e Γ(ν, iaβ) + e−iaβ Γ(ν, −iaβ)
ν
ν
x (x + β)
2β
[a > 0, |Re ν| < 1,
|arg β| < π]
ET II 221(52)
3.766 1.10
0
∞
xμ−1 sin ax
dx
1 + x2
= −a2−μ Γ(μ − 2) 1 F 2 1;
πμ
3 − μ 4 − μ a2
πμ π
,
;
+ sec
sinh(a)
sign(a) sin
2
2
4
2
2
2
MC
[Im a = 0, −1 < Re μ < 3]
3.768
Trigonometric functions and powers
∞
xμ−1 cos(ax)
μπ
π
cosh a
dx = cosec
2
1+x
2
2
1
μπ
+ cos
Γ(μ) {exp [−a + iπ(1 − μ)] γ(1 − μ, −a) − ea γ(1 − μ, a)}
2
2
[a > 0, 0 < Re μ < 3]
ET I 319(24)
∞
x2μ+1 sin(ax) dx
π
= − b2μ sec(μπ) sinh(ab)
x2 + b2
2
sin(μπ)
+
Γ(2μ) [ 1 F 1 (1; 1 − 2μ; ab) + 1 F 1 (1; 1 − 2μ; −ab)]
2a2μ
a > 0, − 32 < Re μ < 12
ET II 220(39)
2.
0
3.9
0
439
∞
4.9
0
1
x2μ+1 cos(ax) dx
π 2(μ+ 12 )
b
cosec
μ
+
=
−
π
cosh(ab)
x2 + b2
2 2
cos μ + 12 π
1
1
Γ 2 μ+
; ab
+
1 F 1 1; 1 − 2 μ +
2(μ+ 12 )
2
2
2a
1
+ 1 F 1 1; 1 − 2 μ +
; −ab
2
a > 0, −1 < Re μ < 12
ET II 221(56)
3.767
∞ xβ−1
1.
sin ax −
βπ
2
π
dx = − γ β−2 e−aγ
2
γ 2 + x2
0
[a > 0,
Re γ > 0,
0 < Re β < 2]
BI (160)(20)
∞ xβ
2.
cos ax −
βπ
2
γ 2 + x2
0
dx =
π β−1 −aγ
γ
e
2
[a > 0,
Re γ > 0,
|Re β| < 1]
BI (160)(21)
∞ xβ−1
3.
sin ax −
x2
0
−
βπ
2
dx =
b2
π β−2
πβ
b
cos ab −
2
2
[a > 0,
b > 0,
0 < Re β < 2]
BI (161)(11)
∞ xβ
4.
cos ax −
x2 − b2
0
βπ
2
π
πβ
dx = − bβ−1 sin ab −
2
2
[a > 0,
b > 0,
|β| < 1]
GW (333)(82)
3.768
∞
1.
u
∞
2.
(x − u)μ−1 sin(ax) dx =
Γ(μ)
μπ
sin au +
μ
a
2
[a > 0,
0 < Re μ < 1]
ET II 203(19)
(x − u)μ−1 cos(ax) dx =
Γ(μ)
μπ
cos au +
aμ
2
[a > 0,
0 < Re μ < 1]
ET II 204(24)
u
1
(1 − x)ν sin(ax) dx =
3.11
0
1 Γ(ν + 1)
−
C ν (a) = a−ν−1/2 s ν+1/2,1/2 (a)
a
aν+1
[a > 0, Re ν > −1]
ET I 11(3)a
440
Trigonometric Functions
3.768
Here C ν (a) is the Young’s function given by:
C ν (a) =
1 ν
2a
Γ(ν + 1)
[ 1 F 1 (1; ν + 1; ia) +
1F 1
(1; ν + 1; −ia)] =
∞
(−1)n aν+2n
.
Γ(ν + 2n + 1)
n=0
i
i −ν−1
a
exp (νπ − 2a) γ (ν + 1, −ia)
2
2
i
− exp − (νπ − 2a) γ (ν + 1, ia)
2
2 n
∞
−a
= Γ(ν + 1)
Γ(ν + 2 + 2n)
n=0
1
(1 − x)ν cos(ax) dx =
4.3
0
[a > 0,
u
xν−1 (u − x)μ−1 sin(ax) dx =
5.
u
μ+ν−1
2i
0
B(μ, ν) [ 1 F 1 (ν; μ + ν; iau) −
[a > 0,
u
xν−1 (u − x)μ−1 cos(ax) dx =
6.
0
xμ−1 (u − x)μ−1 sin(ax) dx =
0
∞
u
xμ−1 (u − x)μ−1 cos(ax) dx =
√ u
π
a
√
u
a
μ−1/2
sin
∞
10.
√
xμ−1 (x − u)μ−1 cos(ax) dx = −
u
1
11.3
0
μ−1/2
u
a
π
0
ν = 0]
ET II 189(26)
uμ+ν−1
B(μ, ν) [ 1 F 1 (ν; μ + ν; iau) + 1 F 1 (ν; μ + ν; −iau)]
2
[a > 0, Re μ > 0, Re ν > 0]
xμ−1 (x − u)μ−1 sin(ax) dx
√
π
=
2
u
9.
Re ν > −1,
(ν; μ + ν; −iau)]
au
au
Γ(μ) J μ−1/2
2
2
[Re μ > 0]
8.
Re μ > 0,
1F 1
ET I 11(3)a
ET II 189(32)
u
7.
Re ν > −1]
π
2
ET II 189(25)
+
au
au ,
au
au
J 1/2−μ
− sin
Y 1/2−μ
Γ(μ) cos
2
2
2
2
a > 0, 0 < Re μ < 12
ET II 203(20)
μ− 12
cos
au
Γ(μ) J μ− 12
2
au
2
[Re μ > 0]
u
a
μ− 12
ET II 189(31)
+
au
au ,
au
au
J 12 −μ
Y 12 −μ
Γ(μ) sin
− cos
2
2
2
2
1
ET II 204(25)
a > 0, 0 < Re μ < 2
i
xν−1 (1 − x)μ−1 sin(ax) dx = − B(μ, ν) [ 1 F 1 (ν; ν + μ; ia) − 1 F 1 (ν; ν + μ; −ia)]
2
[Re μ > 0, Re ν > −1, ν = 0]
ET I 68 (5)a, ET I 317(5)
1
xν−1 (1 − x)μ−1 cos(ax) dx =
12.3
0
1
B(μ, ν) [ 1 F 1 (ν; ν + μ; ia) +
2
1F 1
[Re μ > 0,
(ν; ν + μ; −ia)]
Re ν > 0]
ET I 11(5)
3.771
Trigonometric functions and powers
√
1
xμ (1 − x)μ sin(2ax) dx =
13.
0
1
xμ (1 − x)μ cos(2ax) dx =
14.
0
π
1
(2a)μ+ 2
Γ(μ + 1) J μ+ 12 (a) sin a
[a > 0,
√
π
1
(2a)μ+ 2
∞
1.
0
0
πiaν−1 e−aβ
(β + ix)−ν − (β − ix)−ν sin(ax) dx = −
Γ(ν)
[a > 0,
πa
x (β + ix)−ν + (β − ix)−ν sin(ax) dx = −
0
(ν − 1 − aβ) −aβ
e
Γ(ν)
[a > 0, Re β > 0,
Re ν > 0]
(−1)
(2n)!πaν−2n−1 e−aβ Lν−2n−1
x2n (β + ix)−ν + (β − ix)−ν cos(ax) dx =
(aβ)
2n
Γ(ν)
[a > 0, Re β > 0, 0 ≤ 2n < Re ν]
n
ET I 13(20)
∞
0
∞
7.
0
0
ν−2
ET I 70(17)
6.
1.
Re ν > 0]
∞
0
Re β > 0,
(−1)n i
(2n)!πaν−2n−1 e−aβ Lν−2n−1
x2n (β − ix)−ν − (β + ix)−ν sin(ax) dx =
(aβ)
2n
Γ(ν)
[a > 0, Re β > 0, 0 ≤ 2n < Re ν]
5.
3.771
Re ν > 0]
ET I 70(16)
0
Re β > 0,
∞
4.
ET I 11(4)
ET I 13(19)
∞
3.
Re μ > −1]
πaν−1 e−aβ
(β + ix)−ν + (β − ix)−ν cos(ax) dx =
Γ(ν)
[a > 0,
ET I 68(4)
ET I 70(15)
∞
2.
Re μ > −1]
Γ(μ + 1) J μ+ 12 (a) cos a
[a > 0,
3.769
441
∞
+
−ν
x2n+1 (β + ix)−ν + (β − ix)
,
(−1)n+1
(2n + 1)!πaν−2n−2 e−aβ Lν−2n−2
(aβ)
2n+1
Γ(ν)
[a > 0, Re β > 0, −1 ≤ 2n + 1 < Re ν] ET I 70(18)
sin(ax) dx =
+
,
(−1)n+1
−ν
(2n + 1)!πaν−2n−2 e−aβ Lν−2n−2
x2n+1 (β + ix)−ν − (β − ix)
(aβ)
cos(ax) dx =
2n+1
Γ(ν)
[a > 0, Re β > 0, 0 ≤ 2n < Re ν − 1] ET I 13(21)
√
ν
2
1
π 2β
1
2 ν− 2
β +x
sin(ax) dx =
Γ ν+
[I −ν (aβ) − Lν (aβ)]
2
a
2
a > 0, Re β > 0, Re ν < 12 , ν = − 12 , − 32 , − 52 , . . . EH II 38a, ET I 68(6)
442
Trigonometric Functions
∞
2.
0
2
ν− 12
1
β + x2
cos(ax) dx = √
π
ν
2β
a
3.771
1
cos(πν) Γ ν +
K −ν (aβ)
2
a > 0, Re β > 0,
μ−1
x2ν−1 u2 − x2
sin(ax) dx
0
=
u
4.
0
[Re μ > 0,
∞
u
6.
ν− 12
2β
1
x x2 + β 2
sin(ax) dx = √ β
a
π
√
2β
= πβ
a
u2 − x2
ν− 12
√
sin(ax) dx =
0
∞
7.
u
8.
2
ν− 12
x − u2
u2 − x2
ν− 12
cos(ax) dx =
∞
9.
u
10.
0
u
ET II 189(29)
cos νπ Γ ν +
ν
1
2
Re ν > 0]
ET II 190(35)
K ν+1 (aβ)
1
K ν+1 (aβ)
Γ 2 −ν
[a > 0, Re β > 0,
1
Γ ν+
1
2
Hν (au)
a > 0,
u > 0,
Re ν < 0]
ET I 69(11)
Re ν > − 12
ET I 69(7), WA 358(1)a
ν
2u
a
Γ ν+
1
2
J −ν (au)
a > 0, u > 0,
|Re ν| <
1
2
EH II 81(12)a, ET I 69(8), WA 187(3)a
π
2
2u
a
√
2
ν− 12
π
x − u2
cos(ax) dx = −
2
ν− 12
x u2 − x2
ν
ν
2u
a
√
0
π
2
√
π
sin(ax) dx =
2
u
1 a2 u 2
;−
2
4
1 3
ν + ; ,μ + ν +
2 2
Re μ > 0, Re ν > − 12
1F 2
μ−1
a u
1
1
x2ν−1 u2 − x2
cos(ax) dx = u2μ+2ν−2 B(μ, ν) 1 F 2 ν; , μ + ν; −
2
2
4
0
a 2μ+2ν−1
1
u
B μ, ν +
2
2
2 2
5.7
WA 191(1)a, GW(333)(78)a
u
3.
1
2
Re ν <
√
π
u
sin(ax) dx =
2
ν
Γ ν+
1
2
J ν (au)
a > 0,
u > 0,
Re ν > − 12
ET I 11(8)
2u
a
ν
Γ ν+
1
Y −ν (au)
2 a > 0, u > 0,
|Re ν| <
1
2
WA 187(4)a, EH II 82(13)a, ET I 11(9)
2u
a
ν
Γ ν+
1
J ν+1 (au)
2 a > 0, u > 0,
Re ν > − 12
ET I 69(9)
3.772
Trigonometric functions and powers
∞
11.
x x −u
2
1
2 ν− 2
√
sin(ax) dx =
u
x u2 − x2
ν− 12
∞
13.
x x −u
u
2
2 ν−1/2
√
πu
cos(ax) dx
2
2
ν−1/2
x + 2βx
sin(ax) dx =
∞
2
ν−1/2
x + 2βx
cos(ax) dx
√
π
2
0
2.
0
√
2u
3.
∞
4.
2u
2u
5.
− 12 < Re ν < 0
1
Γ ν+
J −ν−1 (au)
2
a > 0, u > 0, 0 < Re ν < 12
ET I 12(11)
ν
2β
1
Γ ν+
[J −ν (aβ) cos(aβ) + Y −ν (aβ) sin(aβ)]
a 2
a > 0, |arg β| < π, 12 > Re ν > − 32
ET I 69(12)
ν
ν−1/2
√
2ux − x2
sin(ax) dx = π
2u
a
ν
√
2
ν−1/2
π
x − 2ux
sin(ax) dx =
2
2β
a
ν
ν−1/2
√
2ux − x2
cos(ax) dx = π
2u
a
ν
0
ν
2β
a
0
2u
a
π
2
=−
1
Y −ν−1 (au)
2
a > 0, u > 0,
u
s (ν−1)ν+1 (au)
aν
√
−1
ν
2u
π
1
1
1
2ν+1
u
u
−
Γ ν+
=
ν+
Hν+1 (au)
2
2
a
2
2
a > 0, u > 0, Re ν > − 12
ET I 12(10)
∞
1.
Γ ν+
ν+1
cos(ax) dx = −
0
3.772
ν
2u
a
ET I 69(10)
u
12.7
π
u
2
443
Γ ν+
Γ ν+
1
[Y −ν (aβ) cos(aβ) − J −ν (aβ) sin(aβ)]
2
a > 0, |Re ν| < 12
ET I 12(13)
1
sin(au) J ν (au)
2
a > 0, u > 0, Re ν > − 12
ET I 69(13)a
Γ ν+
Γ ν+
1
[J −ν (au) cos(au) − Y −ν (au) sin(au)]
2
a > 0, u > 0, |Re ν| < 12
ET I 70(14)
1
J ν (au) cos(au)
2
a > 0, u > 0, Re ν > − 12
ET I 12(4)
∞
6.
2
ν−1/2
x − 2ux
cos(ax) dx
2u
=−
√
π
2
2u
a
ν
1
Γ ν+
[J −ν (au) sin(au) + Y −ν (au) cos(au)]
2
a > 0, u > 0, |Re ν| < 12
ET I 12(12)
444
3.773
1.
Trigonometric Functions
∞
x2ν
8
μ+1
(x2 + β 2 )
0
3.773
sin(ax) dx
1 2ν−2μ
3 β 2 a2
β
a B (1 + ν, μ − ν) 1 F 2 ν + 1; ν + 1 − μ, ;
2√
2
4
β 2 a2
πa2μ−2ν+1 Γ(ν − μ)
3
,
μ
−
ν
+
1;
μ
+
1;
μ
−
ν
+
+
F
1
2
3
4μ−ν+1 Γ μ − ν + 2
2
4
!
√
2 2 ! −ν + 1
π
a β !
2
β 2ν−2μ−1 G 21
=
!
13
2 Γ(μ + 1)
4 ! μ − ν + 12 , 12 , 0
[a > 0, Re β > 0, −1 < Re ν < Re μ + 1] ET I 71(28)a, ET II 234(17)
=
2.
∞
8
x2m+1 sin(ax)
(z +
0
n+1
x2 )
dx =
√
(−1)n+m π dn
m −a z
·
e
z
n!
2 dz n
[a > 0,
√
∞ 2m+1
x
sin(ax) dx
(−1)m+1 π d2m+1 n
=
[a K n (aβ)]
n+ 1
2n β n Γ n + 12 da2m+1
0
(β 2 + x2 ) 2
[a > 0,
0 ≤ m ≤ n,
|arg z| < π]
ET I 68(39)
3.
2ν
x
cos(ax) dx
(x2
0
∞
5.
+
μ+1
β2)
x2m cos(ax) dx
n+1
(z + x2 )
0
−1 ≤ m ≤ n]
ET I 67(37)
∞
4.
Re β > 0,
∞
6.7
0
2 2
1
1 1 β a
1 2ν−2μ−1
1
1
β
B ν + ,μ − ν +
1F 2 ν + ; ν − μ + , ;
2
2
2
2
2 2 4
√ 2μ−2ν+1 1
Γ ν−μ− 2
πa
3 β 2 a2
+
1 F 2 μ + 1; μ − ν + 1, μ − ν + ;
μ−ν+1
4
Γ(μ − ν +1)
2 4
!
√
2 2 ! −ν + 1
π
a β !
2
β 2ν−2μ−1 G 21
=
!
13
2 Γ(μ + 1)
4 ! μ − ν + 12 , 0, 12
a > 0, Re β > 0, − 12 < Re ν < Re μ + 1
ET I 14(29)a, ET II 235(19)
=
= (−1)m+n
√
1
dn
π
· n z m− 2 e−a z
2 · n! dz
[a > 0,
√
x cos(ax) dx
π
(−1)
d2m
=
{an K n (aβ)}
·
1
n+
da2m
2n β n Γ n + 12
(β 2 + x2 ) 2
a > 0,
2m
n + 1 > m ≥ 0,
|arg z| < π]
ET I 10(28)
m
Re β > 0,
0≤m<n+
1
2
ET I 14(28)
3.774
1.
0
∞
sin(ax) dx
π
i
i
νπ
I ν (ab) + Jν (iab) − Jν (−iab)
sin
√
√
ν = ν
b sin(νπ)
2
2
2
x2 + b2 x + x2 + b2
[a > 0, b > 0, Re ν > −1]
ET I 70(19)
3.775
Trigonometric functions and powers
∞
2.
0
445
1
cos(ax) dx
π
1
νπ
Jν (iab) + Jν (−iab) − cos
I ν (ab)
√
√
ν = ν
b sin(νπ) 2
2
2
x2 + b2 x + x2 + b2
[a > 0, b > 0, Re ν > −1]
ET I 12(15)
∞
3.
0
x + x2 + β 2
x (x2 + β 2 )
ν
sin(ax) dx =
aπ ν
β I 14 − ν2
2
aβ
2
K 14 + ν2
a > 0,
aβ
2
Re β > 0,
Re ν <
3
2
ET I 71(23)
∞
4.
0
x2 + β 2 − x
x (x2 + β 2 )
ν
cos(ax) dx =
aπ ν
β I − 14 + ν2
2
aβ
2
aβ
2
Re β > 0,
K − 14 − ν2
a > 0,
Re ν > − 32
ET I 12(17)
∞
5.
0
ν
x2 + β 2
1
sin(ax) dx =
1
ν+
β
x 2 x2 + β 2
β+
2
Γ
a
3 ν
−
4 2
W ν2 , 14 (aβ) M − ν2 , 14 (aβ)
a > 0, Re β > 0, Re ν < 32
ET I 71(27)
∞
6.
0
ν
x2 + β 2
1
cos(ax) dx = √ Γ
1
β 2a
xν+ 2 β 2 + x2
β+
1 ν
−
4 2
W ν2 ,− 14 (aβ) M − ν2 ,− 14 (aβ)
a > 0, Re β > 0, Re ν < 12
ET I 12(18)
3.775
∞
1.
0
ν
x2 + β 2 + x −
x2 + β 2 − x
x2 + β 2
ν
νπ
K ν (aβ)
2
[a > 0, Re β > 0, |Re ν| < 1]
sin(ax) dx = 2β ν sin
ET I 70(20)
2.
0
∞
ν
x2 + β 2 + x +
x2 + β 2 − x
x2 + β 2
ν
cos(ax) dx = 2β ν cos
[a > 0,
νπ
K ν (aβ)
2
Re β > 0, |Re ν| < 1]
ET I 13(22)
3.
√
√
ν ν
∞
+
x + x2 − u2 + x − x2 − u2
νπ
νπ ,
√
− Y ν (au) sin
sin(ax) dx = πuν J ν (au) cos
2
2
x2 − u2
u
[a > 0, u > 0, |Re ν| < 1]
4.
√
√
ν ν
∞
+
x + x2 − u2 + x − x2 − u2
νπ
νπ ,
√
+ J ν (au) sin
cos(ax) dx = −πuν Y ν (au) cos
2
2
x2 − u2
u
[a > 0, u > 0, |Re ν| < 1]
ET I 70(22)
ET I 13(25)
446
Trigonometric Functions
3.776
√
√
ν ν
x + i u2 − x2 + x − i u2 − x2
π
νπ
√
[Jν (au) − J−ν (au)]
sin(ax) dx = uν cosec
2 − x2
2
2
u
0
ET I 70(21)
[a > 0, u > 0]
√
√
ν ν
u
x + i u2 − x2 + x − i u2 − x2
π
νπ
√
[Jν (au) + J−ν (au)]
cos(ax) dx = uν sec
2 − x2
2
2
u
0
[a > 0, u > 0, |Re ν| < 1]
5.
6.
u
ET I 13(24)
√
√
ν ν
∞
x + x2 − u2 + x − x2 − u2
sin(ax) dx
2 − u2 )
x
(x
u
π 3 ν+
au
au
au
au ,
=−
Y 1/4−ν/2
+ J 1/4−ν/2
Y 1/4+ν/2
au J 1/4+ν/2
2
2
2
2
2
ET I 71(25)
a > 0, u > 0, |Re ν| < 32
7.6
8.
∞
6
u
∞
√
√
ν ν
x + x2 − u2 + x − x2 − u2
cos(ax) dx
2 − u2 )
x
(x
π 3 ν+
au
au
au
au ,
=−
Y −1/4−ν/2
+ J −1/4−ν/2
Y −1/4+ν/2
au J −1/4+ν/2
2
2
2
2
2
ET I 13(26)
a > 0, u > 0, |Re ν| < 32
x+β+
9.
x2 + 2βx
0
ν
+ x+β−
x2 + 2βx
ν
sin(ax) dx
x2 + 2βx
+
νπ
νπ ,
+ J ν (βa) cos βa −
2
2
|arg β| < π, |Re ν| < 1] ET I 71(26)
= πβ ν Y ν (βa) sin βa −
[a > 0,
∞
10.
0
2u
11.
0
3.776 1.
∞
0
2.
0
∞
x+β+
x2 + 2βx
ν
+ x+β−
x2 + 2βx
x2 + 2βx
ν
cos(ax) dx
+
νπ
νπ ,
= πβ ν J ν (βa) sin βa −
− Y ν (βa) cos βa −
2
2
[a > 0, |arg β| < π, |Re ν| < 1] ET I 13(23)
√
4ν √
4ν
√
√
2u + x + i 2u − x
+
2u + x − i 2u − x
√
cos(ax) dx
4u2 x − x3
a
2ν 3/2
J ν−1/4 (au) J −ν−1/4 (au)
= (4u) π
2
ET I 14(27)
[a > 0, u > 0]
a2 (b + x)2 + p(p + 1)
a
sin(ax) dx = p
p+2
(b + x)
b
[a > 0,
b > 0,
p > 0]
BI (170)(1)
a2 (b + x)2 + p(p + 1)
p
cos(ax) dx = p+1
(b + x)p+2
b
[a > 0,
b > 0,
p > 0]
BI (170)(2)
3.784
Rational and trigonometric functions
447
3.78–3.81 Rational functions of x and of trigonometric functions
3.781
∞
sin x
1
−
x
1+x
1.
0
∞
cos x −
2.
0
3.782
1.
0
u
∞
0
3.
3.783
∞
0
∞
2.
0
∞
1.
0
∞
2.
0
∞
3.
0
∞
4.
0
∞
5.
0
∞
6.
0
7.
8.
BI (173)(7)
∞
u
cos x
dx = C + ln u
x
1 − cos ax
aπ
dx =
x2
2
BI (173)(8)
[u > 0]
GW (333)(31)
[a ≥ 0]
BI (158)(1)
∞
1.
3.784
sin ab
1 − cos ax
dx = π
x(x
−
b)
b
−∞
(cf. 3.784 4 and 3.781 2)
dx
= −C
x
1
1+x
1 − cos x
dx −
x
2.
dx
=1−C
x
cos x − 1
1
+
x2
2(1 + x)
cos x −
1
1 + x2
[a > 0,
b real,
b = 0]
ET II 253(48)
1
3
dx
= C−
x
2
4
BI (173)(19)
dx
= −C
x
EH I 17, BI(273)(21)
b
cos ax − cos bx
dx = ln
x
a
[a > 0,
b > 0]
FI II 635, GW(333)(20)
a sin bx − b sin ax
a
dx = ab ln
2
x
b
[a > 0,
b > 0]
FI II 647
cos ax − cos bx
(b − a)π
dx =
x2
2
[a ≥ 0,
b ≥ 0]
BI(158)(12), FI II 645
sin x − x cos x
dx = 1
x2
cos ax − cos bx
1
dx =
x(x + β)
β
BI (158)(3)
b
ci(aβ) cos aβ + si(aβ) sin aβ − ci(bβ) cos bβ − si(bβ) sin bβ + ln
a
[a > 0, b > 0, |arg β| < π] ET II 221(49)
cos ax + x sin ax
dx = πe−a
1 + x2
[a > 0]
GW (333)(73)
∞
sin ax − ax cos ax
π
dx = a2 sign a
3
x
4
0
∞
π (b − a)β + e−bβ − e−aβ
cos ax − cos bx
dx =
x2 (x2 + β 2 )
2β 3
0
LI (158)(5)
[a > 0,
b > 0,
|arg β| < π]
BI(173)(20)a, ET II 222(59)
448
Trigonometric Functions
∞
9.10
0
n
π
cos mx
√
=
e−m sin u sinh φ (cos β sin u cosh φ + sin β cos u sinh φ)
1 + a2 T n (x)
2n 1 + a2 k=1
[u = (2k − 1) π/ (2n) ,
∞
3.785
0
3.786
∞
1.
0
∞
2.11
0
3.
∞
11
0
3.785
1
ak cos bk x dx = −
ak ln bk
x
n
n
k=1
k=1
φ = arcsinh(1/a),
%
β = m cos u cosh φ,
&
n
ak = 0
bk > 0,
0 < |a| < 1]
FI II 649
k=1
(1 − cos ax) sin bx
b b 2 − a2
a a+b
dx = ln
+ ln
2
x
2
b2
2 a−b
(1 − cos ax) cos bx
dx = ln
x
|a2 − b2 |
b
(1 − cos ax) cos bx
π
dx = (a − b)
x2
2
=0
[a > 0,
b > 0]
[a > 0,
b > 0,
ET I 81(29)
a = b]
FI II 647
[a na > 0]
[na > m]
BI(155)(7)
∞
2.
0
sin2 ax − sin2 bx
1 a
dx = ln
x
2 b
[a > 0,
b > 0]
GW (333)(20b)
∞
x3 − sin3 x
13
π
dx =
5
x
32
0
∞
3 − 4 sin2 ax sin2 ax
1
dx = ln 2
4.
x
2
0
π/2
1
π
− cot x dx = ln
3.788
x
2
0 π/2 2
4x cos x + (π − x)x
dx = π 2 ln 2
3.789
sin x
0
3.791
π/2
x dx
= ln 2
1.
1 + sin x
0
π
x cos x
dx = π ln 2 − 4G
2.
0 1 + sin x
π/2
x cos x
dx = π ln 2 − 2G
3.
1 + sin x
0
3.
BI (158)(6)
[a real, a = 0]
HBI (155)(6)
GW (333)(61)a
LI (206)(10)
GW (333)(55a)
GW (333)(55c)
GW (333)(55b)
3.792
Rational and trigonometric functions
π
449
π
π/2 π
− x cos x
2 − x cos x
dx = 2
dx = π ln 2 + 4G = 5.8414484669 . . .
1 − sin x
1 − sin x
0
2
4.
0
BI(207)(3), GW(333)(56c)
π/2
5.
0
π
6.
0
x2 dx
= 4π ln 2
1 − cos x
π/2
7.
0
0
9.
10.
11.
12.
3.
BI (219)(1)
p+1
π
+
2
p
(p + 1)
∞
1
2 −
ζ(2k)
p
42k−1 (p + 2k)
k=1
LI (207)(4)
π
x dx
= − ln 2
1 + cos x
2
GW (333)(55a)
π/2
π
x sin x dx
= ln 2 + 2G
1
−
cos
x
2
0
π
x sin x dx
= 2π ln 2
0 1 − cos x
π/2
π
π
x − sin x
x − sin x
dx = +
dx = 2
1
−
cos
x
2
1
− cos x
0
0
π/2
π
x sin x
dx = − ln 2 + 2G
1 + cos x
2
0
3.792 1.
2.
π
xp+1 dx
=−
1 − cos x
2
BI (207)(3)
[p > 0]
π/2
8.
x2 dx
π2
=−
+ π ln 2 + 4G = 3.3740473667 . . .
1 − cos x
4
GW (333)(56a)
GW (333)(56b)
GW (333)(57a)
GW (333)(55b)
2
dx
2π
a
=
2
2
1−a
−π 1 − 2a cos x + a
π/2
∞
x cos x dx
a2k
π
k
ln(1
+
a)
−
=
(−1)
1 + 2a sin x + a2
2a
(2k + 1)2
0
k=0
2
a
π
2
x sin x dx
π
a
= ln(1 + a)
2
a
0 1 − 2a cos x + a
2
1
π
a
= ln 1 +
a
a
π
2π
0
5.
0
<1
FI II 485
< 1,
<1
LI (241)(2)
a=0
BI (221)(2)
4.
<1
2π
x sin x dx
2π
ln(1 − a)
=
1 − 2a cos x + a2
a
1
2π
ln 1 −
=
a
a
%
a2 < 1,
2
a >1
n−1
a=0
&
−n
x sin nx dx
a−k − ak
2π
n
ln(1
−
a)
+
=
−
a
a
1 − 2a cos x + a2
1 − a2
n−k
k=1
2
a < 1, a = 0
BI (223)(4)
BI (223)(5)
450
Trigonometric Functions
∞
6.
0
!
!
π !! 1 + a !!
sin x
dx
−1
=
·
1 − 2a cos x + a2 x
4a ! 1 − a !
3.792
[a real,
a = 0,
a = 1]
GW (333)(62b)
∞
7.8
0
0
∞
9.
0
∞
10.3
0
∞
11.
0
∞
12.
0
∞
13.
0
0
0
sin x cos bx
π
dx
=
a[b]
·
1 − 2a cos x + a2 x
2(1 − a)
π
π
=
ab + ab−1
2(1 − a)
4
[0 < a < 1,
[b = 0, 1, 2, . . .]
[b = 1, 2, 3, . . .] ;
b > 0] ; (for b = 0, see 3.792 6)
(1 − a cos x) sin bx dx π 1 − a[b]+1
= ·
·
1 − 2a cos x + a2
x
2
1−a
π 1 − ab
πab
= ·
+
2 1−a
4
ET I 19(5)
[b = 1, 2, 3, . . .]
[b = 1, 2, 3, . . .]
[0 < a < 1,
b > 0]
ET I 82(33)
−bβ
1 + ae
1
dx
π
=
2
2
2
2
1 − 2a cos bx + a β + x
2β (1 − a ) 1 − ae−bβ
a2 < 1,
b≥0
1
dx
sin bβ
aπ
=
1 − 2a cos bx + a2 β 2 − x2
β (1 − a2 ) 1 − 2a cos bβ + a2
2
a < 1,
b>0
e−βbc − ac
sin bcx
x dx
π
=
1 − 2a cos bx + a2 β 2 + x2
2 (1 − ae−bβ ) (1 − aebβ )
2
a < 1,
b > 0,
1
sin bx
x dx
π
=
1 − 2a cos bx + a2 β 2 + x2
2 ebβ − a
1
π
=
2a aebβ − 1
a2 < 1,
b>0
a2 > 1,
b>0
BI (192)(1)
BI (193)(1)
c>0
BI (192)(8)
a − cos βbc
sin bcx
x dx
π
=
2
2
2
1 − 2a cos bx + a β − x
2 1 − 2a cos βb + a2
c
2
a2 < 1,
b > 0,
c>0
1 − a sin βbc + 2ac+1 sin βb
cos bcx
dx
π
=
1 − 2a cos bx + a2 β 2 − x2
2β (1 − a2 )
1 − 2a cos βb + a2
2
a < 1, b > 0, c > 0
∞
1 − a cos bx
dx
π e
=
1 − 2a cos bx + a2 1 + x2
2 eb − a
0
16.
[0 < a < 1]
∞
15.
[b = 1, 2, . . .] ;
BI (192)(2)
∞
14.
[b = 0, 1, 2, . . .]
ET I 81(26)
∞
8.
sin bx
dx π 1 + a − 2a[b]+1
=
·
1 − 2a cos x + a2 x
2 (1 − a2 ) (1 − a)
π 1 + a − ab − ab+1
=
2 (1 − a2 ) (1 − a)
b
a2 < 1,
b>0
BI (193)(5)
BI (193)(9)
FI II 719
3.794
Rational and trigonometric functions
∞
17.
0
π eβ−βb + aeβb
cos bx
dx
·
=
1 − 2a cos x + a2 x2 + β 2
2β (1 − a2 ) (eβ − a)
451
[0 ≤ b < 1,
|a| < 1,
Re β > 0]
ET I 21(21)
∞
18.
0
sin bx sin x
dx
·
1 − 2a cos x + a2 x2 + β 2
π sinh bβ
[0 ≤ b < 1]
2β eβ − a +
,
π
=
am eβ(m+1−b) − e(1−b)β
4β (aeβ − 1)
,
+
π
m −(m+1−b)β
−(1−b)β
[m ≤ b ≤ m + 1]
a
e
−
e
−
4β (ae−β − 1)
[0 < a < 1, Re β > 0]
ET I 81(27)
=
∞
19.
0
∞
20.
(cos x − a) cos bx
dx
π cosh βb
·
=
1 − 2a cos x + a2 x2 + β 2
2β (eβ − a)
sin x
(1 − 2a cos 2x +
0
n+1
a2 )
dx
=
x
=
0
|a| < 1,
Re β > 0]
ET I 21(23)
∞
0
[0 ≤ b < 1,
∞
tan x
(1 − 2a cos 2x +
tan x
n+1
a2 )
(1 − 2a cos 4x +
n+1
a2 )
dx
x
n
n
π
dx
=
2n+1
2
k
x
2 (1 − a )
k=0
2
a2k
BI (187)(14)
3.793
2π
1.3
0
k=1
[|a| < 1]
2π
2.
0
3.794
1.3
2.
3.3
%
&
n
sin nx − a sin[(n + 1)x]
1
n
x dx = −2πa ln(1 − a) +
1 − 2a cos x + a2
kak
cos nx − a cos[(n + 1)x]
x dx = 2πan
1 − 2a cos x + a2
∞
a2 < 1
BI (223)(9)
BI (223)(13)
a2k+1
x dx
π2
4
=
+
2
2 (1 − a2 ) (1 − a2 )
(2k + 1)2
0 1 + a + 2a cos x
k=0
2
a <1
⎡
√
√
n n
2π
1 − a2 − 1 − 1 − a2
2π
x sin nx
n 1+
⎣
dx = √
(∓1)
1 ± a cos x
an
1 − a2
0
√
√
⎤
√
n−1
(∓1)k 1 + 1 − a2 k − 1 − 1 − a2 k
2 1±a
⎦
√
× ln √
+
ak
1 + a + 1 − a k=0 n − k
2
a <1
BI (223)(2)
√
n
2π
2
x cos nx
2π 2
1 − 1 − a2
dx = √
a <1
BI (223)(3)
2
1 ± a cos x
∓a
1−a
0
π
452
Trigonometric Functions
√
x sin x dx
π a + a2 − b 2
= ln
b
2(a − b)
0 a + b cos x
√
2π
2π a + a2 − b2
x sin x dx
=
ln
a + b cos x
b
2(a + b)
0
∞
dx
π
sin x
·
= √
x
2 a2 − b 2
0 a ± b cos 2x
4.
5.
6.
3.795
π
=0
[a > |b| > 0]
GW (333)(53a)
[a > |b| > 0]
GW (333)(53b)
a2 > b 2
a2 0]
3.796
π/2
π
cos x ± sin x
x dx = ∓ ln 2 − G
1.
cos x ∓ sin x
4
0
π/4
π
1
cos x − sin x
x dx = ln 2 − G
2.
cos
x
+
sin
x
4
2
0
3.797
1.
2.
3.
3.798
∞
1.8
0
tan x
dx
π
·
= √
2
a + b cos 2x x
2 a − b2
=0
2.8
0
[b > c > 0]
BI (202)(18)
BI (207)(8, 9)
BI (204)(23)
π/4
π
1
π2
π π
− x tan x tan x dx = ln 2 +
− + ln 2
4
2
32
4
8
0
π/4 π
π
1
4 − x tan x dx
= − ln 2 + G
cos
2x
8
2
0
π/4 π
π
1
4 − x tan x
dx = ln 2 + G
cos
2x
8
2
0
[c > b > 0]
BI (204)(8)
BI (204)(19)
BI (204)(20)
[0 0]
BI (208)(5)
3.812
Rational and trigonometric functions
π/4
2.
x dx
2
(cos x + a sin x)
0
=
453
1−a
1
1+a π
ln √ + ·
1 + a2
4 (1 + a) (1 + a2 )
2
[a > 0]
π
3.
a cos x + b
2
2x
(a + b cos x)
0
dx =
2(a − b)
2π
√
ln
b
a + a2 − b 2
BI (204)(24)
[a > |b| > 0]
GW (333)(58a)
3.811
π
1.
0
t1
x dx
t1 + t2
t1 − t2 1 + tan 2
sin x
·
= π cosec
cosec
ln
t2
1 − cos t1 cos x 1 − cos t2 cos x
2
2
1 + tan
2
(cf. 3.794 4)
π/2
2.
0
π/4
3.
0
π/4
4.
0
π/4
5.
0
3.812
1.
0
2.
3.
4.11
π
π
x dx
= ln 2 + G
(cos x ± sin x) sin x
4
BI (208))(16, 17)
π
x dx
= − ln 2 + G
(cos x + sin x) sin x
8
BI (204)(29)
π
x dx
= ln 2
(cos x + sin x) cos x
8
BI (204)(28)
sin x
x dx
π
π 1
= − ln 2 + − ln 2
2
sin x + cos x cos x
8
4
2
BI (204)(30)
π
x sin x dx
b
= √ arctan
2
a + b cos x
a√
ab
√
π
a + −b
√
= √
ln √
2 −ab
a − −b
√
π/2
π 1+ 1+a
x sin 2x dx
= ln
1 + a cos2 x
a
2
0
√
π/2
π 2 1+a− 1+a
x sin 2x dx
= ln
a
2
1 + a sin2 x
0
π
2
π
x dx
= √
2 − cos2 x
a
2a
a2 − 1
0
=0
= divergent
!
!
π
!1 + a!
π
x sin x dx
!
!
=
ln
2
2
2a ! 1 − a !
0 a − cos x
[a > 0,
b > 0]
[a > −b > 0]
GW (333)(60a)
[a > −1,
a = 0]
BI (207)(10)
[a > −1,
a = 0]
BI (207)(2)
a2 > 1
principal value for 0 < a2 < 1
[a = 0]
BI (219)(10)
5.7
BI (222)(5)
[0 < a < 1]
divergent if a = 0
BI (219)(13)
454
Trigonometric Functions
6.
π
11
0
x sin 2x dx
= π ln 4 1 − a2
a2 − cos2 x
,
+
= 2π ln 2 1 − a2 + a a2 − 1
= divergent
3.813
principal value for 0 ≤ a2 < 1
2
a >1
[|a| = 1]
BI (219)(19)
π/2
7.
0
8.
9.
10.
11.
12.∗
13.∗
BI (207)(1)
π
x sin x dx
= π(π − 2t) cosec 2t
2
2
0 1 − cos t sin x
π
∞
x cos x dx
sin(2k + 1)t
=
4
cosec
t
2
2
(2k + 1)2
0 cos t − cos x
k=0
π
π
x sin x dx
= (π − 2t) cot t
2
2
2
0 tan t + cos x
∞
t
x (a cos x + b) sin x dx
= 2aπ ln cos + πbt tan t
2
2x
2
cot
t
+
cos
0
%
&
π
x sin x cos x
a−1
dx = −π ln 2 + ln 1 +
a
a − sin2 x
0
π/2
√
ln a − sin2 x dx = −π ln 2 + iπ ln arccos a
0
14.∗
∞
x sin x dx
sin(2k + 1)t
=
−2
cosec
t
2
2
(2k + 1)2
cos t − sin x
k=0
π/2
PV
BI (219)(12)
BI (219)(17)
BI (219)(14)
BI (219)(18)
[a > 1]
[0 < a < 1]
!
!
ln !a − sin2 x! dx = −π ln 2
[0 < a < 1]
!
!
ln !a − cos2 x! dx = −π ln 2
[0 < a < 1]
0
15.∗
π/2
PV
0
3.813
1.
0
π
x dx
1
=
2
2
2
2
4
a cos x + b sin x
∞
2.
0
0
0
x dx
π2
=
2ab
a2 cos2 x + b2 sin2 x
[a > 0,
b > 0]
GW (333)(36)
GW(333)(81), ET II 222(63)
sin x dx
π
=
2
2
2
2ab
sin x + b cos x
[ab > 0]
sin2 x dx
π
=
2
2
2
2
2b(a + b)
x a cos x + b sin x
[a > 0,
x
0
2π
β
γ
2
1
π sinh(2aδ)
dx
− −
=
·
2
2
2
2
γ
β
sinh(2aδ)
β 2 sin2 ax + γ 2 cos2 ax x2 + δ 2
4δ β sinh (aδ) !
− γ cosh
! (aδ)
!
!
β
!arg ! < π, Re δ > 0, a > 0
!
γ!
∞
3.
4.
∞
a2
BI (181)(8)
b > 0]
BI (181)(11)
3.814
Rational and trigonometric functions
5.
455
π/2
π
x sin 2x dx
a+b
= 2
[a > 0, b > 0, a = b] GW (333)(52a)
ln
a − b2
2b
a2 cos2 x + b2 sin2 x
π
x sin 2x dx
a+b
2π
[a > 0, b > 0, a = b] GW (333)(52b)
ln
= 2
2
2 cos2 x + b2 sin2 x
a
−
b
2a
a
0
∞
π
sin 2x
dx
=
[a > 0, b > 0]
BI (182)(3)
·
2
2
2
2
x
a(a
+ b)
0 a cos x + b sin x
∞
β−γ
sin 2ax
π
x dx
−2aδ
−
e
=
·
2
2
2
2
2
2
β+γ
2 β 2 sinh2 (aδ) − γ 2 cosh2 (aδ)
0 β sin ax + γ cos ax x + δ
!
!
!
!
β
!
!
a > 0, !arg ! < π, Re δ > 0
γ
0
6.
7.
8.
ET II 222(64), GW(333)(80)
∞
9.
0
∞
11.
0
b > 0]
BI (182)(7)a
dx
π
sin x cos2 x
·
=
2
2
2
2a(a + b)
cos x + b sin x x
[a > 0,
b > 0]
BI (182)(4)
π
2
sin3 x
dx
=
·
·
2
2
2
2
2b a + b
a cos x + b sin x x
[a > 0,
b > 0]
BI (182)(1)
a2
0
3.814
[a > 0,
∞
10.
dx
π
(1 − cos x) sin x
·
=
2b(a + b)
a2 cos2 x + b2 sin2 x x
π/2
1.
0
π/4
2.
0
5.
dx
π
tan x
=
2
2
2ab
x + b sin x x
b > 0]
BI (181)(9)
[a > 0,
b > 0]
LI (208)(20)
[a > 0,
b > 0]
GW (333)(59)
[a > 0,
b > 0]
BI (182)(6)
π
tan x
dx
=
·
2ab
a2 cos2 2x + b2 sin2 2x x
[a > 0,
b > 0]
BI (181)(10)a
dx
π
1
sin2 2x tan x
·
=
·
2
2
2
2
2b a + b
a cos 2x + b sin 2x x
[a > 0,
b > 0]
BI (182)(2)a
π
1
cos2 2x tan x
dx
=
·
·
2a a + b
a2 cos2 2x + b2 sin2 2x x
[a > 0,
b > 0]
BI (182)(5)a
cos2
π/2
x cot x dx
a+b
π
= 2 ln
2
2
2
2a
b
cos x + b sin x
0
π/2 π
π π
1
2 − x tan x dx
2 − x tan x dx
=
a2 cos2 x + b2 sin2 x 2 0 a2 cos2 x + b2 sin2 x
0
a+b
π
= 2 ln
2b
a
a2
∞
6.
∞
7.
0
∞
8.
0
9.
0
2
a2
0
BI (204)(30)
[a > 0,
a2
0
BI (206)(9)
π
π 1
x tan x dx
= − ln 2 + − ln 2
(sin x + cos x) cos x
8
4
2
∞
3.
4.
(1 − x cot x) dx
π
=
2
4
sin x
∞
π
sin x tan x
dx
=
·
2
2
2
2b(a + b)
cos x + b sin x x
456
Trigonometric Functions
∞
10.
0
11.
12.
13.
14.
15.
3.815
dx
sin2 x cos x
π
a
·
=−
2
2
2
2
2
8b a + b2
a cos 2x + b sin 2x x cos 4x
∞
sin x
2 cos2 x + b2 sin2 x
a
0
∞
sin x cos x
2
2
2
2
0 a cos x + b sin x
∞
sin x cos2 x
2
2
2
2
0 a cos x + b sin x
∞
sin3 x
2
2
2
2
0 a cos x + b sin x
∞
1 − cos x
2
2
2
2
0 a cos x + b sin x
π/2
1.
0
π/2
2.
0
3.815
[a > 0,
b > 0]
BI (186)(12)a
·
dx
π b 2 − a2
=
·
x cos 2x
2ab b2 + a2
[a > 0,
b > 0]
BI (186)(4)a
·
π
b
dx
=
·
x cos 2x
2a a2 + b2
[a > 0,
b > 0]
BI (186)(7)a
·
π
b2
dx
=
· 2
x cos 2x
2ab a + b2
[a > 0,
b > 0]
BI (186)(8)a
·
π
a
dx
=− · 2
x cos 2x
2b a + b2
[a > 0,
b > 0]
BI (186)(10)
·
dx
π
=
x sin x
2ab
[a > 0,
b > 0]
BI (186)(3)a
x sin 2x dx
π
=
ln
2
2
a
−
b
1 + a sin x 1 + b sin x
√
√
1+ 1+b
1+a
√
·√
1+ 1+a
1+b
[a > 0, b > 0]
√
√
1+ 1+n 1+a
π
x sin 2x dx
√
=
ln
a + ab + b
1+ 1+a
1 + a sin2 x (1 + b cos2 x)
[a > 0, b > 0]
(cf. 3.812 3)
BI (208)(22)
(cf. 3.812 2 and 3)
BI (208)(24)
√
π/2
π
1+ 1+a
x sin 2x dx
√
=
ln
(1 + a cos2 x) (1 + b cos2 x)
a−b 1+ 1+b
0
3.
[a > 0,
b > 0]
(cf. 3.812 2)
BI (208)(23)
π/2
4.
0
t1
cos
x sin 2x dx
2π
2
=
ln
cos2 t1 − cos2 t2 cos t2
1 − sin2 t1 cos2 x 1 − sin2 t2 cos2 x
2
[−π < t1 < π,
−π < t2 < π]
BI (208)(21)
3.816
1.
π
2
2.
(a2
0
3.
dx = π 2
(a2 − cos2 x)
π 2
a − 1 − sin2 x cos x
0
7
√
x2 sin 2x
11
0
π
−
cos2
2
x)
a2 − 1 − a
a (a2 − 1)
!
!
π !! 1 − a !!
x dx = ln !
2
1 + a!
2
[a > 1]
a2 > 1
LI (220)(9)
(cf. 3.812 5)
√ √
a cos 2x − sin2 x 2
2 x dx = −2π ln 2 −a + a a + 1
2
a + sin+ x
,
√ √
a < −1 and a > 0. When a > 0, can write a a + 1 as a(a + 1).
BI (220)(12)
LI (220)(10)
3.818
Rational and trigonometric functions
4.11
0
π
√ √
a cos 2x + sin2 x 2
2 x dx = 2π ln 2 a − a a + 1
2
a +− sin x
,
√ √
a < 0 and a > 1. When a > 1, can write a a + 1 as a(a + 1).
457
(cf. 3.812 6)
LI (220)(11)
3.817
∞
1.
0
∞
2.
0
∞
3.
0
∞
4.
0
∞
5.
0
∞
6.
0
∞
7.
0
∞
8.
0
3.818
∞
1.
0
∞
2.
0
∞
3.
0
∞
4.
0
5.
0
∞
sin x
π a2 + b 2
dx
=
· 3 3
·
2
x
4
a b
a2 cos2 x + b2 sin2 x
[ab > 0]
BI (181)(12)
π
sin x cos x
dx
= 3
2 ·
2
2
2
2
x
4a
b
a cos x + b sin x
[ab > 0]
BI (182)(8)
π
sin3 x
dx
=
2 ·
2
x
4ab3
a2 cos2 x + b2 sin x
[ab > 0]
BI (181)(15)
π
sin x cos2 x
dx
= 3
2 ·
2
2
2
2
x
4a
b
a cos x + b sin x
[ab > 0]
BI (182)(9)
π a2 + b 2
tan x
dx
=
· 3 3
·
2
x
4
a b
a2 cos2 x + b2 sin2 x
[ab > 0]
BI (181)(13)
π a2 + b 2
tan x
dx
=
·
2
x
4 a3 b 3
a2 cos2 2x + b2 sin2 2x
[ab > 0]
BI (181)(14)
π
sin2 x tan x
dx
=
2 ·
2
x
4ab3
a2 cos2 x + b2 sin x
[ab > 0]
BI (182)(11)
tan x cos2 2x
π
dx
= 3
2 ·
2
2
2
2
x
4a
b
a cos 2x + b sin 2x
[ab > 0]
BI (182)(10)
π 3a4 + 2a2 b2 + 3b4
sin x
dx
=
·
·
3
x
16
a5 b 5
a2 cos2 x + b2 sin2 x
[ab > 0]
BI (181)(16)
[ab > 0]
BI (182)(13)
[ab > 0]
BI (182)(14)
π 3a2 + b2
dx
=
·
·
3
x
16
a3 b 5
a2 cos2 x + b2 sin2 x
[ab > 0]
LI (181)(19)
π 3a2 + b2
sin3 x cos x
dx
=
·
·
3
x
64
a3 b 5
a2 cos2 2x + b2 sin2 2x
[ab > 0]
BI (182)(17)
2
π a + 3b
sin x cos x
dx
=
·
3 ·
2
x
16
a5 b 3
a2 cos2 x + b2 sin x
2
π a2 + 3b2
sin x cos2 x
dx
=
·
·
3
x
16
a5 b 3
a2 cos2 x + b2 sin2 x
sin3 x
458
Trigonometric Functions
∞
6.
0
∞
7.
0
∞
8.
0
∞
9.
0
3.819
∞
1.
0
∞
0
0
∞
4.
0
∞
5.
0
∞
6.
0
∞
7.
0
∞
8.
0
9.
0
π 3a2 + b2
sin2 x tan x
dx
=
·
·
3
x
16
a3 b 5
a2 cos2 x + b2 sin2 x
[ab > 0]
BI (181)(17)
[ab > 0]
BI (182)(16)
π 3a4 + 2a2 b2 + 3b4
tan x
dx
=
·
·
3
x
16
a5 b 5
a2 cos2 2x + b2 sin2 2x
2
2
π a + 3b
tan x cos 2x
dx
=
·
3 ·
2
x
16
a5 b 3
a2 cos2 2x + b2 sin 2x
[ab > 0]
BI (181)(18)
[ab > 0]
BI (182)(15)
2
π 5a6 + 3a4 b2 + 3a2 b4 + 5b6
sin x
dx
=
·
·
4
x
32
a7 b 7
a2 cos2 x + b2 sin2 x
∞
BI (181)(20)
π a4 + 2a2 b2 + 5b4
sin x cos x
dx
=
·
·
4
x
32
a7 b 5
a2 cos2 x + b2 sin2 x
[ab > 0]
∞
3.
tan x
π 3a4 + 2a2 b2 + 3b4
dx
=
·
3
x
16
a5 b 5
a2 cos2 x + b2 sin2 x
[ab > 0]
2.
3.819
BI (182)(18)
π a4 + 2a2 b2 + 5b4
sin x cos2 x
dx
=
·
·
4
x
32
a7 b 5
a2 cos2 x + b2 sin2 x
[ab > 0]
BI (182)(19)
[ab > 0]
BI (181)(23)
[ab > 0]
BI (182)(26)
π a2 + 5b2
sin x cos3 x
dx
=
·
·
4
x
32
a7 b 3
a2 cos2 x + b2 sin2 x
[ab > 0]
BI (182)(23)
π a2 + b 2
sin3 x cos2 x
dx
=
· 5 5
·
4
x
32
a b
a2 cos2 x + b2 sin2 x
[ab > 0]
BI (182)(27)
π a2 + 5b2
sin x cos4 x
dx
=
·
·
4
x
32
a7 b 3
a2 cos2 x + b2 sin2 x
[ab > 0]
BI (182)(24)
[ab > 0]
BI (181)(24)
3
4
2 2
π 5a + a b + b
sin x
dx
=
·
4 ·
2
x
32
a5 b 7
a2 cos2 x + b2 sin x
3
2
π a +b
sin x cos x
dx
=
· 5 5
4 ·
2
x
32
a b
a2 cos2 x + b2 sin x
sin5 x
4
2
π 5a2 + b2
dx
=
·
4 ·
2
x
32
a3 b 7
a2 cos2 x + b2 sin x
3.821
Powers and trigonometric functions
∞
10.
0
∞
11.
0
∞
12.
0
∞
13.
0
∞
14.
0
∞
15.
0
∞
16.
0
459
sin3 x cos x
π 5a4 + 2a2 b2 + b4
dx
=
·
·
4
x
128
a5 b 7
a2 cos2 2x + b2 sin2 2x
π 5a2 + b2
sin5 x cos3 x
dx
=
·
·
4
x
512
a3 b 7
a2 cos2 2x + b2 sin2 2x
[ab > 0]
BI (182)(22)
[ab > 0]
BI (182)(30)
π 5a4 + 2a2 b2 + b4
sin2 x tan x
dx
=
·
4 ·
2
x
32
a5 b 7
a2 cos2 x + b2 sin x
4
2
sin x tan x
π 5a + b
dx
=
·
4 ·
2
2
2
2
x
32
a3 b 7
a cos x + b sin x
[ab > 0]
BI (182)(21)
[ab > 0]
BI (182)(29)
2
π a4 + 2a2 b2 + 5b4
cos2 2x tan x
dx
=
·
·
4
x
32
a7 b 5
a2 cos2 2x + b2 sin2 2x
3
2
π a +b
sin 4x tan x
dx
= · 5 5
4 ·
2
2
2
2
x
8
a b
a cos 2x + b sin 2x
[ab > 0]
BI (182)(29)
[ab > 0]
BI (182)(28)
[ab > 0]
BI (182)(25)
2
cos4 2x tan x
π a2 + 5b2
dx
=
·
·
4
x
32
a7 b 3
a2 cos2 2x + b2 sin2 2x
3.82–3.83 Powers of trigonometric functions combined with other powers
3.821 π
Γ(p + 1)
π2
x sinp x dx = p+1 +
1.
,2
p
2
0
+1
Γ
2
rπ
2
(2m
− 1)!! 2
π
·
r
2.
x sinn x dx =
2
(2m)!!
0
(2m)!!
r
= (−1)r+1 π
(2m + 1)!!
[p > −1]
BI(218)(7), LO V 121(71)
[n = 2m]
[n = 2m + 1]
[r is a natural number]
π/2
x cosn x dx =
3.11
0
=
4.
5.
m−1
1
π 2 (n − 1)!!
− n−2
8 (n)!!
2
1
π (n − 1)!!
− n−1
2 (n)!!
2
k=0,m−k odd
m−1
k=0
n
k
n
k
1
(n − 2k)2
1
(n − 2k)2
GW (333)(8c)
[n = 2m]
[n = 2m − 1]
GW (333)(9b)
π (2m − 1)!!
2 (2m)!!
0
sπ
(2m − 1)!!
π2 2
s − r2
x cos2m x dx =
2
(2m)!!
rπ
π
x cos2m x dx =
2
BI (218)(10)
BI (226)(3)
460
Trigonometric Functions
√
Γ p2
sinp x
π
p p
dx =
· p+1 = 2p−2 B
,
x
2 Γ 2
2 2
[p is a fraction with odd numerator and denominator]
∞
6.
0
∞
7.
0
∞
8.
0
∞
9.
0
∞
10.
0
∞
11.
0
3.822
∞
LO V 278, FI II 808
(2n − 1)!! π
sin2n+1 x
dx =
·
x
(2n)!!
2
BI (151)(4)
sin2n x
dx = ∞
x
BI (151)(3)
sin2 ax
aπ
dx =
2
x
2
[a > 0]
LO V 307, 312, FI II 632
sin2m ax
(2m − 3)!! aπ
·
dx =
x2
(2m − 2)!! 2
[a > 0]
GW (333)(14b)
a2 π
sin2m+1 ax
(2m − 3)!!
(2m + 1)
dx =
3
x
(2m)!!
4
[a > 0]
GW (333)(14d)
sinp x
dx
xm
12.
∞ p−1
sin
x
p
cos x dx
=
m−1
m−1 0
x
∞ p−2
∞ p
sin
x
sin x
p2
p(p − 1)
dx
−
dx
=
(m − 1)(m − 2) 0
xm−2
(m − 1)(m − 2) 0 xm−2
0
[p > m − 1 > 0]
[p > m − 1 > 1]
GW (333)(17)
∞
2n
sin px
√
dx = ∞
x
13.
0
∞
1
dx
sin2n+1 px √ = 2n
2
x
14.
0
3.822
BI (177)(5)
π/2
xp cosm x dx = −
1.
0
π (−1)k
2p
n
k=0
p(p − 1)
m2
2n + 1
n+k+1
π/2
xp−2 cosm x dx +
0
√
1
2k + 1
m−1
m
π/2
xp cosm−2 x dx
0
[m > 1,
∞
2.
x−1/2 cos2n+1 (px) dx =
0
∞
3.823
0
xμ−1 sin2 ax dx = −
1
22n
π
2p
n
k=0
Γ(μ) cos μπ
2
2μ+1 aμ
2n + 1
n+k+1
BI (177)(7)
p > 1]
1
√
2k + 1
[a > 0,
GW (333)(9a)
BI (177)(8)
−2 < Re μ < 0]
ET I 319(15), GW(333)(19c)a
3.824
1.
2.
∞
sin2 ax
π 1 − e−2aβ
dx =
2 + β2
x
4β
0
∞
cos2 ax
π 1 + e−2aβ
dx =
2
2
4β
0 x +β
[a > 0,
Re β > 0]
BI (160)(10)
[a > 0,
Re β > 0]
BI (160)(11)
3.824
Powers and trigonometric functions
∞
3.7
2m
sin
0
∞
4.7
dx
(−1)m π
x 2
= 2m+1 ·
2
a +x
2
2
sin2m+1 x
0
2
2m
sinh
2m
a−2
m
461
k
(−1)
k=0
dx
(−1)m−1
=
a2 + x2
22m+2 a
2m
k
sinh[2(m − k)a]
[a > 0]
e(2m+1)a
+ e−(2m+1)a
2m+1
BI (160)(12)
2m + 1 −2ka
Ei[(2k − 2m − 1)a]
e
k
2m + 1 2ka
e Ei[(2m + 1 − 2k)a]
k
(−1)k
k=0
2m+1
(−1)k−1
k=0
[a > 0]
∞
5.7
sin2m+1 x
0
6.
∞
7
cos2m x
0
x dx
π
= 2m+1 e−(2m+1)a
2
+x
2
a2
2m
dx
π
= 2m+1
2
2
a +x
2
a m
+
m
(−1)m+k
k=0
m
π 22m
k=1
BI (160)(14)
2m + 1 2ka
e
k
+
π,
|arg a| <
,
2
m = 0, 1, 2, . . .
2m
e−2ka
m+k
[a > 0]
∞
7.
dx
π
= 2m+1
2
2
a +x
2
a
m
cos2m+1 x
0
k=1
BI (160)(16)
2m + 1
e−(2k+1)a
m+k+1
[a > 0]
∞
8.
cos2m+1 x
0
2m+1
x dx
e−(2m+1)a =
−
a2 + x2
22m+2
−
0
10.
0
22m+2
k=0
2m + 1 2ka
e Ei[(2m − 2k + 1)a]
k
2m + 1 −2ka
Ei[(2k − 2m − 1)a]
e
k
BI (160)(18)
∞
9.
e
k=0
(2m+1)a 2m+1
BI (160)(17)
∞
2
cos ax
π
sin 2ab
dx =
2
2
b −x
4b
[a > 0,
b > 0]
sin2 ax cos2 bx
π
1 2(b−a)β
1 −2(a+b)β
−2bβ
−2aβ
dx =
+e
− e
−e
1− e
β 2 + x2
8β
2
2
π
1 − e−4aβ
=
16β
π
1 2(a−b)β
1 −2(a+b)β
−2bβ
−2aβ
=
+e
− e
−e
1− e
8β
2
2
[a > 0, b > 0] ,
(cf. 3.824 1
BI (161)(10)
[a > b]
[a = b]
[a < b]
and 3)
BI (162)(6)
462
Trigonometric Functions
∞
11.
0
3.825
,
x sin 2ax cos2 bx
π + −2aβ
−2(a+b)β
2(b−a)β
2e
dx
=
+
e
+
e
β 2 + x2
8
π −4aβ
=
e
+ 2e−2aβ
8
,
π + −2aβ
+ e−2(a+b)β − e2(a−b)β
=
2e
8
[a > 0]
[a = b]
[a < b]
LI (162)(5)
3.825
∞
1.
0
π b − c + ce−2ab − be−2ac
sin2 ax dx
=
(b2 + x2 ) (c2 + x2 )
4bc (b2 − c2 )
∞
π b − c + be−2ac − ce−2ab
cos2 ax dx
=
2
2
2
2
4bc (b2 − c2 )
0 (b + x ) (c + x )
2.
∞
3.3
0
(b2
π (c sin 2ab − b sin 2ac)
sin2 ax dx
=
− x2 ) (c2 − x2 )
4bc (b2 − c2 )
(b2
π (b sin 2ac − c sin 2ab)
cos ax dx
=
2
2
2
− x ) (c − x )
4bc (b2 − c2 )
[a > 0,
b > 0,
c > 0]
BI (174)(15)
[a > 0,
b > 0,
c > 0]
BI (175)(14)
[a > 0,
b > 0,
c > 0,
b = c]
LI (174)(16)
∞
4.3
0
2
[a > 0,
b > 0,
c > 0,
b = c]
LI (175)(15)
3.826
∞
1.
0
∞
2.
0
3.827
∞
1.8
0
π
sin2 ax dx
1
−2ab
= 2 2a −
1−e
x2 (b2 + x2 )
4b
b
[a > 0,
b > 0]
BI (172)(13)
π
sin2 ax dx
= 2
x2 (b2 − x2 )
4b
[a > 0,
b > 0]
BII (172)(14)
2a −
1
sin 2ab
b
sin3 ax
3 − 3ν−1 ν−1
νπ
a
Γ(1 − ν)
dx =
cos
ν
x
4
2
[a < Re ν < 4, ν = 1, 2, 3]
GW (333)(19f)
∞
2.8
0
∞
3.
0
∞
4.8
0
∞
5.
0
6.
0
∞
3
π
sin ax
dx =
x
4
LO V 277
sin3 ax
3
dx = a ln 3
2
x
4
BI (156)(2)
sin3 ax
3
dx = a2 π
x3
8
sin4 ax
aπ
dx =
2
x
4
sin4 ax
dx = a2 ln 2
x3
BI(156)(7)a,LO V 312
[a > 0]
BI (156)(3)
BI (156)(8)
3.828
Powers and trigonometric functions
∞
7.
0
∞
8.
0
∞
9.
0
∞
10.
0
∞
11.
0
∞
12.
0
∞
13.
0
∞
14.
0
∞
15.
0
3.828
1.8
2.8
3.8
sin4 ax
a3 π
dx
=
x4
3
[a > 0]
sin5 ax
5
a (3 ln 3 − ln 5)
dx =
x2
16
sin5 ax
5 2
a π
dx =
3
x
32
[a > 0]
sin5 ax
5 3
a (25 ln 5 − 27 ln 3)
dx =
x4
96
∞
0
BI (156)(12)
sin5 ax
115 4
a π
dx =
x5
384
[a > 0]
BI(156)(13), LO V 312
sin6 ax
3
aπ
dx =
2
x
16
[a > 0]
BI (156)(5)
sin6 ax
3 2
a (8 ln 2 − 3 ln 3)
dx =
x3
16
BI (156)(10)
sin6 ax
1 4
a (27 ln 3 − 32 ln 2)
dx =
5
x
16
BI (156)(14)
sin6 ax
11 5
a π
dx =
x6
40
sin2 ax cos 2bx
1
dx = π max(0, a − b)
2
x
2
∞
π
sin 2ax cos2 bx
dx =
x
2
3
= π
8
π
=
4
0
6.
0
1 4a2 − b2
sin2 ax cos bx
dx = ln
x
4
b2
∞
5.8
BI (156)(9)
[a > 0]
LO V 312
FI II 647
BI (157)(1)
BI (151)(10)
4.8
BI(156)(11), LO V 312
BI (156)(4)
In 3.828 1–21 the restrictions a > 0, b > 0, c > 0 apply.
!
!
∞
sin ax sin bx
1 !! a + b !!
[a = b]
dx = ln !
x
2
a − b!
0
∞
dx
1
sin ax sin bx 2 = π min(a, b)
x
2
0
∞ 2
π
sin ax sin bx
dx =
[b < 2a]
x
4
0
π
=
[b = 2a]
8
=0
[b > 2a]
463
[2a = b]
BI (151)(12)
[a > b]
[a = b]
[a < b]
BI (151)(9)
464
Trigonometric Functions
∞
7.8
0
3.828
sin2 ax sin bx sin cx
π
(|b − 2a − c| − |2a − b − c| + 2c)
dx =
2
x
16
[a > 0, 0 < c ≤ b]
!
!
∞
1 !! (b + c)2 (2a − b + c)(2a + b − c) !!
sin2 ax sin bx sin cx
dx = ln !
x
8
(b − c)2 (2a + b + c)(2a − b − c) !
0
[b = c, 2a + c = b, 2a + b = c,
BI(157)(9)a, ET I 79(15)
8.8
∞
9.
0
sin2 ax sin2 bx
π
dx = a
x2
4
π
= b
4
2a = b + c]
LI (152)(2)
[0 ≤ a ≤ b]
[0 ≤ b ≤ a]
BI (157)(3)
∞
10.8
0
11.
∞
8
0
∞
12.
0
∞
13.
0
14.10
0
∞
sin ax sin bx
1
dx = π min a2 , b2 [3 max(a, b) − min(a, b)]
4
x
6
2
2
BI (157)(27)
sin2 ax cos2 bx
1
dx = π [a + max(0, a − b)]
2
x
4
sin3 ax sin 3bx
a3 π
dx =
4
x
2
π 3
8a − 9(a − b)3
=
16
9bπ 2
a − b2
=
8
sin3 ax cos bx
dx = 0
x
π
=−
16
π
=−
8
π
=
16
π
=
4
BI (157)(6)
[b > a]
[a ≤ 3b ≤ 3a]
BI (157)(28)
[3b ≤ a]
LI (157)(28)
[b > 3a]
[b = 3a]
[3a > b > a]
[b = a]
[a > b]
⎛
[a > 0,
b > 0]
BI (151)(15)
sin3 ax cos 3bx
3 ⎝
a ln 81 − 2(a − 3b) ln(a − 3b) + 2(a − b) ln(a − b)
dx =
2
x
16
⎞
+ 2(a + b) ln(a + b) − 2(a + 3b) ln(a + 3)⎠
[Im a = 0,
Im b = 0]
MC
3.828
Powers and trigonometric functions
∞
15.
0
sin3 ax cos bx
π 2
3a − b2
dx =
3
x
8
πb2
=
4
π
=
(3a − b)2
16
=0
[b < a]
[a = b]
[a 0,
∞
16.
0
sin3 ax sin bx
bπ 2
9a − b2
dx =
4
x
24
π 24a3 − (3a − b)3
=
48
πa3
=
2
465
b > 0]
BI(157)(19), ET I 19(10)
[0 3a]
[2b = 3a]
[3a > 2b > a]
[2b = a]
[a > 2b]
[a > 0, b > 0]
!
!
! (2a + b)3 (b − 2a)3 (2a + 3b)(3b − 2a) !
1
sin ax cos3 bx
!
dx =
ln !!
!
x
16
9b8
[2a = b, 2a = 3b]
BI (151)(14)
2
2
2
BI (151)(13)
2
sin ax sin bx sin cx
dx
x
π
=
4 sign(c) − 2 sign(2b + c) + 2 sign(2b − c) + sign(2a − 2b + c) − sign(2a − 2b − c)
32
+ 2 sign(2a − c) + sign(2a + 2b + c) − sign(2a + 2b − c) − 2 sign(2a + c)
20.
0
[Im a = 0,
∞
Im b = 0,
Im c = 0]
MC
sin2 ax sin2 bx sin 2cx dx
x2
a−b−c
a+b+c
a+b−c
ln 4(a − b − c)2 −
ln 4(a + b + c)2 +
ln 4(a + b − c)2
=
16
16
16
a−b+c
a+c
a−c
ln 4(a − b + c)2 +
ln 4(a + c)2 −
ln 4(a − c)2
−
16
8
8
b+c
b−c
1
ln 4(b + c)2 −
ln 4(b − c)2 − c ln 2c
+
8
8
2
[a > 0, b > 0, c > 0]
BI (157)(10)
466
Trigonometric Functions
∞
21.8
0
sin2 ax sin3 bx
3b2 π
dx
=
x3
16
a2 π
=
12
6b2 − (3b − 2a)2
π
=
32
a2 π
=
4
3.829
[2a > 3b]
[2a = 3b]
[3b > 2a > b]
[b ≥ 2a]
BI (157)(18)
3.829
1.
2.
3.831
1.
2.
3.
4.
[(n−1)/2]
n
xn − sinn x
π
(n − 2k)n+1
dx
=
(−1)k
n+2
n (n + 1)!
k
x
2
0
k=0
∞
∞
dx
dx
mπ 2m
1 − cos2m−1 x 2 =
1 − cos2m x 2 = 2m
m
x
x
2
0
0
∞
(2n − 1)!! b
sin2n ax − sin2n bx
dx =
ln
[ab > 0,
x
(2n)!!
a
0
∞
cos2n ax − cos2n bx
(2n − 1)!!
b
dx = 1 −
[ab > 0,
ln
x
(2n)!!
a
0
∞
b
cos2m+1 ax − cos2m+1 bx
dx = ln
[ab > 0,
x
a
0
∞
1
cosm ax cos max − cosm bx cos mbx
b
dx = 1 − m ln
x
2
a
0
π/2
1.
x cos
0
p−1
x sin ax dx =
π
2p+1
Γ(p)
p+a+1 − ψ p−a+1
2
2
p−a+1
p+a+1
Γ
Γ
2
2
ψ
0
0
FI II 651
m = 0, 1, . . .]
FI II
m = 0, 1, . . .]
LI (155)(8)
−(p + 1) < a 0,
∞
3.
FI II 651
BI (205)(6)
∞
2.3
n = 1, 2, . . .]
[p > 0,
BI (158)(7, 8)
∞
[ab > 0,
3.832
GW (333)(63)
sin2m−1 x sin[(2m + 1)x]
BI (162)(17)
2m−1
π
dx
(−1)
1 − e−2a
=
a2 + x2
22m a
[a > 0,
∞
m = 0, 1. . . .]
m+1
a2
m = 1, 2, . . .]
BI (162)(11)
2m−1
dx
(−1)m−1 π −2a e
1 − e−2a
=
2
2m
+x
2 a
[a > 0,
m = 1, 2, . . .]
BI (162)(12)
3.832
Powers and trigonometric functions
5.
6.3
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.3
17.
18.3
∞
dx
(−1)m π −3(2m+1)a
e
=
sinh2m+1 a
a2 + x2
2a
0
[a > 0]
∞
+
m
,
x dx
(−1) π a −2a 2m
−2a
1
−
e
sin2m x sin[(2m − 1)x] 2
=
e
−
1
+
e
a + x2
22m+1
0
[a ≥ 0, m = 0, 1, . . .]
∞
,
+
x dx
(−1)m π −2a 2m
1
−
e
sin2m x sin(2mx) 2
=
−
1
a + x2
22m+1
0
[a > 0, m = 0, 1, . . .]
∞
m
2m
x dx
(−1) π
sin2m x sin[(2m + 2)x] 2
= 2m+1 e−2a 1 − e−2a
2
a +x
2
0
[a > 0, m = 0, 1, . . .]
∞
m
x dx
(−1) π −4ma
e
sin2m x sin 4mx 2
=
sinh2m a
2
a
+
x
2
0
[a > 0, m = 1, 2, . . .]
∞
dx
(2m − 3)!! π
·
[m = 1, 2, . . .]
sin2m x cos x 2 =
x
(2m)!!
2
0
∞
,
dx
(−1)m π +
−2a 2m−1
1
−
e
sin2m x cos[(2m − 1)x] 2
=
−
1
sinh a
a + x2
22m a
0
[a > 0, m = 1, 2, . . .]
∞
2m
dx
(−1)m π sin2m x cos(2mx) 2
= 2m+1 1 − e−2a
2
a +x
2
a
0
[a > 0, m = 0, 1, . . .]
∞
m
2m
dx
(−1) π
sin2m x cos[(2m + 2)x] 2
= 2m+1 e−2a 1 − e−2a
2
a +x
2
a
0
[a > 0, m = 0, 1, . . .]
∞
m
dx
(−1) π −4ma
e
sin2m x cos 4mx 2
=
sinh2m a
2
a
+
x
2a
0
[a > 0, m = 0, 1, . . .]
∞
(2m − 1)!! π
dx
=
·
[m = 0, 1, . . .]
sin2m+1 x cos x
x
(2m + 2)!! 2
0
∞
dx
(2m − 3)!! π
·
[m = 1, 2, . . .]
sin2m+1 x cos x 3 =
x
(2m)!!
2
0
∞
,
x dx
(−1)m π +
−2a 2m−1
1
−
e
sin2m−1 x cos[(2m − 1)x] 2
=
−
1
a + x2
22m
0
[m = 1, 2, . . . , a > 0]
∞
+
,
m−1 π a x dx
(−1)
−2a 2m+1
−a
e
1
−
e
sin2m+1 x cos 2mx 2
=
−
1
−
e
a + x2
22m+2
0
[m = 0, 1, . . . , a ≥ 0]
467
sin2m+1 x sin[3(2m + 1)x]
BI (162)(18)
BI (162)(13)
BI (162)(14)
BI (162)(15)
BI (162)(16)
GW (333)(15a)
BI (162)(25)
BI (162)(26)
BI (162)(27)
BI (162)(28)
GW (333)(15)
GW (333)(15b)
BI (162)(23)
BI (162)(29)
468
Trigonometric Functions
∞
19.
sin2m−1 x cos[(2m + 1)x]
0
3.832
2m−1
x dx
(−1)m π −2a 1 − e−2a
=
e
a2 + x2
22m
[m = 1, 2, . . . ,
∞
20.
sin2m+1 x cos[2(2m + 1)x]
0
∞
21.
0
cosm x sin mx
x dx
(−1)m−1 π −2(2m+1)a
e
=
sinh2m+1 a
2
+x
2
22.
cosn sx sin nsx
0
0
x dx
π
= n+1
2
+x
2
a2
cosn sx sin nsx
cosm−1 x sin[(m + 1)x]
cosm x sin[(m + 1)x]
0
∞
26.3
cosm x sin[(m − 1)x]
0
27.11
28.
29.
30.
Re a > 0,
n ≥ 0]
∞
BI (166)(10)
m−1
x dx
π
= m e−2a 1 + e−2a
2
+x
2
m = 1, 2, . . .]
x dx
π
= m cosh a
2
+x
2
+
a2
[m = 0, 1, . . . ,
,
m−1
1 + e−2a
−1
a > 0]
BI (163)(10)
[m = 0, 1, . . . ,
a ≥ 0]
BI (163)(7)
31.
0
cosm x sin(3mx)
[m = 1, 2, . . . ,
∞
cosm x cos[(m − 1)x]
BI (163)(6)
m
x dx
π
= m+1 e−a 1 + e−2a
a2 + x2
2
x dx
π
= e−3ma coshm a
[a > 0, m = 1, 2, . . .]
2 + x2
a
2
0
∞
n
dx
π cosn sx cos nxs 2
= n+1 1 + e−2as
[n = 0, 1, . . .]
2
a +x
2
a
0
∞
dx
π
cosn as sin nas
cosn sx cos nsx 2
=
[n = 0, 1, . . .]
a − x2
2a
0
∞
m−1
dx
π
cosm−1 x cos[(m + 1)x] 2
= m e−2a 1 + e−2a
2
a +x
2 a
0
BI (163)(9)
a2
[a > 0,
25.
BI (162)(8)
x dx
π −n
2 − cosn as cos nas
=
a2 − x2
2
0
∞
BI (162)(30)
+
,
n
1 + e−2as − 1
[n = 0, 1, . . .]
∞
24.
k=1
a > 0]
m −2ka
e
Ei(2ka) − e2ka Ei(−2ka)
k
[s > 0,
∞
23.
x dx
1
= m+1
+ x2
2
a
a2
m
[a > 0]
∞
BI (162)(24)
a2
[m = 0, 1, . . . ,
a > 0]
a2
dx
π
= m+1 e
2
+x
2
a
+
a
1 + e−2a
m
− 1 − e−2a
[m = 0, 1, . . . ,
BI (163)(11)
BI (163)(16)
a > 0]
,
BI (163)(14)
a > 0]
BI (163)(15)
3.834
Powers and trigonometric functions
∞
32.
cosm x cos[(m + 1)x]
0
∞
33.
0
∞
34.
cos2m x cos 2nx sin x
cosp ax sin bx cos x
0
∞
36.
dx
x=
x
a > 0]
BI (163)(17)
[p > q − 1 > 0]
[p > q − 1 > 1]
GW (333)(18)
∞
cos2m−1 x cos 2nx sin
0
∞
37.
0
π
dx
=
x
2
cosp ax sin pax cos x
0
[m = 0, 1, . . . ,
2m
m+n
π
dx
x = 2m+1
x
2
BI (152)(5, 6)
∞
35.
∞
sinp−1 x
dx
p
p + 1 ∞ sinp+1 x
=
dx −
dx
q
q−1
x
q−1 0
x q−1 0
xq−1
∞
p(p − 1)
dx
=
sinp−2 x cos x q−2
(q − 1)(q − 2) 0
x
∞
(p + 1)2
dx
−
sinp x cos x q−2
(q − 1)(q − 2) 0
x
0
m
dx
π
= m+1 e−a 1 + e−2a
a2 + x2
2
a
sinp x cos x
469
dx
x2
n
-
k=1
p > −1]
[b > ap,
π
dx
= p+1 (2p − 1)
x
2
[p > −1]
%
π
cospk ak x sin bx sin x =
2
b>
BI (153)(12)
BI (153)(2)
n
&
ak pk ,
ak > 0,
pk > 0
k=1
BI (157)(15)
3.833
1.
∞
10
2m+1
sin
0
x cos
2n
dx
=
x
x
∞
sin2m+1 x cos2n−1 x
0
(2m − 1)!!(2n − 1)!!
dx
=
π
x
2m+n+1 (m + n)!
BI (151)(24, 25)
1
1
1
= B m + ,n +
2
2
2
GW (333)(24)
∞
2.
sin2m+1 2x cos2n−1 2x cos2 x
0
3.834
1.
0
∞
π (2m − 1)!!(2n − 1)!!
dx
= ·
x
2
(2m + 2n)!!
sin2m+1 x
dx (−1)m π(1 + a)4m
=
·
2
1 − 2a cos x + a
x
22m+2 a2m+1
−
2m
k=0
k
(−1)
m−
k
LI (152)(4)
!
!
! 1 − a !2m−1
!
!
!1 + a!
1
2
4a
(1 + a)2
k
[|a| = 1]
GW (333)(62a)
470
Trigonometric Functions
∞
2.
0
sin2m+1 x cosn x
dx
p ·
2
x
(1 − 2a cos x + a )
=
3.835 1.
∞
0
∞
2.
0
3.836 1.
2.
n
(−1)k (2m + 2n − 2k + 1)!!(2m + 2k − 1)!!
n!π
2n+1 (2m + n + 1)!(1 + a)2p
k!(n − k)!
k=0
4a
3
× F m + n − k + , p; 2m + n + 2;
2
(1 + a)2
[a = ±1]
GW (333)(62)
π b2m−1
cos2m x cos 2mx sin x dx
=
·
2
x
2 a(a + b)2m
a2 cos2 x + b2 sin x
[ab > 0]
BI (182)(31)a
π b2m−1
cos2m−1 x cos 2mx sin x dx
=
·
x
2a (a + b)2m
a2 cos2 x + b2 sin2 x
[ab > 0]
LI (182)(32)a
[m ≥ n]
LI (159)(12)
∞
0
3.835
∞
11
0
sin x
x
n
sin x
x
n
π
sin mx
dx =
x
2
nπ
cos mx dx = n
2
1
2 (m+n)
k=0
(−1)k (n + m − 2k)n−1
k!(n − k)!
[0 ≤ m < n]
[m ≥ n ≥ 2]
=0
π
=
4
[m = n = 1]
GI(159)(14), ET I 20(11)
∞
3.
0
∞
4.8
0
sin x
x
n−1
sin x
x
n
sin nx cos x
π
dx
=
x
2
⎡
π⎢
sin(anx)
1
dx = ⎣1 − n−1
x
2
2
n!
[n ≥ 1]
⎤
1
2 n(1+a)
BI (159)(20)
(−1)k
k=0
n
⎥
(n + an − 2k)n ⎦
k
[all real a, n ≥ 1]
5.10
I n (b) =
0
π/2
1.
0
2.
0
n
−1 sin x
n
(n − b − 2k)n−1
cos bx dx = n 2n−1 n!
(−1)k
k
x
0
k=0
where 0 ≤ b < n, n ≥ 1, r = (n − b)/2, and r is the largest integer contained in r
r
∞
LO V 340(14)
∞
6.11
3.837
2
π
ET I 20(11)
π/4
sin x
x
n
cos anx dx = 0
[a ≤ −1 or a ≥ 1,
x2 dx
= π ln 2
sin2 x
π
x2 dx
π2
+ ln 2 + G = 0.8435118417 . . .
=
−
2
16
4
sin x
n ≥ 2;
for n = 1 see 3.741 2]
BI (206)(9)
BI (204)(10)
3.839
Powers and trigonometric functions
π/4
3.
0
π/4
4.
0
π2
π
x2 dx
=
+ ln 2 − G
2
cos x
16
4
π
xp+1
dx = −
4
sin2 x
p+1
471
GW (333)(35a)
π
+ (p + 1)
4
p
∞
1
1 1
−
ζ(2k)
p 2
42k−1 (p + 2k)
k=1
[p > 0]
π/2
5.
0
π/2
6.
0
∞
7.
0
∞
8.
0
1
9.
0
π
10.3
0
11.3
0
3.838
π
BI (206)(7)
3
x3 cos x
π3
+ π ln 2
dx = −
3
16 2
sin x
BI (206)(8)
cos 2nx
dx
sin2n+1 x m = 0
cos x
x
1
x dx
= cosec a · ln sec a
cos ax cos[a(1 − x)]
a
m−1
,
n>
2
m−2
,
n>
2
+
π,
a<
2
1
x sin(2n + 1)x
dx = π 2
sin x
2
[n = 0, 1, 2, . . .]
x sin 2nx
dx = −4
(2k − 1)−2
sin x
[n = 1, 2, 3, . . .]
π/2
π/4
2.
0
4.
m>0
BI (180)(16)
m>0
BI (180)(17)
BI (149)(20)
n
πp
x cosp−1 x
π
sec
dx =
2p
2
sinp+1 x
π
1
x sinp−1 x
dx =
−
β
p+1
cos
x
4p 2p
[p < 1]
p+1
2
[p > −1]
m−1
π
1 (−1)k−1
x sin2m−1 x
dx
=
(1
−
cos
mπ)
+
cos2m+1 x
8m
2m
2m − 2k − 1
0
k=0
%
&
π/4
m−1
(−1)k−1
1
π
x sin2m x
dx =
+ (−1)m−1 ln 2 +
cos2m+2 x
2(2m + 1) 2
m−k
0
3.
k=1
0
π2
x2 cos x
dx = −
+ 4G = 1.1964612764 . . .
2
4
sin x
cos 2nx
dx
sin2n x m = 0
cos x
x
1.
LI (204)(14)
BI (206)(13)a
LI (204)(15)
π/4
BI (204)(17)
BI (204)(16)
k=0
3.839
1.
π/4
11
x tan2 x dx =
1
π π2
−
− ln 2
4
32 2
BI (204)(3)
x tan3 x dx =
1
π 1 π
− + ln 2 − G
4
2
8
2
BI (204)(7)
0
π/4
2.
0
3.
0
π/4
1
π π2
x2 tan x
dx
=
ln
2
−
+
cos2 x
2
4
16
(cf. 3.839 1)
BI (204)(13)
472
Trigonometric Functions
π/4
4.
0
1
x2 tan2 x
dx =
cos2 x
3
1−
π/2
x cosp x tan x dx =
5.
0
π/2
x sinp x cot x dx =
6.
0
∞
7.
sin2n x tan x
0
∞
8.
∞
0
∞
10.
coss rx tan qx
0
q2
Γ(p + 1)
·+
,2
p
+1
Γ
2
2p−1
π
−
B
2p
p
BI (204)(12)
[p > −1]
BI (205)(3)
[p > −1]
BI (206)(11)
p+1 p+1
,
2
2
π
dx
=
x
2
cos[(2n − 1)x]
·
cos x
tanr px
π
2p+1 p
(cf. 3.839 2)
π (2n − 1)!!
dx
= ·
x
2
(2n)!!
0
9.
π π2
π
ln 2 − +
+G
4
2
16
3.841
sin x
x
[s > −1]
2n
dx = (−1)n−1
rπ
dx
π
sec
tanhr pq
=
2
+x
2q
2
3.84 Integrals containing
GW (333)(16)
BI (151)(26)
22n − 1 2n−1
·2
π|B2n |
(2n)!
2
r <1
BI (180)(15)
BI (160)(19)
√
1 − k2 sin2 x, 1 − k2 cos2 x, and similar expressions
√
Notation: k = 1 − k 2
3.841
∞
dx
= E(k)
sin x 1 − k 2 sin2 x
1.
x
0
∞
dx
= E(k)
2.
sin x 1 − k 2 cos2 x
x
0
∞
dx
= E(k)
3.
tan x 1 − k 2 sin2 x
x
0
∞
dx
= E(k)
4.
tan x 1 − k 2 cos2 x
x
0
3.842
1.
∞
11
0
dx
sin x
2
1 + sin x x
∞
tan x
dx
=
·
2
x
0 ∞ 1 + sin x
∞
sin x
tan x
1
dx
dx
√
√
=
=
=√ K
2
2
2
1 + cos x x
1 + cos x x
0
0
BI (154)(8)
BI (154)(20)
BI (154)(9)
BI (154)(21)
1
√
2
≈ 1.3110287771
BI (183)(4, 5, 9, 10)
2.
u
π
2
x cos x dx
sin x − sin u
2
2
=
π
ln (1 + cos u)
2
BI (226)(4)
3.844
Integrals containing
∞
3.
0
1 − k 2 sin2 xf unction]squareroot and similar expressions
∞
dx
tan x
=
2
2
2
2
x
1 − k sin x
0 ∞ 1 − k sin x
sin x
√
=
1
−
k 2 cos2 x
0
sin x
473
dx
x
∞
tan x
dx
dx
√
=
= K(k)
2 cos2 x x
x
1
−
k
0
BI (183)(12, 13, 21, 22)
π/2
4.
0
π/2
5.
0
6.
0
7.
3.843
1.
2.
3.11
α
dx
= E(k)
tan x 1 − k 2 sin2 2x
x
0
∞
dx
= E(k)
tan x 1 − k 2 cos2 2x
x
0
∞
∞
dx
1
tan x
tan x
dx
√
=
=√ K
2
2
2
1 + cos 2x x
0
0
1 + sin 2x x
∞
∞
1.
0
4.
5.
6.
BI (214)(1)
LO III 284
LO III 284
∞
0
3.
x sin x cos x
1
√
dx = 2 [π − 2 E(k)]
2
2
2k
1 − k cos x
π sin2 α2
x sin x dx
=
cos2 α
cos2 x sin2 α − sin2 x
4.
2.
BI (211)(1)
cos α + 1 − sin2 α sin2 β
π ln
β
2 cos β cos2 α2
x sin x dx
=
0
1 − sin2 α sin2 x
sin2 β − sin2 x
2 cos α 1 − sin2 α sin2 β
3.844
x sin x cos x
1
dx = 2 [−πk + 2 E(k)]
2
2
2k
1 − k sin x
tan x
dx
=
2
1 − k2 sin 2x x
0
BI (154)(10)
BI (154)(22)
1
√
2
≈ 1.3110287771
BI (183)(6, 11)
∞
√
tan x
dx
= K(k)
2
2
1 − k cos 2x x
1
sin x cos x dx
√
= 2 [K(k) − E(k)]
2
2
k
1 − k cos x x
BI (183)(14, 23)
BI (185)(20)
∞
1
sin x cos2 x
dx
√
= 2 [K(k) − E(k)]
·
2
2
x
k
1 − k cos x
0
∞
sin x cos3 x
1 dx
√
= 4 2 + k 2 K(k) − 2 1 + k2 E(k)
·
2
2
3k
1 − k cos x x
0
∞
4
1 sin x cos x
dx
√
= 4 2 + k 2 K(k) − 2 1 + k2 E(k)
·
2
2
x
3k
1 − k cos x
0
∞
1 sin3 x cos x
dx
√
= 4 1 + k 2 E(k) − 2k 2 K(k)
·
3k
1 − k 2 cos2 x x
0
∞
3
2
1 sin x cos x
dx
√
= 4 1 + k 2 E(k) − 2k 2 K(k)
·
2
2
3k
1 − k cos x x
0
BI (185)(21)
BI (185)(22)
BI (185)(23)
BI (185)(24)
BI (185)(25)
474
Trigonometric Functions
7.
8.
3.845
1.11
2.11
3.11
∞
sin2 x tan x
1 dx
√
= 2 E(k) − k2 K(k)
·
2
2
k
1 − k cos x x
0
∞
4
1 sin x tan x
dx
√
= 4 2 + 3k 2 k 2 K(k) − 2 k 2 − k 2 E(k)
·
2
2
x
3k
1 − k cos x
0
√ &
% √ 1
sin x cos x
2
2
dx √
√
= 2 E
− K
≈ 0.5990701174
·
2
x
2
2
2
1 + cos x
0
√ &
% √ ∞
sin x cos2 x dx √
2
2
1
√
= 2 E
·
− K
≈ 0.5990701174
2
x
2
2
2
1 + cos x
0
√ &
% √ ∞
sin2 x tan x dx √
2
2
√
= 2 K
−E
≈ 0.7119586598
·
2
x
2
2
1 + cos x
0
3.846 1.
3.845
BI (184)(16)
BI (184)(18)
∞
BI (185)(6)
BI (185)(7)
BU (184)(8)
∞
1 sin x cos x
dx
= 2 E(k) − k2 K(k)
BI (185)(9)
·
2
2
x
k
0
1 − k sin x
∞
1 sin x cos2 x
dx
= 2 E(k) − k2 K(k)
2.
BI (185)(10)
·
2
2
x
k
0
1 − k sin x
∞
1 sin x cos3 x
dx
= 4 2 − 3k2 k 2 K(k) − 2 k 2 − k 2 E(k)
3.
BI (185)(11)
·
2
2
x
3k
0
1 − k sin x
∞
1 sin x cos4 x
dx
= 4 2 − 3k2 k 2 K(k) − 2 k 2 − k 2 E(k)
4.
BI (185)(12)
·
2
2
x
3k
0
1 − k sin x
∞
1 sin3 x cos x
dx
= 4 1 + k2 E(k) − 2k 2 K(k)
5.
BI (185)(13)
·
2
2
x
3k
0
1 − k sin x
∞
1 sin3 x cos2 x
dx
= 4 1 + k2 E(k) − 2k 2 K(k)
6.
BI (185)(14)
·
2
3k
0
1 − k2 sin x x
∞
1
sin2 x tan x
dx
= 2 [K(k) − E(k)]
7.
BI (184)(9)
·
2
k
0
1 − k2 sin x x
∞
1 sin4 x tan x
dx
= 4 2 + k2 K(k) − 2 1 + k 2 E(k)
8.
BI (184)(11)
·
2
3k
0
1 − k2 sin x x
% √ √ &
∞
∞
sin x cos x
sin x cos2 x dx √
2
2
dx
11
=
= 2 K
3.847
·
·
−E
≈ 0.7119586598
2
2
x
x
2
2
0
0
1 + sin x
1 + sin x
BI (185)(3, 4)
3.848
1.
2.
∞
sin3 x cos x
1
dx
= 2 [K(k) − E(k)]
·
2
2
x
4k
0
1 − k sin 2x
∞
1 cos2 2x tan x
dx
= 2 E(k) − k2 K(k)
·
k
0
1 − k2 sin2 2x x
BI (185)(15)
BI (184)(12)
3.852
Trigonometric functions with powers
3.
4.
5.
6.
7.
3.849
1.11
2.11
3.11
475
∞
cos4 2x tan x
1 dx
= 4 2 − 3k2 k 2 K(k) − 2 k 2 − k 2 E(k)
·
2
3k
0
1 − k2 sin 2x x
∞
2
4 sin 4x tan x
dx
= 4 1 + k2 E(k) − 2k 2 K(k)
·
2
3k
0
1 − k2 sin 2x x
∞
3
sin x cos x
1 dx
√
= 2 E(k) − k2 K(k)
·
2
2
4k
1 − k cos 2x x
0
∞
2
1
cos 2x tan x
dx
√
= 2 [K(k) − E(k)]
·
2
2
x
k
1 − k cos 2x
0
∞
1 cos4 2x tan x
dx
√
= 4 2 + k2 K(k) − 2 1 + k 2 E(k)
·
2
2
3k
1 − k cos 2x x
0
√ &
% √ 1
sin3 x cos x
2
2
dx
√
= √ K
−E
≈ 0.1779896649
·
2
x
2
2
2 2
1 + cos 2x
0
√ √ &
√ %
∞
sin3 x cos x
2
2
2
dx
=
2E
−K
≈ 0.1497675293
·
2
x
8
2
2
0
1 + sin 2x
√ &
% √ ∞
cos2 2x tan x dx √
2
2
= 2 K
−E
≈ 0.7119586598
·
2
x
2
2
0
1 + sin 2x
BI (184)(13)
BI (184)(17)
BI (185)(26)
BI (184)(19)
BI (184)(20)
∞
BI (185)(8)
BI (185)(5)
BI (184)(7)
3.85–3.88 Trigonometric functions of more complicated arguments combined with
powers
3.851
∞
5.
0
dx
bπ
sin ax2 cos(bx) 2 =
x
2
S
b
√
2 a
−C
b
√
2 a
+
b2
π
+
4a
4
b > 0] ,
(cf. 3.691 7)
√
aπ sin
[a > 0,
ET I 23(3)a
3.852
∞
1.
0
∞
2.
0
sin ax2
aπ
dx =
x2
2
√
dx 1 π √
sin ax2 cos bx2 2 =
a+b+ a−b
x
2 2
1√
πa
=
2
√
1 π √
a+b− b−a
=
2 2
[a ≥ 0]
[a > b > 0]
[b = a ≥ 0]
[b > a > 0] ,
3.
0
∞
√
sin2 a2 x2
2 π 3
a
dx
=
x4
3
BI (177)(10)a
[a ≥ 0]
(cf. 3.852 1)
BI (177)(23)
GW (333)(19e)
476
Trigonometric Functions
√
sin3 a2 x2
a π
dx =
3− 3
2
x
4 2
0
∞
2
1 π
2
2 dx
sin x − x cos x
=
x4
3 2
0
∞
dx
1
1
=− C
cos2 x −
2
1+x
x
2
0
4.
10
5.
6.
3.853
∞
∞
1.
0
2.
3.
4.
3.854
3.853
Im a2 = 0
MC
BI (178)(8)
BI (173)(22)
√ √
√ ,
sin ax2
π +√
π
π
C
S
dx =
2 sin aβ 2 +
aβ − 2 cos aβ 2 +
aβ − sin aβ 2
2
2
β +x
2β
4
4
ET II 219(33)a
[a > 0, Re β > 0]
+
2 √
√ √
√ ,
cos ax2
π
π
π
2
2
cos
C
S
aβ
−
dx
=
2
cos
aβ
+
aβ
−
2
sin
aβ
+
aβ
β 2 + x2
2β
4
4
0
ET II 221(51)a
[a > 0, Re β > 0]
2
∞ 2
x sin ax
dx
β 2 + x2
0
√ √
√ ,
π
π
βπ + 2 √
sin aβ − 2 sin aβ 2 +
C
S
aβ + 2 cos aβ 2 +
aβ
=
2 4
4
1 π
−
2 2a
ET II 219(32)a
[a > 0, Re β > 0]
2
∞ 2
√ √
x cos ax
βπ
1 π
π
−
cos aβ 2 − 2 cos aβ 2 +
C
dx =
aβ
2
2
β +x
2 2a
2
4
0
√
√ π
S
aβ
− 2 sin aβ 2 +
4
ET II 221(50)a
[a > 0, Re β > 0]
∞
∞
1.
cos ax2 − sin ax2
0
dx
πe−ab
√
=
x4 + b4
2b3 2
2
[a > 0,
b > 0]
LI (178)(11)a, BI (168)(25)
∞
2.
0
∞
3.
x2 dx
πe−ab
√
cos ax2 + sin ax2
=
x4 + b4
2b 2
2
cos ax2 + sin ax2
0
x2 dx
(x4
+
2
b4 )
[a > 0, b > 0]
=
πe−ab
√
4 2b3
=
πe−ab
√
4 2b
2
a+
1
2b2
[a > 0,
4.
0
∞
cos ax2 − sin ax2
x4 dx
(x4
+
2
b4 )
2
LI (178)(12)
b > 0]
LI (178)(14)
b > 0]
BI (178)(15)
1
−a
2b2
[a > 0,
3.856
3.855
1.
2.
3.
4.
Trigonometric functions with powers
sin ax2
aβ
aβ
1 aπ
I 14
dx =
K 14
2
4
2
2
2
2
β +x
0
2
∞
cos ax
aβ
aβ
1 aπ
I − 14
dx =
K 14
2
4
2
2
2
2
β +x
0
u
2
sin a2 x2
a2
a π3
√
dx =
2
J 14
4
2
u2
u4 − x4
0
∞
sin a2 x2
a2 u 2
a2 u 2
a π3
√
J 14
dx = −
Y 14
4
2
2
2
x4 − u4
u
5.
6.
∞
2
cos a2 x2
a2 u 2
a π3
√
dx =
J − 14
4
2
2
u4 − x4
0
2 2
∞
cos a x
a2 u 2
a π3
√
J − 14
dx = −
Y − 14
4
2
2
x4 − u4
u
∞
1.
0
∞
2.
0
[a > 0,
Re β > 0]
ET I 66(28)
[a > 0,
Re β > 0]
ET I 9(22)
[a > 0]
ET I 66(29)
[a > 0]
ET I 66(30)
u
3.856
477
∞
3.
0
β 4 + x4 + x2
β 4 + x4
β 4 + x4 + x2
β 4 + x4
β 4 + x4 − x2
β 4 + x4
ν
2 2
sin a x
a
dx =
2
ν
a
cos a2 x2 dx =
2
ν
a
cos a2 x2 dx =
2
ET I 9(23)
a2 u 2
2
π 2ν
β I 14 − ν2
2
ET I 10(24)
a2 β 2
2
K 14 + ν2
3
Re ν < ,
2
a2 β 2
2
|arg β| <
π
4
a2 β 2
a2 β 2
π 2ν
β I − 14 − ν2
K − 14 + ν2
2
2
2
π
3
Re ν < , |arg β| <
2
4
ET I 71(23)
ET I 12(16)
a2 β 2
a2 β 2
π 2ν
β I − 14 + ν2
K − 14 − ν2
2
2
2
π
3
Re ν > − , |arg β| <
2
4
ET I 12(17)
2 2
∞
sin a2 x2 dx
sinh a 2β
<
= √
K0
2β 2
0
β 4 + x4 x2 + β 4 + x4
4.
5.
0
∞
cos a2 x2 dx
4
4
x2 + β 4 + x4
β +x
2
3
2
a2 β 2
2
sinh a β
= √ 2 K1
2 2β 4
+
π,
|arg β| <
4
ET I 66(32)
a2 β 2
2
+
π,
|arg β| <
4
ET I 10(27)
478
Trigonometric Functions
∞
6.
0
3.857 1.
∞
0
∞
2.
0
3.858 1.
<
β 4 + x4 + x2
a2 β 2
π
sin a2 x2 dx = √ e− 2 I 0
2 2
β 4 + x4
x2
R1 R2
x2
R1 R2
3.857
a2 β 2
2
+
π,
|arg β| <
4
1
R2 − R1
sin ax2 dx = √ K 0 (ac) sin ab
R2 +
2 b
R1 <
<
2
2
2
2
R1 = c + (b − x ) , R2 = c2 + (b + x2 ) ,
1
R2 + R1
cos ax2 dx = √ K 0 (ac) cos ab
R2 −
R
2
b
1
<
<
2
2
2
2
R1 = c + (b − x ) , R2 = c2 + (b + x2 ) ,
ET I 67(33)
a > 0,
c>0
ET I 67(34)
a > 0,
c>0
ET I 10(26)
√
2 √
ν ν
x + x4 − u4 + x2 − x4 − u4
√
sin a2 x2 dx
4
4
u
x − u a2 u 2
a2 u 2
a2 u 2
a2 u 2
a π 3 2ν
u
=−
J 14 + ν2
Y 14 − ν2
+ J 14 − ν2
Y 14 + ν2
4
a
2
2
2
2
3
Re ν < 2
ET I 71(25)
√
ν ν
∞ 2 √ 4
x + x − u4 + x2 − x4 − u4
√
2.
cos a2 x2 dx
4
4
x −u
u
a2 u 2
a2 u 2
a2 u 2
a2 u 2
a π 3 2ν
1
ν
1
ν
1
ν
1
ν
u
=−
J−4+2
Y −4−2
+ J−4−2
Y −4+2
4
a
2
2 2
2
3
Re ν < 2
ET I 13(26)
∞
n
dx
1
1
= − nC
3.859
BI (173)(24)
cos x2 −
n+1
2
x
2
1+x
0
3.861
√ m− 1
∞
n+1
2 dx
1
2n + 1
πa 2
2n+1
ax
sin
=
±
(−1)k−1
1.
(2k − 1)m− 2
1
2m
2n−m+
n
+
k
x
2 (2m − 1)!!
2
0
k=1
the + sign is taken when m ≡ 0 (mod 4) or m ≡ 1 (mod 4),
BI (177)(19)a
the − sign is taken when m ≡ 2 (mod 4) or m ≡ 3 (mod 4)
√ m− 1
∞
n
dx
1
2n
πa 2
2.
sin2n ax2 2m = ± 2n−2m+1
(−1)k
k m− 2
n+k
x
2
(2m − 1)!!
0
k=1
the + sign is taken when m ≡ 0 (mod 4) or m ≡ 3 (mod 4),
BI (177)(18)a, LI (177)(18)
the − sign is taken when m ≡ 2 (mod 4) or m ≡ 1 (mod 4)
∞
2√ 2 √ sin2 x n
cos ax n + sin ax n
3.862
dx
x2
0
√
n
√ n− 12
n π
√
n − 2k + a n
=
(−1)k
k
(2n − 1)!! 2 k=0
√
a> n>0
BI (178)(9)
∞
3.866
3.863
Trigonometric functions with powers
∞
1.
0
∞
2.
0
3.864
1.
2.
π
x cos ax4 sin 2bx2 dx = −
8
2
π
x2 cos ax4 cos 2bx2 dx = −
8
b3
sin
a3
b2
π
−
2a
8
479
b2
2a
J − 14
[a > 0,
b3
sin
a3
b2
π
+
2a
8
√
√
dx
π
b
sin ax
= Y 0 2 ab + K 0 2 ab
x
x
2
0
∞
√
√
dx
π
b
= − Y 0 2 ab + K 0 2 ab
cos cos ax
x
x
2
0
+ cos
b2
π
−
2a
8
J 34
b > 0]
b2
2a
J − 34
b2
2a
ET I 25(22)
+ cos
b2
π
+
2a
8
J − 14
b2
2a
[a > 0,
b > 0]
ET I 25(23)
[a > 0,
b > 0]
WA 204(3)a
[a > 0,
b > 0]
∞
sin
WA 204(4)a, ET I 24 (12)
3.865
u
1.
0
∞
2.
u
u2 − x2
x2μ
μ−1
sin
(x − u)
x2μ
μ−1
sin
√
π
a
dx =
x
2
a
dx =
x
2
a
μ− 12
a
u
[a > 0,
3
uμ− 2 Γ(μ) J 12 −μ
u > 0,
0 < Re μ < 1]
ET II 189(30)
π 1 −μ
a
a 2 Γ(μ) sin
J 1
u
2u μ− 2
a
2u
[a > 0,
u > 0,
Re μ > 0]
ET II 203(21)
u
3.
0
μ−1
√
u2 − x2
π
a
dx
=
−
cos
x2μ
x
2
μ− 12
2
a
3
Γ(μ)uμ− 2 Y
1
2 −μ
[a > 0,
∞
4.
u
(x − u)
x2μ
μ−1
cos
a
dx =
x
a
u
u > 0,
0 < Re μ < 1]
ET II 190(36)
π 1 −μ
a
a 2 Γ(μ) cos
J 1
u
2u μ− 2
a
2u
[a > 0,
u > 0,
Re μ > 0]
ET II 204(26)
3.866
∞
1.
xμ−1 sin
0
2.
0
b2
π
sin a2 x dx =
x
4
b
a
μ
cosec
μπ
[J μ (2ab) − J −μ (2ab) + I −μ (2ab) − I μ (2ab)]
2
[a > 0, b > 0, |Re μ| < 1]
ET I 322(42)
∞
xμ−1 sin
b2
π
cos a2 x dx =
x
4
b
a
μ
sec
μπ
[J μ (2ab) + J −μ (2ab) + I μ (2ab) − I −μ (2ab)]
2
[a > 0, b > 0, |Re μ| < 1]
ET I 322(43)
480
Trigonometric Functions
∞
3.
0
xμ−1 cos
b2
π
cos a2 x dx =
x
4
b
a
μ
cosec
3.867
μπ
[J −μ (2ab) − J μ (2ab) + I −μ (2ab) − I μ (2ab)]
2
[a > 0, b > 0, |Re μ| < 1]
ET I 322(44)
3.867
1
1.
0
2.
0
3.868
1
cos ax − cos xa
1
dx =
1 − x2
2
cos ax + cos xa
1
dx =
1 + x2
2
∞
1.
b2
x
sin a2 x +
0
∞
3.
∞
4.
∞
0
[a > 0]
GW (334)(7a)
[a > 0]
GW (334)(7b)
cos ax + cos xa
π
dx = e−a
1 + x2
2
dx
= π J 0 (2ab)
x
[a > 0,
b > 0]
∞
2
dx
= −π Y 0 (2ab)
x
[a > 0,
b > 0]
GW (334)(11a)
sin a2 x −
b2
x
dx
=0
x
[a > 0,
b > 0]
GW (334)(11b)
cos a2 x −
b2
x
dx
= 2 K 0 (2ab)
x
[a > 0,
b > 0]
GW (334)(11b)
b
x dx
π
= exp −αβ −
2
+x
2
β
[a > 0,
b > 0,
0
sin ax −
0
b
x
0
3.869 1.
0
cos ax − cos xa
π
dx = sin a
1 − x2
2
cos a2 x +
0
∞
GW (334)(11a), WA 200(16)
∞
2.
b
x
β2
Re β > 0]
ET II 220(42)
∞
2.
0
cos ax −
b
x
dx
b
π
exp −aβ −
=
β 2 + x2
2β
β
[a > 0,
b > 0,
Re β > 0]
ET II 222(58)
3.871
∞
1.
0
2.
0
∞
+
b2
μπ
μπ ,
− Y μ (2ab) sin
xμ−1 sin a x +
dx = πbμ J μ (2ab) cos
x
2
2
[a > 0, b > 0, Re μ < 1]
+
b2
μπ
μπ ,
μ−1
+ Y μ (2ab) cos
x
cos a x +
dx = −πbμ J μ (2ab) sin
x
2
2
[a > 0, b > 0, |Re μ| < 1]
∞
b2
μπ
xμ−1 sin a x −
dx = 2bμ K μ (2ab) sin
x
2
0
ET I 321(35)
3.
ET I 319(17)
[a > 0,
b > 0,
|Re μ| < 1]
ET I 319(16)
3.874
Trigonometric functions with powers
∞
4.
μ−1
x
0
481
b2
μπ
cos a x −
dx = 2bμ K μ (2ab) cos
x
2
[a > 0,
b > 0,
|Re μ| < 1]
ET I 321(36)
3.872
1.
2.
3.873
1.
2.
3.874
dx
1
1
sin a x +
sin a x −
x
x 1 − x2 0
dx
1 ∞
1
π
1
=
sin a x +
= − sin 2a
sin a x −
2 0
x
x
1 − x2
4
[a ≥ 0]
BI (149)(15), GW (334)(8a)
1
dx
1
1
cos a x +
cos a x −
2
x
x
1
+
0
∞x
dx
1
1
π
1
=
cos a x +
= e−2a
cos a x −
2 0
x
x
1 + x2
4
[a ≥ 0]
GW (334)(8b)
1
√
π a2
2 2 dx
√
sin(2ab) + cos(2ab) + e−2ab
cos
b
x
=
2
2
x
x
4 2a
0
[a > 0,
√
∞
2
dx
π a
cos(2ab) − sin(2ab) + e−2ab
cos 2 cos b2 x2 2 = √
x
x
4 2a
0
[a > 0,
∞
sin
∞
1.
sin a2 x2 +
0
∞
2.
cos a2 x2 +
0
∞
3.
∞
∞
0
∞
2
cos a2 x2 −
b2
x2
b
x
2
sin ax −
b
x
2
cos ax −
0
6.
b
x2
b
x2
0
5.
2
sin a2 x2 −
0
4.
b2
x2
√
dx
π
π
sin 2ab +
=
2
x
2b
4
[a > 0,
√
dx
π
= − √ e−2ab
x2
2 2b
√
dx
π
= √ e−2ab
x2
2 2b
√
dx
2π
=
x2
4b
√
dx
2π
=
x2
4b
ET I 24(15)
b > 0]
ET I 24(16)
b > 0]
BI (179)(6)a, GW(334)(10a)
√
dx
π
π
cos 2ab +
=
2
x
2b
4
b > 0]
[a > 0,
b > 0]
GI (179)(8)a, GW(334)(10a)
[a ≥ 0,
b > 0]
GW (335)(10b)
[a ≥ 0,
b > 0]
GW (334)(10b)
[a > 0,
b > 0]
BI (179)(13)a
[a > 0,
b > 0]
BI (179)(14)a
482
3.875
Trigonometric Functions
∞
1.
u
∞
2.
u
∞
3.6
3.876
1.
0
2.
√
x sin p x2 − u2
π
cos bx dx = exp −p a2 + u2 cosh ab
2
2
x +a
2
[0 < b < p]
√
x sin p x2 − u2
π
cos bx dx = e−ap cos b u2 − a2
2
2
2
a +x −u
2
√
sin p a2 + x2
(a2 + x2 )
0
∞
3.875
3/2
cos bx dx =
πp −ab
e
2a
√
sin p x2 + a2
π
√
cos bx dx = J 0 a p2 − b2
2
2
2
x +a
=0
√
∞
cos p x2 + a2
π
√
cos bx dx = − Y 0 a p2 − b2
2
2
2 x +a
0
= K 0 a b2 − p2
√
∞
cos p x2 + a2
π −bc
e
cos
bx
dx
=
cos
p
a2 − c 2
x2 + c2
2c
0
ET I 27(39)
[0 0]
ET I 27(38)
[0 0]
ET I 26(29)
[0 p > 0]
[a > 0]
ET I 26(30)
[0 p > 0]
[a > 0]
ET I 26(34)
3.
[c > 0, b > p]
√
√
∞
sin p x2 + a2
π e−bc sin p a2 − c2
√
√
cos bx dx =
[c = a]
2
2
2
2
2c
a2 − c 2
0 (x + c ) x + a
p
π
[c = a]
= e−ba
2
a
[b > p, c > 0]
√
∞
cos p x2 + a2
π −ab
e
cos bx dx =
[b > p > 0; a > 0]
2 + a2
x
2a
0
√
∞
x cos p x2 + a2
π
sin bx dx = e−ab
[a > 0, b > p > 0]
2
2
x +a
2
0
√
u
cos p u2 − x2
π
√
cos bx dx = J 0 u b2 + p2
2
2
2
u −x
0
√
∞
2
2
cos p x − u
√
[0 |p|]
2
ET I 26(33)
4.
5.6
6.6
7.
8.
ET I 26(31)a
ET I 27(35)a
ET I 85(29)a
ET I 28(42)
ET I 28(43)
3.881
3.877
Trigonometric functions with powers
u
1.
0
483
√
,
,
+u +u sin p u2 − x2
π3 p
<
J 14
cos bx dx =
b2 + p2 − b J 14
b2 + p2 + b
8
2
2
3
4
(u2 − x2 )
[b > 0, p > 0]
ET I 27(40)
√
∞
+u
,
+u
,
sin p x2 − u2
π3 p
<
J 14
cos bx dx = −
b − b2 − p2 Z 14
b + b2 − p2
8
2
2
3
4
u
(x2 − u2 )
2.
[b > p > 0]
ET I 27(41)
√
u
,
,
+u +u cos p u2 − x2
π3 p
<
J − 14
cos bx dx =
p2 + b2 − b J − 14
p2 + b2 + b
8
2
2
3
4
0
(u2 − x2 )
3.
4.
[u > 0, p > 0]
ET I 28(44)
√
∞
+u
,
+u
,
cos p x2 − u2
π3 p
<
J − 14
b − b2 − p2 Y 14
b + b2 − p2
cos bx dx = −
8
2
2
3
4
u
(x2 − u2 )
[b > p > 0]
3.878
∞
1.
0
2.
3.
√
sin p x4 + a4
π
1
2
√
cos bx dx =
4
4
2
2
x +a
√
∞
cos p x4 + a4
√
cos bx2 dx
x4 + a4
0
π
1
=−
2
2
∞
3.879
0
sin axp
x sin (a tan x) dx =
0
3.
[p > b > 0]
3
π
dx
=
x
2p
π/2
1.
2.
b J − 14
a2
p − p2 − b2
2
J 14
a2
p + p2 − b2
2
ET I 26(32)
2
a2
a
2
2
2
2
b J − 14
Y 14
p− p −b
p+ p −b
2
2
[a > 0, p > b > 0]
ET I 27(36)
√
u
cos p u4 − x4
u2 2
u2 2
π 3
1
2
2
2
√
cos bx dx =
b J − 14
p + b − p J − 14
p +b +p
2
2
2
2
u4 − x4
0
3.881
3
ET I 28(45)
[p > 0,
b > 0]
ET I 28(46)
[a > 0,
p > 0]
GW (334)(6)
π −a C + ln 2a − e2a Ei(−2a)
e
4
[a > 0]
BI (205)(9)
[a > 0]
BI (151)(6)
[a > 0]
BI (151)(19)
∞
π
dx
=
1 − e−a
x
2
0
∞
π
dx
=
1 − e−a
sin (a tan x) cos x
x
2
0
sin (a tan x)
484
Trigonometric Functions
∞
4.
cos (a tan x) sin x
0
π
dx
= e−a
x
2
∞
5.
∞
[a > 0]
BI (152)(11)
cos (a tan x) sin3 x
1 − a −a
dx
=
πe
x
4
[a > 0]
BI (151)(23)
[a > 0]
BI (152)(13)
∞
sin (a tan x) tan
0
π/2
8.
π
x dx
= − Ei(−a)
sin 2x
4
[a > 0]
BI (206)(15)
sin (a cot x)
x dx
1 − e−a
π
=
2a
sin2 x
[a > 0]
LI (206)(14)
π/2
9.
0
π/2
10.
0
1 + a −a
x
dx
cos2 x
=
πe
2
x
4
cos (a tan x)
0
π
x cos (a tan x) tan x dx = − e−a C + ln 2a + e2a Ei(−2a)
4
∞
11.
cos (a tan x) tan x
0
∞
12.
π
dx
= e−a
x
2
cos (a tan x) sin2 x tan x
0
∞
13.
∞
17.
[a > 0]
BI (151)(22)
[a > 0]
BI (152)(13)
[a > 0]
BI (152)(10)
[a > 0]
BI (180)(6)
sin (a tan 2x) cos2 2x tan x
1 + a −a
dx
=
πe
x
4
∞
∞
0
x dx
π
exp (−a tanh b) − e−a
sin a tan2 x 2
=
b + x2
2
[a > 0,
∞
2.
cos a tan2 x cos x
0
3.
0
BI (152)(15)
π
dx
= e−a
x
2
π
dx
= e−a
x
2
0
∞
π
dx
=
1 − e−a
sin (a tan 2x) tan x cot 2x
x
2
0
[a > 0]
cos (a tan 2x) tan x
sin (a tan 2x) tan x tan 2x
3.882 1.
BI (151)(21)
BI (152)(9)
0
16.
[a > 0]
[a > 0]
0
15.
BI (205)(10)
π
dx
= e−a
x
2
∞
14.
1 − a −a
dx
=
πe
x
16
[a > 0]
sin (a tan x) tan2 x
0
BI (151)(20)
1 + a −a
dx
=
πe
x
2
0
7.
[a > 0]
sin (a tan x) sin 2x
0
6.
3.882
b2
BI (160)(22)
dx
π cosh b exp (−a tanh b) − e−a sinh b
=
2
+x
2b
[a > 0,
∞
b > 0]
b > 0]
BI (163)(3)
b > 0]
BI (191)(10)
x dx
π
exp (−a tanh b)
cos a tan2 x cosec 2x 2
=
b + x2
2 sinh 2b
[a > 0,
3.892
Trigonometric functions and exponentials
4.
5.11
6.
∞
−a
x dx
π
e cosh b − exp (−a tanh b) sinh b
cos a tan2 x tan x 2
=
2
b
+
x
2
cosh
b
0
[a > 0, b > 0]
∞
x dx
π
coth b exp (−a tanh b) − e−a
cos a tan2 x cot x 2
=
2
b +x
2
0
[a > 0, b > 0]
∞
x dx
π
coth 2b exp (−a tanh b) − e−a
cos a tan2 x cot 2x 2
=
2
b +x
2
0
[a > 0, b > 0]
3.883 1.
1
cos (a ln x)
0
xμ−1 sin (β ln x) dx = −
0
3.88411
BI (163)(4)
BI (163)(5)
BI (191)(11)
dx
aπ
=
(1 + x)2
2 sinh aπ
1
2.
3.
485
BI (404)(4)
β
β 2 + μ2
[Re μ > |Im β|]
ET I 319(19)
1
μ
[Re μ > |Im β|]
+ μ2
0
∞
+
,
sin a |x|
sign x dx = π exp −a |−b| + exp −a |b|
x−b
−∞
[a > 0, Im b = 0]
xμ−1 cos (β ln x) dx =
ET I 321(38)
β2
ET II 253(46)
3.89–3.91 Trigonometric functions and exponentials
3.891 1.
2π
eimx sin nx dx = 0
[m = n; or m = n = 0]
0
= πi
[m = n = 0]
2π
eimx cos nx dx = 0
2.
[m = n]
0
=π
[m = n = 0]
= 2π
[m = n = 0]
3.892 π
1.11
eiβx sinν−1 x dx =
0
π
2ν−1 ν B
π
2
2.
−π
2
eiβx cosν−1 x dx =
2ν−1 ν B
πeiβ 2
ν+β+1 ν−β+1
,
2
2
[Re ν > −1]
NH 158, EH I 12(29)
[Re ν > −1]
GW (335)(19)
π
ν+β+1 ν−β+1
,
2
2
486
Trigonometric Functions
π/2
ei2βx sin2μ x cos2ν x dx =
3.6
0
3.893
exp iπ β − ν − 12 B (β − μ − ν, 2ν + 1)
× F (−2μ, β − μ − ν; 1 + β − μ + ν; −1) + exp iπ μ + 12
1
22μ+2ν+1
× B(β − μ − ν, 2μ + 1) F (−2ν, β − μ − ν; 1 + β + μ − ν; −1)
Re μ > − 12 , Re ν > − 12
EH I 80(6)
π exp [iπ(β − ν)] F (−2ν, β − μ − ν; 1 + β + μ − ν; −1)
4μ+ν (2μ + 1) B(1 − β + μ + ν, 1 + β + μ − ν)
π
ei2βx sin2μ x cos2ν x dx =
4.
0
EH I 80(8)
π/2
π
ei(μ+ν)x sinμ−1 x cosν−1 x dx = eiμ 2 B(μ, ν)
5.
0
=
3.893
∞
1.8
e−px sin(qx + λ) dx =
0
∞
2.8
e−px cos(qx + λ) dx =
0
∞
3.
2μ+ν−1
e
iμ π
2
1
1
F (1 − ν, 1; μ + 1; −1) + F (1 − μ, 1; ν + 1; −1)
μ
ν
[Re μ > 0, Re ν > 0]
EH I 80(7)
1
(q cos λ + p sin λ)
+ q2
[Re p > 0]
BI (261)(3)
1
(p cos λ − q sin λ)
p2 + q 2
[Re p > 0]
BI (261)(4)
p2
e−x cos t cos (t − x sin t) dx = 1
0
∞ −βx
e
4.8
1
0
sin ax
dx = Re
sin bx
1
ψ
2bi
BI (261)(7)
β
a+b
−i
2b
2b
−ψ
β
b−a
−i
2b
2b
[Re β > 0,
∞ −2px
e
5.8
0
6.
1
sin[(2n + 1)x]
dx =
+
sin x
2p
e
0
k=1
p
p2 + k 2
sin 2nx
1
dx = 2p
sin x
p2 + (2k + 1)2
∞ −px
8
n
∞
[Re p > 0]
BI (267)(15)
[Re p > 0]
GW (335)(15c)
(−1)k (2k + 1)
2n + 1
n
+
(−1)
2
p2 + (2n + 1)2
p2 + (2k + 1)2
n−1
k=0
[p > 0]
,ν
+
2π Γ(ν + 1) P m
ν (β)
β + β 2 − 1 cos x einx dx =
Γ(ν + m + 1)
−π
[Re β > 0]
1.
0
GW (335)(15)
n−1
e−px cos[(2n + 1)x] tan x dx =
0
3.895
b = 0]
k=0
7.
3.894
LI (267)(16)
π
∞
e−βx sin2m x dx =
ET I 157(15)
(2m)!
β(β 2 + 22 ) (β 2 + 42 ) · · · [β 2 + (2m)2 ]
[Re β > 0]
FI II 615, WA 620a
3.895
Trigonometric functions and exponentials
2.
π
10
e−px sin2m x dx =
0
3.
π/2
10
=
(2m)!
2 + 22 ) (p2 + 42 ) · · · [p2 + (2m)2 ]
p (p
%
2
2
2
&
2
2
2
2
2
2
p
p
p
p
·
·
·
p
+
2
+
2
+
(2m
−
2)
pπ
p
× 1 − e− 2 1 +
+
+ ··· +
2!
4!
(2m)!
BI (270)(4)
∞
4.
e−βx sin2m+1 x dx =
0
5.10
6.
(β 2
+
12 ) (β 2
(2m + 1)!
+ 32 ) · · · [β 2 + (2m + 1)2 ]
[Re β > 0]
π
e−px sin2m+1 x dx =
0
GW (335)(4a)
e−px sin2m x dx
0
(2m)! (1 − e−pπ )
p (p2 + 22 ) (p2 + 42 ) · · · [p2 + (2m)2 ]
487
π/2
8
FI II 615, WA 620a
−pπ
(p2
+
)
(2m + 1)! (1 + e
2
2
+ 3 ) · · · [p + (2m + 1)2 ]
12 ) (p2
GW (335)(4b)
e−px sin2m+1 x dx
0
=
(2m + 1)!
+%32 ) · · · [p2 + (2m + 1)2 ]
+
2
&
p + 12 p2 + 32 · · · p2 + (2m − 1)2
−pπ
p2 + 12
2
× 1 − pe
+ ··· +
1+
3!
(2m + 1)!
(p2
12 ) (p2
BI (270)(5)
∞
7.
e−px cos2m x dx =
0
(2m)!
p (p2
+
× 1+
22 ) · · · [p2
2
p
p2
+
2!
+ (2m)2 ]
p2 p2 + 22 · · · p2 + (2m − 2)2
p2 + 22
+ ··· +
4!
(2m)!
[p > 0]
π/2
8.10
e−px cos2m x dx
0
=
(2m)!
p (p2
+
22 ) · · · [p2
× −e−p 2
π
9.7
BI (262)(3)
+ (2m)2 ]
p2 p2 + 22
p2 p2 + 22 · · · p2 + (2m − 2)2
p2
+
+ ··· +
+1+
2!
4!
(2m)!
BI (270)(6)
∞
e−px cos2m+1 x dx
0
=
(2m + 1)!p
2 + 12 ) (p2 + 32 ) · · · [p2 + (2m + 1)2 ]
(p
2
2
p + 12 p2 + 32
p + 12 p2 + 32 · · · p2 + (2m − 1)2
p2 + 12
+
+ ··· +
× 1+
3!
5!
(2m + 1)!
[p > 0]
BI (262)(4)
488
Trigonometric Functions
π/2
10.11
3.896
e−px cos2m+1 x dx
0
=
∞
11.8
e−βx sinn ax
0
e−ax cos2 mx dx =
0
15.
16.
3.896
1.
2.
3.
4.
−n−2
dx =
1
2
e 4 (1∓1+2n)πi
a(n
+
1)
⎧
−1 ⎫
⎨ b+na+iβ −1
⎬
b+na−iβ
2a
2a
×
± (−1)n
⎩
⎭
n+1
n+1
b > 0,
a2 + 2m2
a (a2 + 4m2 )
∞
∞
−ax
2
x2
DW61 (861.15)
DW61 (861.14)
DW61 (861.10)
DW61 (861.13)
√
p2
π − 4q
e 2 sin pλ
q
−∞
√
∞
p2
2 2
π − 4q
e 2 cos pλ
e−q x cos[p(x + λ)] dx =
q
−∞
∞
2
1 3 b2
b2
b
e−ax sin bx dx =
exp −
; ;
1F 1
2a
4a
2 2 4a
0
2
b
3
b
=
1 F 1 1; ; −
2a
2 4a
∞
k−1
1
b b2
=
−
2a
(2k − 1)!!
2a
k=1
∞
2
b2
1 π
exp −
e−βx cos bx dx =
2 β
4β
0
e−q
Re β > 0]
DW61 (861.06)
a a2 + m2 + n2
e
cos mx cos nx dx = 2
(a + (m − n)2 ) (a2 + (m + n)2 )
0
∞
m a2 + m2 − n2
−ax
e
sin mx cos nx dx = 2
(a + (m − n)2 ) (a2 + (m + n)2 )
0
∞
2m
[a > 0]
e−ax sin2 mx dx =
a (a2 + 4m2 )
0
∞
2amn
e−ax sin mx sin nx dx = 2
2 ] [a2 + (m + n)2 ]
[a
+
(m
−
n)
0
14.
sin bx
cos bx
BI (270)(7)
[a > 0,
∞
12.
13.
(2m + 1)!
2 + 12 ) (p2 + 32 ) · · · [p2 + (2m + 1)2 ]
(p
%
2
&
p + 1 p2 + 32 · · · p2 + (2m − 1)2
p2 + 12
−p π
× e 2 +p 1+
+ ··· +
3!
(2m + 1)!
sin[p(x + λ)] dx =
BI (269)(2)
BI (269)(3)
ET I 73(18)
[a > 0]
[Re β > 0]
FI II 720
BI (263)(2)
3.911
3.897
1.8
2.
Trigonometric functions and exponentials
γ − ib
π
(γ − ib)2
√
exp
1−Φ
β
2 β
0
4β
(γ + ib)2
γ + ib
√
− exp
1−Φ
4β
2 β
[Re β > 0]
∞
2 2
(γ − ib)
γ − ib
1 π
√
exp
e−βx −γx cos bx dx =
1−Φ
4 β
4β
2 β
0
γ + ib
(γ + ib)2
√
+ exp
1−Φ
4β
2 β
[Re β > 0]
3.898 1.
∞
e−βx
2
∞
e
−βx2
0
∞
2.
−γx
sin bx dx = −
i
4
∞
3.8
e−βx cos ax cos bx dx =
2
e−px sin2 ax dx =
2
0
1.
7
2.
1
sin ax sin bx dx =
4
0
3.899
489
1
4
1
4
π
β
π
β
e
− (a−b)
4β
e−
2
−e
(a−b)2
4β
a2
π
1 − e− p
p
− (a+b)
4β
+ e−
2
(a+b)
4β
ET I 74(27)
ET I 15(16)
[Re β > 0]
2
BI (263)(4)
[Re β > 0]
BI (263)(5)
[Re p > 0]
BI (263)(6)
&
√ %
n
sin[(2n + 1)x]
π 1 −( kp )2
dx =
+
e
[p > 0]
sin x
p 2
0
k=1
&
√ %
∞ −p2 x2
2n
2
e
cos[(4n + 1)x]
π 1 k −( k
)
dx =
+
(−1) e p
cos x
p 2
0
∞ p2 x2
e
BI (267)(17)
k=0
∞
3.
0
<
π
[p > 0]
∞
e−px dx
p
k2
1 k
+
=
a
exp
−
2
1 − 2a cos x + a
1 − a2 2
4p
k=1
<
π
∞
p
1 −k
k2
+
= 2
a exp −
a −1 2
4p
2
BI (267)(18)
a2 < 1,
p>0
a2 > 1,
p>0
EI (266)(1)
LI (266)(1)
k=1
3.911 1.
∞
0
∞
2.
0
3.11
0
∞
1
π
sin ax
dx =
−
eβx + 1
2a 2β sinh aπ
β
sin ax
π
dx =
coth
eβx − 1
2β
πa
β
−
sin ax x/2
1
e dx = π tanh(aπ)
x
e −1
2
1
2a
[a > 0,
Re β > 0]
BI (264)(1)
[a > 0,
Re β > 0]
BI (264)(2), WH
[a > 0]
ET I 73(13)
490
Trigonometric Functions
∞
4.
0
sin ax
1
dx =
γx
−e
2i(β − γ)
eβx
0
∞
0
ET 73(15)
a > 0]
ET I 15(10)
e
2.11
[Re β > −1]
1
β − ia
β + ia
cos ax dx =
B ν,
+ B ν,
2γ
γ
γ
[Re β > 0, Re γ > 0, Re ν > 0,
−βx
1−e
−γx ν−1
iβx
ν
2 ix
cos x β e
2 −ix μ
+ν e
dx =
−π
2
π 2 F 1 −μ, β2 −
2ν (ν + 1) B 1 +
ν
2
− μ2 ; 1 +
β
2
+
ν
2
2
− μ2 ; βν 2
ν
μ
β
ν
μ
β
+ − ,1 − + +
2
2
2
2
2
2
[Re ν > −1,
π
2
GW (335)(8)
ET I 73(17)
π
2
1.
Re γ > 0]
a > 0]
e
−ψ
BI (264)(8)
ν−1
i
β − ia
β + ia
e−βx 1 − e−γx
sin ax dx = −
B ν,
− B ν,
2γ
γ
γ
[Re β > 0, Re γ > 0, Re ν > 0,
0
β + ia
β−γ
[a > 0]
β − ia
β−γ
i
sin ax dx
= [ψ(β + ia) − ψ(β − ia)]
eβx (e−x − 1)
2
∞
2.
3.913
ψ
[Re β > 0,
∞
6.
n−1
∞
0
3.912 1.
sin ax −nx
a
1
π
π
−
+
−
e
dx
=
1 − e−x
2
2a e2πa − 1
a2 + k 2
k=1
5.
3.912
|ν| > |β|]
EH I 81(11)a
ν
e−iux cosμ x a2 eix + b2 e−ix dx
−π
2
=
;1 +
πb2ν 2 F 1 −ν, − u+μ+ν
2
πa2ν 2 F 1 −ν, u−μ−ν
;1 +
2
2μ (μ + 1) B 1 +
∞
e−β
√
γ 2 +x2
cos bx dx = 0
2.
0
βγ
β 2 + b2
K1 γ
μ−ν+u b2
; a2
2
μ+ν−u
u+μ−ν
,1 +
2
2
[Re μ > −1]
γ 2 + x2 e−β
√
γ 2 +x2
2 2
cos bx dx =
for a2 < b2
for b2 < a2
ET I 122(31)a
β 2 + b2
[Re β > 0,
∞
u+μ+ν
u+ν−μ
,1 +
2μ (μ + 1) B 1 −
2
2
=
3.914 1.
μ−ν−u a2
; b2
2
β γ
K 0 (γA) +
A2
2
Re γ > 0]
2β γ
γ
−
K 1 (γA)
A3
A
,
+
A = β 2 + b2
ET I 16(26)
3.915
Trigonometric functions and exponentials
∞
3.
√ 2 2
2
γ + x2 e−β γ +x cos bx dx
0
=
∞
1
5.
β
0
∞
0
∞
7.
(γ 2
+
3/2
x2 )
+
1
2
γ + x2
xe−β
√
γ 2 +x2
sin bx dx =
∞
8.
e−β
10.
3.915
Re γ > 0,
√
γ 2 +x2
cos bx dx =
1 2
β + b2 K 1 γ β 2 + b2
βγ
ET I 175(35)
−
bγ 2
4bβ 2 γ 2
+
A2
A4
√ 2 2
2
γ + x2 e−β γ +x x sin bx dx =
2bγ
8bβ 2 γ
bβ 2 γ 3
+
K 0 (γA) + − 3 +
5
A
A
A3
,
+
A = β 2 + b2
12bβγ 2
24bβ 3 γ 2
bβ 3 γ 4
+
+
K 0 (γA)
4
6
A
A
A4
24bβγ
48bβ 3 γ
3bβγ 3
8bβ 3 γ 3
+ −
+
−
+
7
A5
A
A3 , A5
+
2
A = β + b2
K 1 (γA)
∞
π
ea cos x sin x dx =
1.
0
K 1 (γA)
−
√ 2 2
xe−β γ +x
γb
sin bx dx = K 1 γ β 2 + b2
γ 2 + x2
β 2 + b2
0
∞
√ 2 2
1
b
1
+ 2
e−β γ +x x sin bx dx = K 0 γ β 2 + b2
2
3/2
γ +x
β
0
β (γ 2 + x2 )
b > 0]
bβγ 2
K 2 γ β 2 + b2
2
2
β +b
0
9.
[Re β > 0,
√ 2 2
x γ 2 + x2 e−β γ +x sin bx dx
=
K 1 (γA)
ET I 16(27)
0
6βγ
8β 3 γ
β3γ3
+
K 0 (γA) + − 3 +
5
A
A3
,A
+
A = β 2 + b2
(6.726(4))
6.
3βγ 2
4β 3 γ 2
+
A2
A4
√ 2 2
γ +x
e
cos bx dx = K 0 γ β 2 + b2
2
2
γ +x
0
−
∞ −β
4.
491
2
sinh a
a
ET I 75(36)
(6.726(3))
GW (337)(15c)
π
eiβ cos x cos nx dx = in π J n (β)
2.
EH II 81(2)
0
3.3
π
2
√
π
2
β
ν
eiβ sin x cos2ν x dx =
e±β cos x sin2ν x dx =
√
π
2
β
ν
−π
2
4.
0
π
Γ ν+
1
2
J ν (β)
Γ ν+
1
2
I ν (β)
Re ν > − 12
Re ν > − 12
EH II 81(6)
GW (337)(15b)
492
Trigonometric Functions
π
eiβ cos x sin2ν x dx =
5.
√
π
0
3.916
π/2
1.
e−p
2
x√
tan x sin 2 cos x
sin 2x
0
π/2
2.
0
2
β
ν
Γ ν+
1
2
J ν (β)
3.916
Re ν > − 12
WA 34(2), WA 60(6)
2 2
1
1
dx = C (p) −
+ S (p) −
2
2
exp (−p tan x) dx
1
= − eap Ei(−ap)
sin 2x + a cos 2x + a
2
[p > 0] ,
NT 33(18)a
(cf. 3552 4 and 6)
BI (273)(11)
π/2
3.
0
1
exp (−p cot x) dx
= − e−ap Ei(ap)
sin 2x + a cos 2x − a
2
[p > 0] ,
(cf. 3.552 4 and 6)
BI (273)(12)
π/2
4.
(1 −
0
1
exp (−p tan x) sin 2x dx
= − e−ap Ei(ap) + eap Ei(−ap)
2
2
2
− 2a cos 2x − (1 + a ) cos 2x
4
a2 )
[p > 0]
π/2
5.
(1 −
0
BI (273)(13)
1
exp (−p cot x) sin 2x dx
= − e−ap Ei(ap) + eap Ei(−ap)
2
2
2
+ 2a cos 2x − (1 + a ) cos 2x
4
a2 )
[p > 0]
3.917
π/2
1.
0
π/2
2.
0
3.918
π/2
1.
0
π/2
2.
0
1
e−2β cot x cosν−1/2 x sin−(ν+1) x sin β − ν −
2
1
e−2β cot x cosν−1/2 x sin−(ν+1)x cos β − ν −
2
√
π
1
Γ ν+
x dx =
J ν (β)
ν
2(2β)
2
WA 186(7)
Re ν > − 12
√
π
1
Γ ν+
x dx =
Y ν (β)
2(2β)ν
2
Re ν > − 12
WA 186(8)
cosμ x i γ(β−μx)−2β cot x
π
iγ
(ε)
(2β)−μ Γ(μ + 1) H μ+ 1 (β)
dx
=
e
2μ+2
2
2
2β
sin
x
ε = 1, 2, γ = (−1)ε+1 , Re β > 0, Re μ > −1
cosμ x sin(β − μx) −2β cot x
1
dx =
e
2
sin2μ+2 x
3.
0
π/2
cosμ x cos(β − μx) −2β cot x
1
e
dx = −
2μ+2
2
sin
x
GW (337)(16)
π
(2β)−μ Γ(μ + 1) J μ+ 12 (β)
2β
[Re β > 0,
BI (273)(14)
Re μ > −1]
WH
π
(2β)−μ Γ(μ + 1) Y μ+ 12 (β)
2β
[Re β > 0,
Re μ > −1]
GW (337)(17b)
3.922
3.919
Trigonometric functions and exponentials
π/2
1.
0
π/2
2.
0
493
sin 2nx
2n − 1
dx
= (−1)n−1
·
2n+2
4(2n + 1)
sin
x exp (2π cot x) − 1
BI (275)(6), LI (275)(6)
dx
sin 2nx
n
n−1
2n+2 exp (π cot x) − 1 = (−1)
2n + 1
sin
x
BI (275)(7), LI (275)(7)
3.92 Trigonometric functions of more complicated arguments combined with
exponentials
3.9216
∞
1.
e
−γx
cos ax (cos γx − sin γx) dx =
2
0
[a > 0,
1
π
exp − tan2n x = − 1
n
2
0
n=1
%
&
% ∞
&
π/2
π/2
∞
1
1
π
2n
2n
sin x =
cos x =
exp −
exp −
n
n
4
0
0
n=1
n=1
2.10
3.10
3.922
1.
2.
γ2
π
exp −
8a
2a
Reγ ≥ |Im γ|]
.
π
β 2 + a2 − β
e−βx
e−βx sin ax2 dx =
8
β 2 + a2
0
−∞
√
1
a
π
arctan
= sin
4
2
2
2
β
2 β +a
[Re β > 0, a > 0]
.
∞
2
π
β 2 + a2 + β
1 ∞ −βx2
e−βx cos ax2 dx =
e
cos ax2 dx =
2 −∞
8
β 2 + a2
0
√
1
a
π
arctan
= cos
4
2
2
2
β
2 β +a
[Re β > 0, a > 0]
ET I 26(28)
∞
π/4 -
∞
2
1
sin ax2 dx =
2
∞
2
FI II 750, BI (263)(8)
FI II 750, BI (263)(9)
[In formulas 3.922 3 and 4, a > 0, b > 0, Re β > 0, and
1 2
1 2
b2
,
B=
A=
β + a2 + β ,
C=
β + a2 − β .
2
2
4 (a + β )
2
2
If a is complex, then Re β > |Im a|.]
∞
2
π
1
e−βx sin ax2 cos bx dx = −
e−Aβ (B sin Aa − C cos Aa)
3.
2 + a2
2
β
0
√
1
a
ab2
π
βb2
arctan −
exp −
= sin
4 (β 2 + a2 )
2
β
4 (β 2 + a2 )
2 4 β 2 + a2
LI (263)(10), GW (337)(5)
494
Trigonometric Functions
∞
4.
e
−βx2
0
3.923
π
e−Aβ (B cos Aa + C sin Aa)
β 2 + a2
√
1
a
ab2
π
βb2
arctan
−
exp
−
= cos
4 (β 2 + a2 )
2
β
4 (β 2 + a2 )
2 4 β 2 + a2
1
cos ax cos bx dx =
2
2
LI (263)(11), GW (337)(5)
3.923
1.
exp − ax2 + 2bx + c sin px2 + 2qx + r dx
−∞
√
a b2 − ac − aq 2 − 2bpq + cp2
π
= 4
exp
a2 + p2
a2 + p2
2
p p q − pr − b2 p − 2abq + a2 r
1
arctan −
× sin
2
a
a2 + p2
2.
∞
[a > 0]
GW (337)(3), BI (296)(6)
∞
exp − ax2 + 2bx + c cos px2 + 2qx + r dx
−∞
√
a b2 − ac − aq 2 − 2bpq + cp2
π
= exp
4
2
a2 + p2
a2 +
p
2
p p q − pr − b2 p − 2abq + a2 r
1
arctan −
× cos
2
a
a2 + p2
[a > 0]
3.924
.
∞
e
1.
−βx4
0
∞
2.
π
sin bx dx =
4
2
.
e−βx cos bx2 dx =
4
0
3.925
∞
2
e
1.
p
−x
2
0
2.
∞
e
0
b2
b
exp −
2β
8β
p2
−x
2
π
4
1
sin 2a x dx =
2
b2
b
exp −
2β
8β
∞
2 2
cos 2a2 x2 dx =
1
2
2
e
p
−x
2
−∞
∞
e
−∞
p2
−x
2
I 14
I − 14
GW (337)(3), BI (269)(7)
b2
8β
[Re β > 0,
b > 0]
ET 73(22)
[Re β > 0,
b > 0]
ET I 15(12)
b2
8β
√
π −2ap
e
sin 2a x dx =
(cos 2ap + sin 2ap)
4a
2 2
[a > 0, b > 0]
√
π −2ap
e
cos 2a2 x2 dx =
(cos 2ap − sin 2ap)
4a
[a > 0,
b > 0]
BI (268)(12)
BI (268)(13)
3.932
Trigonometric and exponential functions
3.926
1.
Notation:
495
.
a2 + β 2 + β
a2 + β 2 − β
,
v=
u=
2
2
∞
√
γ
2
π
1
√
√
e−(βx + x2 ) sin ax2 dx =
e−2u γ [v cos (2v γ) + u sin (2v γ)]
2
2
2 a +β
0
∞
2.
e−(βx
2
.
+ xγ2 )
cos ax2 dx =
0
1
2
[Re β > 0,
a2
∞
3.927
p
e− x sin2
0
3.928
∞
1.
0
∞
2.
2a p
p2
a
dx = a arctan
+ ln 2
x
p
4 p + 4a2
q2
b2
exp − p2 x2 + 2
sin a2 x2 + 2
x
x
exp − p2 x2 +
0
2
q
x2
BI (268)(14)
√
π
√
√
e−2u γ [u cos (2v γ) − v sin (2v γ)]
2
+β
[Re β > 0,
Re γ > 0]
dx =
[a > 0,
Re γ > 0]
BI (268)(15)
p > 0]
LI (268)(4)
√
π −2rs cos(A+B)
e
sin {A + 2rs sin(A + B)}
2r
BI (268)(22)
cos a2 x2 +
2
b
x2
√
π −2rs cos(A+B)
e
cos {A + 2rs sin(A + B)}
dx =
2r
BI (268)(23)
∞
3.929
+
,
√ 2
e−x cos p x + pe−x sin px dx = 1
0
Notation: For the formulas in 3.928: a2 + p2 > 0, r =
B=
LI (268)(3)
a2
1
4
a4 + p4 , s = 4 b4 + q 4 , A = arctan 2 , and
2
p
1
b2
arctan 2 .
2
q
3.93 Trigonometric and exponential functions of trigonometric functions
3.931
π/2
1.
0
π
2.
e−p cos x sin (p sin x) dx = Ei(−p) − ci(p)
e−p cos x sin (p sin x) dx = −
−π
0
π/2
3.
π/2
4.
0
3.932
e−p cos x cos (p sin x) dx =
1
2
π
ep cos x sin (p sin x) sin mx dx =
1.
0
e−p cos x sin (p sin x) dx = −2 shi(p)
GW (337)(11b)
e−p cos x cos (p sin x) dx = − si(p)
0
0
NT 13(27)
2π
NT 13(26)
e−p cos x cos (p sin x) dx = π
GW (337)(11a)
0
1
2
2π
ep cos x sin (p sin x) sin mx dx =
0
π pm
·
2 m!
BI (277)(7), GW (337)(13a)
496
Trigonometric Functions
π
ep cos x cos (p sin x) cos mx dx =
2.
0
1
2
3.933
2π
ep cos x cos (p sin x) cos mx dx =
0
π pm
·
2 m!
BI (277)(8), GW (337)(13b)
π
ep cos x sin (p sin x) cosec x dx = π sinh p
3.933
BI (278)(1)
0
3.934
π
1.
ep cos x sin (p sin x) tan
x
dx = π (1 − ep )
2
BI (271)(8)
ep cos x sin (p sin x) cot
x
dx = π (ep − 1)
2
BI (272)(5)
0
π
2.
0
ep cos x cos (p sin x)
0
3.936
p2k+1
sin 2nx
dx = π
sin x
(2k + 1)!
n−1
π
3.935
2π
π
ep cos x cos (p sin x − mx) dx = 2
1.
[p > 0]
LI (278)(3)
k=0
0
ep cos x cos (p sin x − mx) dx =
0
2πpm
m!
BI (277)(9), GW (337)(14a)
2π
2.
ep sin x sin (p cos x + mx) dx =
mπ
2πpm
sin
m!
2
[p > 0]
GW (337)(14b)
ep sin x cos (p cos x + mx) dx =
mπ
2πpm
cos
m!
2
[p > 0]
GW (337)(14b)
0
2π
3.
0
2π
ecos x sin (mx − sin x) dx = 0
4.
WH
0
5.
π
eβ cos x cos (ax + β sin x) dx = β −a sin(aπ) γ(a, β)
EH II 137(2)
0
3.937 Notation: In formulas 3.937 1 and 2, (b−p)2 +(a+q)2 > 0, m = 0, 1, 2, . . . , A = p2 −q 2 +a2 −b2 ,
B = 2(pq + ab), C = p2 + q 2 − a2 − b2 , and D = 2(ap + bq).
1.11
0
2π
exp (p cos x + q sin x) sin (a cos x + b sin x − mx) dx
√
√
− m = iπ (b − p)2 + (a + q)2 2 (A + iB)m/2 I m
C − iD − (A − iB)m/2 I m
C + iD
GW (337)(9b)
2π
exp (p cos x + q sin x) cos (a cos x + b sin x − mx) dx
√
√
− m m
m
= π (b − p)2 + (a + q)2 2 (A + iB) 2 I m
C − iD + (A − iB) 2 I m
C + iD
2.
0
GW (337)(9a)
2π
exp (p cos x + q sin x) sin (q cos x − p sin x + mx) dx =
3.
0
m
q
2π 2
p + q2
sin m arctan
m!
2
p
GW (337)(12)
3.944
Trigonometric and exponentials functions and powers
2π
exp (p cos x + q sin x) cos (q cos x − p sin x + mx) dx =
4.
0
497
m
q
2π 2
p + q2
cos m arctan
m!
2
p
GW (337)(12)
3.938
π
er(cos px+cos qx) sin (r sin px) sin (r sin qx) dx =
1.
0
∞
1
π
r(p+q)k
2
Γ(pk + 1) Γ(qk + 1)
k=1
π
er(cos px+cos qx) cos (r sin px) cos (r sin qx) dx =
2.
0
π
2
2+
∞
k=1
r(p+q)k
Γ(pk + 1) Γ(qk + 1)
BI (277)(14)
BI (277)(15)
3.939
π
eq cos x
1.
0
∞
sin rx
π (pq)kr
sin
(q
sin
x)
dx
=
1 − 2pr cos rx + p2r
2pr
Γ(kr + 1)
k=1
%
π
eq cos x
2.3
0
1 − pr cos rx
π
2+
cos (q sin x) dx =
1 − 2pr cos rx + p2r
2
[r > 0,
∞
k=1
0 0, 0 < p < 1]
q−1
cos (p sin 2x) dx
π
exp p
=
2q
q+1
cos2 x + q 2 sin2 x
BI (278)(15)
BI (278)(16)
π/2 p cos 2x
e
3.
0
BI (273)(8)
3.94–3.97 Combinations involving trigonometric functions, exponentials, and powers
3.941 1.
∞
e−px sin qx
q
dx
= arctan
x
p
e−px cos qx
dx
=∞
x
e−px cos px
√
x dx
π
= 2 exp −bp 2
b4 + x4
4b
[p > 0,
b > 0]
BI (386)(6)a
x dx
π
= 2 e−bp sin bp
4
−x
4b
[p > 0,
b > 0]
BI (386)(7)a
0
∞
2.
0
3.942 1.
∞
0
∞
2.
e−px cos px
0
3.943
0
∞
b4
e−βx (1 − cos ax)
1 a2 + β 2
dx
= ln
x
2
β2
[p > 0]
BI (365)(1)
BI (365)(2)
[Re β > 0]
BI (367)(6)
3.944 u
i
i
−μ
−μ
xμ−1 e−βx sin δx dx = (β + iδ) γ [μ, (β + iδ) u] − (β − iδ) γ [μ, (β − iδ) u]
1.
2
2
0
[Re μ > −1]
ET I 318(8)
498
Trigonometric Functions
∞
2.
xμ−1 e−βx sin δx dx =
u
xμ−1 e−βx cos δx dx =
0
xμ−1 e−βx cos δx dx =
u
∞
xμ−1 e−βx sin δx dx =
0
∞
xμ−1 e−βx cos δx dx =
0
0
Γ(μ)
(β 2
+
0
∞
10.
0
[Re μ > −1,
Re β > |Im δ|]
Re β > |Im δ|]
[Re μ > 0,
xμ−1 exp (−ax cos t) sin (ax sin t) dx = Γ(μ)a−μ sin(μt)
+
Re μ > −1,
xμ−1 exp (−ax cos t) cos (ax sin t) dx = Γ(μ)a−μ cos(μt)
+
Re μ > −1,
xp−1 e−qx sin (qx tan t) dx =
1
Γ(p) cosp t sin pt
qp
xp−1 e−qx cos (qx tan t) dx =
1
Γ(p) cosp (t) cos pt
qp
0
11.
δ
β
Γ(μ)
δ
cos μ arctan
(δ 2 + β 2 ) μ2
β
0
sin μ arctan
a > 0,
|t| <
π,
2
a > 0,
|t| <
π,
2
EH I 13(35)
∞
9.
μ/2
δ2)
ET I 320(29)
EH I 13(36)
∞
8.
1
1
−μ
−μ
(β + iδ) Γ [μ, (β + iδ) u] + (β − iδ) Γ [μ, (β − iδ) u]
2
2
FI II 812, BI (361)(10)
∞
7.
ET I 320(28)
FI II 812, BI (361)(9)
6.
1
1
−μ
−μ
(β + iδ) γ [μ, (β + iδ) u] + (β − iδ) γ [μ, (β − iδ) u]
2
2
[Re β > |Im δ|]
5.11
ET I 318(9)
[Re μ > 0]
∞
4.
i
i
−μ
−μ
(β + iδ) Γ [μ, (β + iδ) u] − (β − iδ) Γ [μ, (β − iδ) u]
2
2
[Re β > |Im δ|]
u
3.
3.944
∞
xn e−βx sin bx dx = n!
β
β 2 + b2
∂n
= (−1)
∂β n
n
n+1
(−1)k
0≤2k≤n
,
+
π
|t| < ,
2
q>0
+
π
|t| < ,
2
q>0
n+1
2k + 1
LO V 288(16)
,
b
β
LO V 288(15)
2k+1
b
2
b + β2
[Re β > 0,
b > 0]
GW (336)(3), ET I 72(3)
3.947
Trigonometric and exponentials functions and powers
∞
12.
xn e−βx cos bx dx = n!
0
β
β 2 + b2
∂n
= (−1)n n
∂β
n+1
(−1)k
0≤2k≤n+1
n+1
2k
∞
13.
xn−1/2 e−βx sin bx dx = (−1)n
0
14.
b
β
2k
β
b2 + β 2
[Re β > 0,
499
n
b > 0]
GW (336)(4), ET I 14(5)
⎛ <
⎞
β 2 + b2 − β
⎠
β 2 + b2
π d ⎝
2 dβ n
[Re β > 0,
⎛ <
⎞
∞
2 + b2 + β
n
β
π d ⎝
⎠
xn−1/2 e−βx cos bx dx = (−1)n
2 dβ n
β 2 + b2
0
b > 0]
ET I 72(6)
[Re β > 0,
b > 0]
ET I 15(6)
3.945 1.
∞
∞
∞
−βx
dx
e
sin ax − e−γx sin bx r
x 0
2
r−1
b
a
2 r−1
sin (r − 1) arctan
sin (r − 1) arctan
= Γ(1 − r) b + γ
− a2 + β 2
2
γ
2
β
[Re β > 0, Re γ > 0, r < 2, r = 1] BI(371)(6)
2.
−βx
dx
e
cos ax − e−γx cos bx r
x
0
2
r−1
a
b
2 r−1
cos (r − 1) arctan
cos (r − 1) arctan
= Γ(1 − r) a + β
− b2 + γ 2
2
β
2
γ
[Re β > 0, Re γ > 0, r < 2, r = 1] BI (371)(7)
3.
0
−βx
dx
1 a2 + γ 2
γ
β
β
γ
ln 2
arccot
−
arccot
ae
sin bx − be−γx sin ax 2 = ab
+
x
2 b + β2
a
a
b
b
[Re β > 0,
3.946
∞
1.
e−px sin2m+1 ax
0
m
2m + 1
(−1)m dx
= 2m
(−1)k
k
x
2
k=0
arctan
Re γ > 0]
(2m − 2k + 1)a
p
[m = 0, 1, . . . ,
∞
2.
0
e−px sin2m ax
m+1 m−1
(−1)
dx
=
x
22m
(−1)k
k=0
2m
k
1.
0
∞
e−βx sin γx sin ax
1 β 2 + (a + γ)2
dx
= ln 2
x
4 β + (a − γ)2
p > 0]
1
ln p2 + (2m − 2k)2 a2 − 2m
2
[m = 1, 2, . . . ,
3.947
BI (368)(22)
[Re β > |Im γ|,
p > 0]
a > 0]
GW (336)(9a)
2m
m
ln p
GW (336)(9b)
BI (365)(5)
500
Trigonometric Functions
2.
∞
11
e−px sin ax sin bx
0
3.
∞
11
0
3.948
e−px sin ax cos bx
3.948
|a + b|
|a − b|
dx |a + b|
|a − b|
arctan
arctan
=
−
x2
2
p
2
p
p2 + (a − b)2
p
+ ln
4
p2 + (a + b)2
[p > 0, for p = 0 see 3.741 3] BI (368)(1), FI II 744
a+b
a−b
dx
= arctan
+ arctan
x
p
p
[a ≥ 0,
∞
1.11
e−βx (sin ax − sin bx)
0
p > 0]
a
br
dx
= arctan − arctan
x
β
β
[Re β > 0] ,
∞
1 b2 + β 2
dx
= ln 2
x
2 a + β2
e−βx (cos ax − cos bx)
0
e−βx (cos ax − cos bx)
0
0
2
2
a
dx
β a +β
b
= ln 2
+ b arctan − a arctan
2
2
x
2
b +β
β
β
∞
0
dx
2b p p + 4a
2a
− b arctan
− ln 2
e−βx sin2 ax − sin2 bx 2 = a arctan
x
p
p
4 p + 4b2
1.
2.8
2
∞
BI (368)(25)
dx
2b p p2 + 4a2
2a
+ b arctan
+ ln 2
e−βx cos2 ax − cos2 bx 2 = −a arctan
x
p
p
4 p + 4b2
[p > 0]
BI (368)(20)
2
[p > 0]
5.
3.949
(cf. 3.951 3)
[Re p > 0]
∞
4.
[Re β > 0] ,
BI (367)(8), FI II 748a
∞
3.
(cf. 3.951 2)
BI (367)(7)
2.
GW (336)(10b)
BI (368)(26)
1
a+b+c 1
a+b−c 1
a−b+c
dx
= − arctan
+ arctan
+ arctan
x
4
p
4
p
4
p
0
−a + b + c
1
+ arctan
4
p
[p > 0]
BI (365)(11)
∞
1
b
1 1
2pb
dx
π
= arctan −
arctan 2
e−px sin2 ax sin bx
+s
2
2
x
2
p 2 2
p %+ 4a− b
2
0
&
1 for p2 + 4a2 − b2 < 0
s=
0 for p2 + 4a2 − b2 ≥ 0
3.
0
∞
e−px sin ax sin bx sin cx
e−px sin2 ax cos bx
1
dx
= ln
x
8
2
p + (2a + b)2 p2 + (2a − b)2
BI (365)(8)
2
(p2 + b2 )
[p > 0]
BI (365)(9)
3.951
Trigonometric and exponentials functions and powers
4.
∞
8
e
−px
0
1
a 1 1
2pa
dx
π
= arctan +
arctan 2
sin ax cos bx
+s
2 − a2
x
2
p 2 2
p %+ 4b
2
&
2
1 for p + 4b2 − a2 < 0
s=
0 for p2 + 4b2 − a2 ≥ 0
2
BI (365)(10)
∞
5.
e−px sin2 ax sin bx sin cx
0
3.951
∞
1.
0
e
0
∞ −γx
e
3.
0
4.11
∞ −γx
e
0
∞
6.
0
∞
7.
0
8.
9.
(β − γ)b
− e−βx
sin bx dx = arctan 2
x
b + βγ
[Re β > 0,
Re γ ≥ 0]
BI (367)(3)
1 b2 + β 2
− e−βx
cos bx dx = ln 2
x
2 b + γ2
[Re β > 0,
Re γ ≥ 0]
BI (367)(4)
Re γ > 0]
BI (368)(21)a
b
− e−βx
b b2 + β 2
b
sin bx dx = ln 2
+ β arctan − γ arctan
2
2
x
2 b +γ
β
γ
2
1
x
bπ
π
cos bx dx = 2 − 2 cosech2
−1
2b
2β
β
eβx
0
1
1
−
ex − 1 x
cos bx dx = ln b −
1 − cos ax dx
a 1 1−e
·
= + ln
2πx
e
−1
x
4 2
a
0
[b > 0]
ET I 15(9)
[a > 0]
BI (387)(10)
−a
[Re β > 0,
Re γ > 0]
∞
1
cos x
−
ex − 1
x
ae−px −
BI (390)(6)
dx = C
e−qx
sin ax
x
BI (367)(10)
1 √
bp 2
2
[p > 0]
∞
0
ET I 15(18)
∞
10.
11.
[Re β > 0]
1
[ψ(ib) + ψ(−ib)]
2
−βx
dx
1 a2 + γ 2
= ln
e
− e−γx cos ax
x
2
β2
0
∞
π
cos px − e−px dx
1 √
= 4 exp − bp 2 sin
4
4
b +x
x
2b
2
0
BI (365)(15)
FI II 745
[Re β > 0,
∞
5.
dx 1 p2 + (b + c)2
= ln 2
x
8 p + (b − c)2
,
2
+ 2
2
2
p
p
+
(2a
−
b
+
c)
+
(2a
+
b
−
c)
1
+ ln 2
16
[p + (2a + b + c)2 ] [p2 + (2a − b − c)2 ]
[p > 0]
√
dx
= ln 2
1 − e−x cos x
x
∞ −γx
2.
501
NT 65(8)
dx
a a2 + q 2
a
= ln
+ q arctan − a
x
2
p2
q
[p > 0,
q > 0]
BI (368)(24)
502
Trigonometric Functions
2m
π
x2m sin bx
1
m ∂
dx = (−1)
coth bπ −
[b > 0]
ex − 1
∂b2m 2
2b
0
∞ 2m+1
2m+1
π
1
x
cos bx
m ∂
dx
=
(−1)
coth
bπ
−
ex − 1
∂b2m+1 2
2b
0
12.
13.
3.952
∞
∞
14.
0
∞
15.
0
∞
16.
0
∞
17.
0
2m
x2m sin bx dx
m ∂
=
(−1)
∂b2m
e(2n+1)cx − e(2n−1)cx
%
2m
x2m sin bx dx
m ∂
=
(−1)
∂b2m
e(2n−2)cx
[b > 0]
bπ b
π
tanh
−
2
4c
2c
b + (2k − 1)2 c2
n
%
[b > 0]
b
bπ π
tanh
−
4c
2c
b2 + (2k − 1)2 c2
n
∞
18.
[b > 0]
b
bπ
1
π
coth
−
−
4c
2c
2b
b2 + (2k)2 c2
n−1
∞
19.
0
x2m+1 cos bx dx
∂ 2m+1
= (−1)m 2m+1
2ncx
(2n−2)cx
∂b
e
−e
%
[b > 0,
c > 0]
bπ
1
b
π
coth
−
−
2
4c
2c
2b
b + (2k)2 c2
n−1
0
∞
21.
0
3.952
∞
1.
bπ
2p
0
GW (336)(14c)
c > 0]
GW (336)(14d)
m
cosh
cos ax − cos bx
dx
1
1
b2 + (2k − 1)2 p2
= ln
−
ln 2
aπ
(2m−1)px
x
2 cosh
2
a + (2k − 1)2 p2
−e
k=1
2p
GW (336)(16a)
bπ
m−1
cos ax − cos bx dx
1 a sinh 2p
1 b2 + 4k 2 p2
=
ln
−
ln 2
2 b sinh aπ
2
a + 4k 2 p2
e2mpx − e(2m−2)px x
2p
k=1
∞
1 (b + 1) sinh[(b − 1)π]
sin x sin bx dx
= ln
·
x
1−e
x
4 (b − 1) sinh[(b + 1)π]
b2 = 1
1
2aπ
sin2 ax dx
= ln
·
1 − ex x
4 sinh 2aπ
xe−p
x2
xe−p
x2
2
sin ax dx =
0
2.
&
k=1
[p > 0]
∞
20.
GW (336)(14b)
&
k=1
[p > 0]
GW (336)(14a)
e(2m+1)px
0
&
k=1
[b > 0,
GW (336)(15b)
&
k=1
x2m+1 cos bx dx
∂ 2m+1
= (−1)m 2m+1
(2n+1)cx
(2n−1)cx
∂b
e
−e
%
GW (336)(15a)
2
cos ax dx =
GW (336)(16b)
LO V 305
LO V 306, BI (387)(5)
√
a π
a2
exp − 2
3
4p
4p
∞
1
a (−1)k k!
−
2p2
4p3
(2k + 1)!
k=0
BI (362)(1)
a
p
2k+1
[a > 0]
BI (362)(2)
3.953
Trigonometric and exponentials functions and powers
∞
3.
x2 e−p
2
x2
sin ax dx =
0
6.3
x2 e−p
2
x2
cos ax dx =
∞
x3 e−p
2
x2
2
2
BI (362)(5)
sin ax dx =
k=0
γ2
∞
7.
μ−1 −βx2
x
e
sin γx dx =
0
γe− 4β
2β
∞
xμ−1 e−βx cos ax dx =
2
0
3.953
Γ
1+μ
2
1F 1
1−
a
2p
BI (365)(21)
μ 3 γ2
; ;
2 2 4β
Re μ > −1]
1 −μ/2
μ −a2 /4β
μ 1 1 a2
β
+ ; ;
e
Γ
1F 1 −
2
2
2
2 2 4β
[Re β > 0, Re μ > 0,
√
a
π
a2
2n −β 2 x2
n
√
x e
cos ax dx = (−1) n+1 2n+1 exp − 2 D 2n
2 √β
8β
β 2
0
a
π
a2
n
exp − 2 H 2n
= (−1)
(2β)2n+1
4β
2β
+
,
π
|arg β| < , a > 0
4
√
∞
2
2 2
a
π
a
√
x2n+1 e−β x sin ax dx = (−1)n n+ 3
exp − 2 D 2n+1
2n+2
8β
β 2
2 √2 β
0
a
π
a2
n
exp − 2 H 2n+1
= (−1)
(2β)2n+2
4β +
2β
,
π
|arg β| < , a > 0
4
10.
μ+1
2
BI (362)(6)
[Re β > 0,
8.10
9.
BI (362)(4)
√ 2p − a
a
π
exp − 2
5
8p
4p
2
√ 6ap2 − a3
a2
π
exp
−
16p7
4p2
0
√ ∞
∞
2k
2 2
a π
a
(−1)k
dx
π
=
e−p x sin ax
= Φ
x
2p
k!(2k
+
1)
2p
2
0
2k+1
[a > 0]
∞
0
5.
a
p
k=0
4.
∞
a
2p2 − a2 (−1)k k!
+
4p4
8p5
(2k + 1)!
503
∞
xμ−1 e−γx−βx sin ax dx
ET I 320(30)
=−
∞
WH, ET I 15(13)
WH, ET I 74(23)
2
0
a > 0]
∞
1.
2.
ET I 318(10)
γ 2 − a2
8β
i
μ exp
2(2β) 2
iaγ
4β
D −μ
Γ(μ) exp −
iaγ
4β
D −μ
xμ−1 e−γx−βx cos ax dx
iaγ
γ + ia
D −μ √
4β
2β
Re β > 0, a > 0] ET I 318(11)
− exp
2
0
=
γ − ia
√
2β
[Re μ > −1,
Γ(μ) exp −
1
μ exp
2(2β) 2
γ 2 − a2
8β
iaγ
γ − ia
γ + ia
√
+ exp
D −μ √
4β
2β
2β
[Re μ > 0, Re β > 0, a > 0] ET I 16(18)
504
Trigonometric Functions
γ − ia
(γ − ia)2
√
xe
(γ − ia) exp −
1−Φ
4β
2 β
0
(γ + ia)2
γ + ia
√
− (γ + ia) exp −
1−Φ
4β
2 β
[Re β > 0, a > 0]
√
∞
2
π
γ − ia
(γ − ia)2
√
xe−γx−βx cos ax dx = − (γ − ia) exp
1−Φ
4β
2 β
8 β3
0
(γ + ia)2
γ + ia
1
√
+ (γ + ia) exp
+
1−Φ
4β
2β
2 β
[Re β > 0, a > 0]
3.
4.
3.954
∞
3.954 1.11
∞
−γx−βx2
√
i π
sin ax dx = 8 β3
e−βx sin ax
2
0
ET I 74(28)
ET I 16(17)
x dx
π βγ 2
a
a
−γa
γa
√
√
e
=
−
Φ
γ
β
−
Φ
γ
β
+
−
e
2
sinh
aγ
+
e
γ 2 + x2
4
2 β
2 β
[Re β > 0, Re γ > 0, a > 0]
ET I 74(26)a
∞
2.11
0
dx
π βγ 2
a
a
−βx2
−γa
γa
e
e
cos ax 2
=
Φ γ β− √
−e Φ γ β+ √
2 cosh aγ − e
γ + x2
4γ
2 β
2 β
[Re β > 0, Re γ > 0, a > 0]
ET I 15(15)
∞
π − β2
e 4 D ν (β)
[Re ν > −1]
EH II 120(4)
2
0
∞
2
dx √ e − 1
3.956
e−x (2x cos x − sin x) sin x 2 = π
BI (369)(19)
x
2e
0
3.957 ∞
−β 2
1.
xμ−1 exp
sin ax dx
4x
0
√
μ
πi √
i
i
i
μπ K μ βe−πi/4 a
= μ β μ a− 2 exp − μπ K μ βe 4 a − exp
2
4
4
[Re β > 0, Re μ < 1, a > 0] ET I 318(12)
3.955
∞
2.
xν e−
xμ−1 exp
0
3.958
x2
2
∞
−∞
π
2
dx =
−β 2
4x
cos ax dx
√
√
μ
i
1
i
μπ K μ βe−πi/4 a
= μ β μ a− 2 exp − μπ K μ βeπi/4 a + exp
2
4
4
[Re β > 0, Re μ < 1, a > 0] ET I 320(32)a
n −(ax2 +bx+c)
x e
1.
cos βx − ν
sin(px + q) dx = −
×
−1
2a
n−2k
j=0
n
π
exp
a
b2 − p2
−c
4a
n − 2k n−2k−j j
p sin
b
j
[a > 0]
n/2
k=0
n!
ak
(n − 2k)!k!
pb
π
−q+ j
2a
2
GW (37)(1b)
3.963
Trigonometric and exponentials functions and powers
∞
n −(ax2 +bx+c)
2.
x e
−1
2a
cos(px + q) dx =
−∞
×
n−2k
j=0
∞
3.959
xe−p
2
x2
tan ax dx =
0
3.961
∞
1.
π
exp
a
b2 − p2
−c
4a
n − 2k j
p cos
j
n/2
k=0
n!
ak
(n − 2k)!k!
pb
π
−q+ j
2a
2
[a > 0]
GW (337)(1a)
[p > 0]
BI (362)(15)
√ ∞
a π
a2 k 2
(−1)k k exp − 2
3
p
p
k=1
exp −β
0
n
505
x dx
aγ
γ 2 + x2 sin ax =
K 1 γ a2 + β 2
γ 2 + x2
a2 + β 2
[Re β > 0, Re γ > 0,
a > 0]
ET I 75(36)
∞
2.
+
exp −β
0
,
dx
γ 2 + x2 cos ax = K 0 γ a2 + β 2
γ 2 + x2
[Re β > 0,
Re γ > 0,
a > 0]
ET I 17(27)
3.962
∞
<
1.
0
γ 2 + x2 − γ exp −β
γ 2 + x2
γ 2 + x2
sin ax dx =
2
2
π a exp −γ a + β
<
2 2
β + a2 β + a2 + β 2
[Re β > 0, Re γ > 0, a > 0]
ET I 75(38)
<
∞ x exp −β γ 2 + x2
2
2
,
+ π β+ a +β
2 + β2
<
exp
−γ
a
cos
ax
dx
=
2
a2 + β 2
0
γ 2 + x2
γ 2 + x2 − γ
[Re β > 0, Re γ > 0, a > 0]
2.
ET I 17(29)
3.963
1.
∞
e− tan
2
√
sin x dx
π
=
cos2 x x
2
x
0
π/2
2.
e−p tan x
x dx
1
= [ci(p) sin p − cos p si(p)]
2
cos x
p
xe− tan
x
sin 4x
xe− tan
x
sin3 2x
0
π/2
3.8
2
0
4.
8
0
π/2
2
BI (391)(1)
[p > 0]
(cf. 3.339)
BI (396)(3)
3√
dx
=−
π
8
cos x
2
BI (396)(5)
√
dx
=2 π
8
cos x
BI (396)(6)
506
3.964
Trigonometric Functions
π/2
1.
xe−p tan x
0
π/2
2.
π/2
xe−p tan
xp
2
0
3.965
1.
2.
3.966
∞
1
p − cos x
dx =
4
cos x cot x
4
2
x
0
3.8
p sin x − cos x
dx = − sin p si(p) − ci(p) cos p
cos3 x
xe−p tan
2
π
p
1 + 2p
− 2 cos2 x
dx =
6
cos x cot x
8p
π
p
∞
xe−βx sin ax2 sin βx dx =
β
4
[p > 0]
LI (396)(4)
[p > 0]
BI (396)(7)
[p > 0]
BI (396)(8)
+
π
|arg β| < ,
4
π − β2
e 2a
2a3
0
∞
π − β2
β
−βx
2
xe
cos ax cos βx dx =
e 2a
4 2a3
0
1.
3.964
[a > 0,
xe−px cos 2x2 + px dx = 0
,
a>0
Re β > |Im β|]
ET I 84(17)
ET 26(27)
[p > 0]
BI (361)(16)
[p > 0]
BI (361)(17)
[p > 0]
BI (361)(18)
0
√
1
p π
exp − p2
xe−px cos 2x2 − px dx =
8
4
0
∞
x2 e−px sin 2x2 + px + cos 2x2 + px dx = 0
2.
3.
∞
0
∞
4.
x2 e−px sin 2x2 − px − cos 2x2 − px dx =
0
√
1
π
2 − p2 exp − p2
16
4
∞
1
e 4a Γ(μ)
μπ
D −μ
xμ−1 e−x cos x + ax2 dx =
cos
μ
4
(2a) 2
∞
e
xμ−1 e−x sin x + ax2 dx =
5.3
0
1
√
a
[Re μ > 0,
6.6
0
1
4a
Γ(μ)
μπ
D −μ
sin
μ
4
(2a) 2
∞
1.
β2
e− x2 sin a2 x2
0
2.
∞
e
0
2
−β
x2
√
√
π −√2aβ
dx
e
=
sin
2aβ
2
x
2β
√
cos a2 x2
dx
π −
e
=
x2
2β
a > 0]
ET I 321(37)
1
√
a
[Re μ > −1,
3.967
BI (361)(19)
[Re β > 0,
a > 0]
ET I 319(18)
a > 0]
ET I 75(30)a, BI(369)(3)a
√
2aβ
cos
√
2aβ
[Re β > 0,
a > 0]
BI (369)(4), ET I 16(20)
3.972
Trigonometric and exponentials functions and powers
∞
3.
2 −βx2
x e
0
√
π
cos ax dx = <
cos
3
4 4 (a2 + β 2 )
2
507
3
a
arctan
2
β
[Re β > 0]
3.968 1.
∞
e
−βx2
0
∞
2.
π
sin ax dx = −
8
4
e−βx cos ax4 dx =
2
0
π
8
β
J 14
a
β
J 14
a
β2
8a
cos
β2
8a
π
+ +Y
8
ET I 14(3)a
β2
8a
1
4
[Re β > 0,
β2
8a
sin
β2
π
+
8a
8
−Y
1
4
β2
8a
[Re β > 0,
β2
8a
sin
π
+
8
a > 0]
cos
β2
8a
+
a > 0]
π
8
ET I 75(34)
ET I 16(24)
3.969 √
∞
π
−p2 x4 +q 2 x2
3
3
2px cos 2pqx + q sin 2pqx
dx =
e
1.
2
0
∞
2 4
2 2 2.
e−p x +q x 2px sin 2pqx3 − q cos 2pqx3 dx = 0
BI (363)(7)
BI (363)(8)
0
3.971 Notation: In formulas 3.971 1 and 2, p ≥ 0, q ≥ 0, r = 4 a2 + p2 , s = 4 b2 + q 2 , A = arctan ap ,
and B = arctan qb .
∞
dx 1 ∞
dx
q
b
q
b
2
2
exp −px − 2 sin ax + 2
=
exp −px2 − 2 sin ax2 + 2
1.
2
x
x
x
2√ −∞
x
x
x2
0
π
exp [−2rs cos(A + B)] sin [A + 2rs sin(A + B)]
=
2s
∞
2.
0
exp −px2 −
q
b
cos ax2 + 2
2
x
x
BI (369)(16, 17)
∞
dx 1
dx
q
b
=
exp −px2 − 2 cos ax2 + 2
2
x
2√ −∞
x
x
x2
π
exp [−2rs cos(A + B)] cos [A + 2rs sin(A + B)]
=
2s
BI (369)(15, 18)
3.972 1.
2.
,
+ dx
exp −β γ 4 + x4 sin ax2 4 + x4
γ
0
2 2
γ
γ
aπ
I 1/4
=
β 2 + a2 − β K 1/4
8
2 +
4
π
Re β > 0, |arg γ| < ,
4
∞
+ ,
dx
exp −β γ 4 + x4 cos ax2 4
4
0
γ + x
2 2
γ
γ
aπ
I −1/4
=
β 2 + a2 − β K 1/4
8
2 +
4
π
Re β > 0, |arg γ| < ,
4
∞
β 2 + a2 + β
,
a>0
ET I 75(37)
β 2 + a2 + β
,
a>0
ET I 17(28)
508
Trigonometric Functions
3.973 1.
∞
exp (p cos ax) sin (p sin ax)
0
π
dx
= (ep − 1)
x
2
∞
2.
exp (p cos ax) sin (p sin ax + bx)
0
exp (p cos ax) cos (p sin ax + bx)
0
a > 0]
x dx
π
= exp −cb + pe−ac
2
+x
2
[a > 0, b > 0,
WH, FI II 725
c2
c > 0,
exp (p cos x) sin (p sin x + nx)
0
∞
5.
exp (p cos x) sin (p sin x) cos nx
0
dx
π
exp −cb + pe−ac
=
c2 + x2
2c
[a > 0, b > 0,
c > 0,
π
dx
= ep
x
2
[p > 0]
∞
pn π π pk
dx
=
· +
x
n! 4
2
k!
k=n+1
∞
6.
exp (p cos x) cos (p sin x) sin nx
0
LI (366)(3)
n−1
π pk
pn π
dx
=
+
x
2
k!
n! 4
k=0
[p > 0]
3.974
p > 0]
BI (366)(2)
[p > 0]
p > 0]
BI (372)(4)
∞
4.
[p > 0,
BI (372)(3)
∞
3.
3.973
∞
1.
0
LI (366)(4)
π ep − exp pe−ab
dx
exp (p cos ax) sin (p sin ax) cosec ax 2
=
b + x2
2b sinh ab
[a > 0, b > 0, p > 0]
π ep − exp pe−ab
x dx
[1 − exp (p cos ax) cos (p sin ax)] cosec ax 2
=
b + x2
2 sinh ab
BI (391)(4)
∞
[a > 0, b > 0, p > 0]
π ep − exp pe−ab − ab
dx
exp (p cos ax) sin (p sin ax + ax) cosec ax 2
=
b + x2
2b sinh ab
BI (391)(5)
∞
[a > 0, b > 0, p > 0]
∞
π ep − exp pe−ab − ab
x dx
exp (p cos ax) cos (p sin ax + ax) cosec ax 2
=
b + x2
2 sinh ab
0
BI (391)(6)
2.
0
3.
0
4.
[a > 0,
∞
5.
exp (p cos ax) sin (p sin ax)
0
6.
exp (p cos ax) cos (p sin ax)
0
p > 0]
BI (391)(7)
x dx
π
= [1 − exp (p cos ab) cos (p sin ab)]
2
−x
2
b2
[p > 0,
∞
b > 0,
a > 0]
BI (378)(1)
dx
π
exp (p cos ab) sin (p sin ab)
=
b2 − x2
2b
[a > 0,
b > 0,
p > 0]
BI (378)(2)
3.981
Trigonometric and hyperbolic functions
∞
7.
exp (p cos ax) sin (p sin ax) tan ax
0
dx
π
· tanh ab exp pe−ab − ep
=
b2 + x2
2b
[a > 0,
∞
8.
exp (p cos ax) sin (p sin ax) cot ax
0
b2
∞
9.
exp (p cos ax) sin (p sin ax) cosec ax
0
∞
10.
p > 0]
BI (372)(14)
b > 0,
p > 0]
BI (372)(15)
dx
π
cosec ab [ep − exp (p cos ab) cos (p sin ab)]
=
b2 − x2
2b
[a > 0,
b > 0,
dx
π
coth ab ep − exp pe−ab
=
2
+x
2b
[a > 0,
509
[1 − exp (p cos ax) cos (p sin ax)] cosec ax
0
b > 0,
p > 0]
BI (391)(12)
x dx
π
= − exp (p cos ab) sin (p sin ab) cosec ab
b2 − x2
2
[a > 0, b > 0, p > 0]
BI (391)(13)
3.975
∞ sin
1.
β arctan γx
(γ 2 + x2 )
0
·
β
2
1
1
dx
γ 1−β
= ζ(β, γ) − β −
−1
2
4γ
2(β − 1)
e2πx
[Re β > 1,
∞
2.
sin (β arctan x)
(1 + x2 )
0
∞
3.976
0
β
2
·
1
ζ(β)
dx
=
− β
+1
2(β − 1)
2
Re γ > 0]
WH, ET I 26(7)
[Re β > 1]
e2πx
EH I 33(13)
β− 12 −px2
e−p
1 + x2
e
cos [2px + (2β − 1) arctan x] dx = β sin πβ Γ(β)
2p
[Re β > 0, p > 0]
WH
3.98–3.99 Combinations of trigonometric and hyperbolic functions
3.981
∞
1.
0
∞
2.
0
[Re β > 0,
π
aπ
i
sin ax
dx = −
tanh
−
ψ
cosh βx
2β
2β
2β
β + ai
4β
−ψ
β − ai
4β
[Re β > 0,
a > 0]
∞
0
a > 0]
cos ax
π
aπ
dx =
sech
cosh βx
2β
2β
[Re β > 0,
all real a]
sin ax
β + γ + ia
2γ
−ψ
∞
4.
0
sinh aπ
π
sinh βx
i
γ
dx =
+
ψ
βπ
sinh γx
2γ cosh aπ
2γ
γ + cos γ
[|Re β| < Re γ,
5.
∞
cos ax
0
BI (264)(16)
GW (335)(12), ET I 88(1)
3.
π
aπ
sin ax
dx =
tanh
sinh βx
2β
2β
πβ
γ
sin
π
sinh βx
dx =
βπ
sinh γx
2γ cosh aπ
γ + cos γ
[|Re β| < Re γ]
β + γ − ia
2γ
a > 0]
BI (264)(14)
ET I 88(5)
BI (265)(7)
510
Trigonometric Functions
6.
7.
βπ
aπ
π sin 2γ sinh 2γ
sinh βx
dx =
sin ax
βπ
cosh γx
γ cosh aπ
0
γ + cos γ
⎡
∞
3γ − β + ia
1 ⎣
sinh βx
dx =
ψ
cos ax
cosh
γx
4γ
4γ
0
∞
3γ + β + ia
4γ
−ψ
8.
9.
∞
[|Re β| < Re γ,
3γ − β − ia
4γ
⎤
2π sin πβ
γ
⎦
+
πa
cos πβ
+
cosh
γ
γ
sinh
π
cosh βx
dx =
·
πβ
sinh γx
2γ cosh πa
0
γ + cos γ
⎡
∞
3γ + β + ia
i ⎣
cosh βx
dx =
ψ
sin ax
−ψ
cosh γx
4γ
4γ
0
3γ − β − ai
4γ
−ψ
∞
10.
cos ax
0
π/2
cos2m x cosh βx dx =
11.11
0
3γ + β − ia
4γ
ET I 31(13)
[|Re β| < Re γ,
a > 0]
BI (265)(4)
3γ − β + ia
4γ
[|Re β| < Re γ,
a > 0]
[|Re β| < Re γ,
all real a]
cos2m+1 x cosh βx dx =
0
3.982
∞
1.
0
∞
2.
0
3.11
0
∞
ET I 88(6)
BI (265)(6)
(2m)! sinh πβ
2
β (β 2 + 22 ) . . . [β 2 + (2m)2 ]
π/2
12.11
a > 0]
[β = 0]
BI (265)(2)
[|Re β| < Re γ,
3γ + β − ai
+ψ
4γ
⎤
2πi sinh πa
γ
⎦
−
βπ
cosh aπ
+
cos
γ
γ
βπ
aπ
π cos 2γ cosh 2γ
cosh βx
dx =
βπ
cosh γx
γ cosh aπ
γ + cos γ
a > 0]
−ψ
+ψ
πa
γ
sin ax
3.982
WA 620a
πβ
2
(2m + 1)! cosh
(β 2 + 12 ) (β 2 + 32 ) . . . [β 2 + (2m + 1)2 ]
aπ
cos ax
dx =
aπ
2
cosh βx
2β 2 sinh
2β
[Re β > 0,
WA 620a
a > 0]
BI (264)(16)
βπ
aπ
aπ
π a sin βπ
2γ cosh 2γ − β cos 2γ sinh 2γ
sinh βx
sin ax
dx =
βπ
cosh2 γx
γ 2 cosh aπ
γ − cos γ
sin2 x cos ax
π
dx =
2
4
sinh hx
[|Re β| < 2 Re γ, a > 0]
a−2
+
= I(a)
1 − e−π(a−2)
a+2
2a
−
1 − e−πa
1 − e−π(a+2)
1
I(0) = (π coth π − 1) ,
2
I(±2) =
ET I 88(9)
1 π
+ (coth 2π − coth π)
4
2
3.984
Trigonometric and hyperbolic functions
511
3.983
∞
1.6
0
π sin βa arccosh cb
cos ax dx
= √
b cosh βx + c
β c2 − b2 sinh aπ
β
=
[c > b > 0]
π sinh βa arccos cb
√
β b2 − c2 sinh aπ
β
[b > |c| > 0]
[Re β > 0,
∞
2.
0
3.3
4.
5.
aγ
β
cos ax dx
π sinh
=
cosh βx + cos γ
β sin γ sinh aπ
β
a > 0]
π Re β < Im βγ,
GW (335)(13a)
a>0
BI (267)(3)
∞
sin ab
cos ax dx
= −π coth aπ
[a > 0, b > 0]
ET I 30(8)
sinh b
0 cosh x − cosh b
π
∞
cos ax dx
2
<
<
ET I 30(9)
=
[a > 0]
2
2
0 1 + 2 cosh
πx
1
+
2
cosh
πa
3
3
+
,
+
,
+
,
+
,
β
β
a
a
∞
π
sin
(π
−
δ)
sinh
(π
+
δ)
−
sin
(π
+
δ)
sinh
(π
−
δ)
γ
γ
γ
γ
sin ax sinh βx
dx =
2πβ
2πa
cosh
γx
+
cos
δ
0
γ sin δ cosh
− cos
γ
[π Re γ > |Re γδ|,
∞
6.
0
cos ax cosh βx
dx =
cosh γx + cos b
γ
|Re β| < Re γ,
a > 0]
BI (267)(2)
+
,
+
,
+
,
+
,
π cos βγ (π − b) cosh γa (π + b) − cos βγ (π + b) cosh γa (π − b)
2πβ
γ sin b cosh 2πa
γ − cos γ
[|Re β| < Re γ, 0 −1,
|arg(β + 1)| < π,
a > 0]
ET I 30(10)
3.984
1.
6
2.6
lim
∞
lim
∞
c↑1 0
c↑1 0
∞
0
∞
4.
0
5.
0
[|b| ≤ π,
sinh ab
cos ax cosh cx
dx = −π cot b
cosh x + cos b
sinh aπ
[0 < |b| < π,
sin ax sinh x2
π sinh aβ
dx =
cosh x + cos β
2 sin β2 cosh aπ
3.8
sin ax sinh cx
cosh ab
dx = π
cosh x + cos b
sinh aπ
∞
π cos aγ
cos ax cosh β2 x
β
dx =
cosh βx + cosh γ
2β cosh γ2 cosh aπ
β
sin ax sinh βx
aπ
dx =
2
cosh 2βx + cos 2ax
4 (a + β 2 )
a real]
[Reβ < π,
a real]
a > 0]
! !
π Re β > !Im βγ !
[a > 0,
Re β > 0]
BI (267)(1)
BI (267)(5)
ET I 80(10)
ET I 31(16)
BI (267)(7)
512
Trigonometric Functions
∞
6.
0
∞
7.8
0
βπ
cos ax cosh βx
dx =
cosh 2βx + cos 2ax
4 (a2 + β 2 )
3.985
[Re β > 0,
a > 0]
BI (267)(8)
sinh2μ−1 x cosh2−2ν+1 x
1
dx = B(μ, ν − μ) 2 F 1 (, μ; ν; β)
2
cosh2 x − β sinh2 x
[Re ν > Re μ > 0]
3.985 1.
∞
0
2ν−2
cos ax dx
=
Γ
coshν βx
β Γ(ν)
ν
ai
+
2 2β
Γ
ν
ai
−
2 2β
[Re β > 0,
Re ν > 0,
EH I 115(12)
a > 0]
ET I 30(5)
2.
3.
3.986
n−1
-
∞
a2
cos ax dx
4n−1 πa
+ k2
=
aπ
2n
2
2
4β
cosh
βx
0
2(2n − 1)!β sinh
2β k=1 2
πa a + 22 β 2 a2 + 42 β 2 · · · a2 + (2n − 2)2 β 2
=
aπ
2(2n − 1)!β 2n sinh
2β
[n ≥ 2, a > 0]
%
&
∞
n
2
2k − 1
cos ax dx
a2
π22n−1
+
=
2n+1
2
βx (2n)!β cosh aπ k=1 4β 2
0 cosh
2β
2
π a + β 2 a2 + 32 β 2 · · · a2 + (2n − 1)2 β 2
=
aπ
2(2n)!β 2n+1 cosh
2β
[Re β > 0, n = 0, 1, . . . , all real a]
∞
1.
0
∞
2.
0
∞
3.
0
∞
4.3
0
∞
[|Im(β + γ)| < Re δ]
BI (264)(19)
π sinh πα
sin αx cos βx
γ
dx =
απ
sinh γx
2γ cosh γ + cosh βπ
γ
[|Im(α + β)| < Re γ]
LI (264)(20)
γπ
cosh βπ
π
cos βx cos γx
2δ cosh 2δ
dx = ·
γπ
cosh δx
δ cosh βπ
δ + cosh δ
[|Im(β + γ)| < Re δ]
BI (264)(21)
β−1
β coth β − 1
sin2 βx
β
+
=
dx =
2
2β
π (e − 1)
2π
2π
sinh πx
2.
0
∞
EH I 44(3)
sin ax (1 − tanh βx) dx =
π
1
−
a 2β sinh απ
2β
[Re β > 0]
ET I 88(4)a
sin ax (coth βx − 1) dx =
aπ
1
π
coth
−
2β
2β
a
[Re β > 0]
ET I 88(3)
0
ET I 30(4)
γπ
sinh βπ
π
sin βx sin γx
2δ sinh 2δ
dx = ·
cosh δx
δ cosh βδ π + cosh γδ π
[|Im β| < π]
3.987 1.
ET I 30(3)
3.991
3.988
Trigonometric and hyperbolic functions
π/2
1.
0
513
cos ax sinh (2b cos x)
π√
√
dx =
πb I α2 + 14 (b) I − a2 + 14 (b)
2
cos x
[a > 0]
π/2
2.
0
ET I 37(66)
cos ax cosh (2b cos x)
π√
√
dx =
πb I a2 − 14 (b) I − a2 − 14 (b)
2
cos x
[a > 0]
∞
3.
0
3.989
∞
1.
0
∞
2.
0
∞
3.
0
4.
5.
π P − 12 +ia (cos b)
cos ax dx
√
√
=
2 cosh aπ
cosh x cos b
2
[a > 0,
b > 0]
ET I 30(7)
[a > 0,
b > 0]
ET I 93(44)
[a > 0,
b > 0]
ET I 93(45)
2
sin a πx sin bx
π
πb2
πb
dx =
sin 2 cosech
sinh ax
2a
4a
2a
2
ET I 37(67)
2
2
πb
cos a πx sin bx
π cosh πb
a − cos 4a2
dx =
sinh ax
2a
sinh πb
2a
a2 π
sin xπ cos ax
π cos 4 − √2
dx =
cosh x
2 cosh aπ
2
2
1
ET I 36(54)
2
2
a π
1
cos xπ cos ax
π sin 4 + √2
dx = ·
cosh x
2
cosh aπ
0
2
∞
∞
+
2
2 ,
sin πax cos bx
dx = −
exp − k + 12 b sin k + 12 πa
cosh πx
0
k=0
% %
&
2 &
∞
k + 12 π
b k + 12
b2
1 π
−
+
exp −
+√
sin
a
4
4πa
a
a
∞
ET I 36(55)
k=0
ET I 36(56)
[a > 0, b > 0]
&
%
∞
2
∞
cos πax2 cos bx
1
1
dx =
(−1)k exp − k +
πa
b cos k +
cosh
πx
2
2
0
k=0
%
% &
2 &
∞
k + 12 π
b k + 12
1 π
b2
+√
cos
−
+
exp −
a
4
4πa
a
a
6.
k=0
[a > 0,
3.991
1.
2.11
∞
π
a
1
tanh sin
+
2
2
4
0
∞
a
1
cos πx2 sin ax coth πx dx = tanh
1 − cos
2
2
0
sin πx2 sin ax coth πx dx =
a2
4π
π
a2
+
4
4π
b > 0]
ET I 36(57)
ET I 93(42)
ET I 93(43)
514
3.992
Trigonometric Functions
∞
1.
0
2
sin πx2 cos ax
2
1 + 2 cosh √ πx
3
π
a
cos 12
− 4π
√
dx = − 3 +
a
4 cosh √ − 2
3
[a > 0,
∞
4
2
ib
1
4+ 2
,
(a)
∞
b > 0]
ET I 37(63)
ET I 93(47)
,
cos (2a sinh x) sin bx
i√
√
dx = −
πa I − 1 − ib (a) K − 1 + ib (a) − I − 1 + ib (a) K − 1 − ib (a)
4
2
4
2
4
2
4
2
2
sinh x
0
[a > 0, b > 0]
ET I 93(48)
√ +
∞
,
sin (2a sinh x) cos bx
πa
√
I 1 − ib (a) K 1 + ib (a) + I 1 + ib (a) K 1 − ib (a)
dx =
4
2
4
2
4
2
4
2
2
sinh x
0
[a > 0, b > 0]
ET I 37(64)
√ +
∞
,
cos (2a sinh x) cos bx
πa
√
I − 1 − ib (a) K − 1 + ib (a) + I − 1 + ib (a) K − 1 − ib (a)
dx =
4
2
4
2
4
2
4
2
2
sinh x
0
7.
+
[a > 0,
sin (a cosh x) sin (a sinh x)
0
3.995
(a) + J 1 − ib (a) Y
[a > 0, b > 0]
+
,
sin (2a sinh x) sin bx
i√
√
dx = −
πa I 1 − ib (a) K − 1 + ib (a) − I 1 + ib (a) K 1 − ib (a)
4
2
4
2
4
2
4
2
2
sinh x
0
6.
ib
1
4− 2
∞
3.
5.
ET I 37(58)
[a > 0, b > 0]
ET I 37(62)
+
,
cos (2a cosh x) cos bx
π√
√
dx = −
aπ J − 1 + ib (a) Y − 1 − ib (a) + J − 1 − ib (a) Y − 1 + ib (a)
4
2
4
2
4
2
4
2
4
cosh x
0
4.
ET I 37(61)
∞
2.
π
12
sin
cos πx2 cos ax
2
1 + 2 cosh √ πx
3
ET I 37(60)
2
a
− 4π
dx = 1 −
2.
a
0
4 cosh √ − 2
3
√
∞ 2
sin x + cos x2
π sin2 a + cos a2
√
√
·
cos ax dx =
3.993
2
cosh ( πx)
cosh ( πa)
0
3.994
∞
sin (2a cosh x) cos bx
π√ +
√
dx = −
1.
aπ J 1 + ib (a) Y
4
2
4
cosh x
0
∞
3.992
π/2
1.
sin 2a cos2 x cosh (a sin 2x)
b2 cos2 x + c2 sin2 x
0
2.
0
π
dx
= sin a
sinh x
2
π/2
cos 2a cos x cosh (a sin 2x)
dx =
2
b2
cos2
x+
c2
2
sin x
dx =
b > 0]
[a > 0]
ET I 37(65)
BI (264)(22)
2ac
π
sin
2bc
b+c
[b > 0,
c > 0]
BI (273)(9)
[b > 0,
c > 0]
BI (273)(10)
2ac
π
cos
2bc
b+c
3.997
3.996
Trigonometric and hyperbolic functions
∞
sin (a sinh x) sinh βx dx = sin
1.
0
βπ
K β (a)
2
∞
2.
cos (a sinh x) cosh βx dx = cos
0
βπ
K β (a)
2
π/2
3.
cos (a sin x) cosh (β cos x) dx =
0
4.
5.
3.997
√
π/2
sinν x sinh (β cos x) dx =
0
π
0
π/2
0
4.
0
[|Re β| < 1,
a > 0]
WA 202(13)
MO 40
ν
2
2
β
π
2
Γ
ν+1
2
[|Re β| < 1,
a > 0]
WA 199(12)
[|Re β| < 1,
a > 0]
WA 199(13)
L ν2 (β)
√
π
2
β
ν
2
Γ
ν+1
2
π/2
EH II 38(53)
I ν2 (β)
[Re ν > −1]
3.
EH II 82(26)
[Re ν > −1]
sinν x cosh (β cos x) dx =
2.
a > 0]
∞
1.
[|Re β| < 1,
π
J0
a2 − β 2
2
π
sin a cosh x − 12 βπ cosh βx dx = J β (a)
2
0
∞
π
cos a cosh x − 12 βπ cosh βx dx = − Y β (a)
2
0
515
∞
√ dx
(−1)k
√
√
= 2π
2k + 1
cosh (tan x) cos x sin 2x
k=0
WH
BI (276)(13)
∞
Γ(q) dx
tanq x
sin kλ
=
(−1)k−1 q
cosh (tan x) + cos λ sin 2x
sin λ
k
k=1
[q > 0]
BI (275)(20)
516
Trigonometric Functions
4.111
4.11–4.12 Combinations involving trigonometric and hyperbolic functions and powers
4.111
∞
1.
0
∂ 2m
sin ax
π
· x2m dx = (−1)m
· 2m
sinh βx
2β ∂a
tanh
aπ
2β
[Re β > 0]
GW (336)(17a)
∞
2.
0
cos ax
π ∂ 2m+1
· x2m+1 dx = (−1)m
sinh βx
2β ∂a2m+1
aπ
2β
tanh
[Re β > 0]
∞
3.
0
(cf. 3.981 1)
∞
4.
0
∂ 2m+1
sin ax
π
· x2m+1 dx = (−1)m+1
· 2m+1
cosh βx
2β ∂a
∂ 2m
cos ax
π
· x2m dx = (−1)m
· 2m
cosh βx
2β ∂a
GW (336)(17b)
1
cosh aπ
2β
[Re β > 0]
1
cosh aπ
2β
(cf. 3.981 1)
(cf. 3.981 3)
GW (336)(18b)
[Re β > 0]
(cf. 3.981 3)
GW (336)(18a)
∞
5.
x
sinh
π2
sin 2ax
dx =
·
2
cosh βx
4β cosh2 aπ
β
[Re β > 0,
a > 0]
BI (364)(6)a
x
π2
cos 2ax
1
dx =
·
2
sinh βx
4β cosh2 aπ
β
[Re β > 0,
a > 0]
BI (364)(1)a
[Re β > 0,
a > 0]
0
∞
6.
0
∞
7.
0
aπ
β
πa
sin ax dx
= 2 arctan exp
cosh βx x
2β
−
π
2
BI (387)(1), ET I 89(13), LI (298)(17)
4.112
1.
2.
∞
2
cos ax
2β 3
x + β2
πx dx = cosh3 aβ
0
cosh
2β
∞
cos ax
6β 4
x x2 + 4β 2
πx dx = cosh4 aβ
0
sinh
2β
[Re β > 0,
a > 0]
ET I 32(19)
[Re β > 0,
a > 0]
ET I 32(20)
4.113
Trigonometric and hyperbolic functions and powers
4.113 1.
∞
0
dx
sin ax
1
πe−aβ
· 2
=− 2 −
2
sinh πx x + β
2β
β sin πβ
1 + 2 2 F 1 1, −β; 1 − β; −e−a +
2β
∞
(−1)k e−ak
πe−aβ
1
−
−
=
2
2β
2β sin πβ
k2 − β 2
2F 1
517
1, β; 1 + β : −e−a
k=1
[Re β > 0,
2.
3.
4.
5.
6.
7.
β = 0, 1, 2, . . . ,
a > 0]
ET I 90(18)
m−1
dx
1 (−1)k e−ka
(−1)m e−ma sin ax
(−1)m ae−ma
· 2
+
+
ln 1 + e−a
=
2
2m
2m
m−k
2m
0 sinh πx x + m
k=1
1 dm−1 (1 + z)m−1
+
ln(1 + z)
2m! dz m−1
z
z=e−a
[a > 0]
ET I 89(17)
∞
∞
sin ax
sin ax dx
dx
a
1
a
·
=
= − cosh a+sinh a ln 2 cosh
GW (336)(21b)
2
2
sinh
πx
1
+
x
2
sinh
πx
1
+
x
2
2
0
−∞
∞
sin ax
dx
dx
1 ∞ sin ax
π
·
·
=
= sinh a − cosh a arctan (sinh a)
π
π
2
2
2 −∞ sinh x 1 + x
2
0 sinh x 1 + x
2
2
∞
GW (336)(21a)
√
√
sin ax
2
dx
π −a sinh a 2 cosh a + 2 √
π · 1 + x2 = − √2 e + √2 ln 2 cosh a − √2 + 2 cosh a arctan 2 sinh a
0 sinh x
4
[a > 0]
LI (389)(1)
√
∞
1
sin ax
x dx
π −a sinh a 2 cosh a + 2 √
π · 1 + x2 = √2 e + √2 ln 2 cosh a − √2 − 2 cosh a arctan √2 sinh a
0 cosh x
4
BI (388)(1)
[a > 0]
∞
x dx
cos ax
1 a
·
= − + e−a + cosh a ln 1 + e−a
2
sinh
πx
1
+
x
2
2
0
∞
∞
8.
0
∞
9.11
0
10.11
0
[a > 0]
BI (389)(14), ET I 32(24)
−a π −a
cos ax
x dx
+ e −1
π · 1 + x2 = 2 sinh a arctan e
2
sinh x
2
[a > 0]
BI (389)(11)
∞
(−1)k e−(k+1/2)a
cos ax
dx
πe−aβ
· 2
−
=
2
cosh πx x + β 2
2β cos(βπ)
k + 1 − β2
k=0
∞
−aβ
2
(−1)m e
aβ +
dx
cos ax
·
=
2
cosh πx x2 + m + 1
2β 2
2
[Re β > 0,
1
2
−
a > 0]
ET I 32(26)
∞
(−1)k e−(k+1/2)a
2
k + 12 − β 2
k=0
[Re β > 0,
a > 0]
ET I 32(25)
518
Trigonometric Functions
∞
11.
0
∞
12.
0
∞
13.
0
4.114
a
a dx
cos ax
a ·
= 2 cosh − ea arctan e− 2 + e−a arctan e 2
cosh πx 1 + x2
2
dx
cos ax
·
= ae−a + cosh a ln 1 + e−2a
π
2
cosh 2 x 1 + x
dx
cos ax
π
2 sinh a
·
arctan
= √ e−a + √
π
2
cosh 4 x 1 + x
2
2
[a > 0]
ET I 32(21)
[a > 0]
BI (388)(6)
1
√
2 sinh a
√
cosh a 2 cosh a + 2
√
− √ ln
2
2 cosh a − 2
[a > 0]
4.114
∞
1.
0
∞
2.
0
4.115
∞
1.
0
0
4.
βπ
aπ
sin ax sinh βx
dx = arctan tan
tanh
x sinh γx
2γ
2γ
[|Re β| < Re γ,
βπ
aπ
cos ax sinh βx
1 cosh 2γ + sin 2γ
dx = ln
βπ
x cosh γx
2 cosh aπ
2γ − sin 2γ
[|Re β| < Re γ]
a > 0]
BI (387)(6)a
ET I 33(34)
∞
π e−ab sin bβ x sin ax sinh βx
ke−ak sin kβ
dx =
+
·
(−1)k
2
2
x + b sinh πx
2 sin bπ
k 2 − b2
k=1
[0 < Re β < π,
a > 0,
b > 0]
BI (389)(23)
∞
2.
3.
BI (388)(5)
∞
∞
1
x sin ax sinh βx
1
·
dx = e−a (a sin β − β cos β) − sinh a sin β ln 1 + 2e−a cos β + e−2a
2
x + 1 sinh πx
2
2
sin β
+ cosh a cos β arctan a
e + cos β
[|Re β| < π, a > 0]
LI (389)(10)
x sin ax sinh βx
·
dx
x2 + 1 sinh π x
0
2
cos β
cosh a + sin β
1
π
− sin β cosh a arctan
= e−a sin β + cos β sinh a ln
2
2
cosh a+ − sin β
sinh
a
,
π
BI (389)(8)
|Re β| < , a > 0
2
∞
∞
π e−ab sin bβ cos ax sinh βx
e−ak sin kβ
dx
=
·
+
·
(−1)k
2
2
sinh πx
2b
sin bπ
k 2 − b2
0 x +b
k=1
[0 < Re β < π, a > 0, b > 0]
5.
0
BI (389)(22)
1
cos ax sinh βx
1
·
dx = e−a (a sin β − β cos β) + cosh a sin β ln 1 + 2e−a cos β + e−2a
2
x + 1 sinh πx
2
2
sin β
− sinh a cos β arctan a
e + cos β
[|Re β| < π, a > 0, b > 0] BI (389)(20)a
4.115
Trigonometric and hyperbolic functions and powers
∞
6.
0
BI (389)(18)
a
β
β sin ax sinh βx
β
a
dx = e− 2 a sin − β cos
·
− sinh sin ln 1 + 2e−a cos β + e−2a
1
cosh πx
2
2
2
2
x2 +
β
sin β
a
4
+ cosh cos arctan
2
2
1 + e−a cos β
[|Re β| < π, a > 0]
ET I 91(26)
∞
−ak
1
sin ax cosh γx
cos kγ
π e−aβ cos βγ k−1 e
dx
=
·
+
·
−
(−1)
2
2
2
2
x +β
sinh πx
2β
2β
sin βπ
k − β2
k=1
[0 ≤ Re β, |Re γ| < π,
0
8.
0
∞
0
11.
∞
a > 0]
BI (389)(21)
9.
10.
cosh a + sin β
cos β
cos ax sinh βx
π
1
·
+ sinh a sin β arctan
dx = e−a sin β − cosh a cos β ln
x2 + 1 sinh π x
2
2
cosh a − sin β
sinh a
2
+
,
π
|Re β| < , a > 0, b > 0
2
∞
7.
519
1
sin ax cosh βx
1
·
dx = − e−a (a cos β + β sin β) + sinh a cos β ln 1 + 2e−a cos β + e−2a
2
x + 1 sinh πx
2
2
sin β
+ cosh a sin β arctan a
e + cos β
[|Re β| < π, a > 0] ET I 91(25), LI (389)(9)
∞
cosh a + sin β
cos β
sin ax cosh βx
π −a
1
·
π dx = − 2 e cos β+ 2 sinh a sin β ln cosh a − sin β +cosh a cos β arctan sinh a
2+1
x
0
sinh x
2
+
,
π
|Re β| < , a > 0
BI (389)(7)
2
∞
∞
−ak
π e−ab cos bβ x cos ax cosh βx
cos kβ
k ke
dx
=
·
+
·
(−1)
2
2
2
x +b
sinh πx
2
sin bπ
k − b2
0
k=1
[|Re β| < π,
∞
12.
0
13.
0
∞
a > 0]
BI (389)(24)
1
x cos ax cosh βx
·
dx = e−a (a cos β + β sin β)
x2 + 1 sinh πx
2
1 1
− + cosh a cos β ln 1 + 2e−a cos β + e−2a
2 2
sin β
+ sinh a sin β arctan a
e + cos β
[|Re β| < π, a > 0]
BI (389)(19)
x cos ax cosh βx
cosh a + sin β
π
1
·
dx = −1 + e−a cos β + cosh a sin β ln
x2 + 1 sinh π x
2
2
cosh a − sin β
cos β
2
+ sinh a cos β arctan
sinh a +
,
π
|Re β| < , a > 0
2
BI (389)(17)
520
Trigonometric Functions
∞
14.
0
4.116
1.
4.116
cos ax cosh βx
e−2a sin 2β
−a
−a
·
dx
=
ae
cos
β
+
βe
sin
β
+
sinh
a
sin
β
arctan
x2 + 1 cosh π x
1 + e−2a cos 2β
1
2
+ cosh a cos β ln 1 + 2e−2a cos 2β + e−4a
2
,
+
π
|Re β| < , a > 0
ET I 34(37)
2
∞
6
x cos 2ax tanh x dx
the integral is divergent
BI (364)(2)
[Re β > 0,
BI (387)(8)
0
∞
2.
cos ax tanh βx
0
4.117
∞
1.
0
2.
3.
4.
5.
aπ
dx
= ln coth
x
4β
sin ax
πx
dx = a cosh a − sinh a ln (2 sinh a)
tanh
1 + x2
2
[a > 0]
BI (388)(3)
∞
π
sin ax
πx
a
dx = − ea + sinh a ln coth + 2 cosh a arctan (ea )
tanh
2
4
2
2
0 1+x
∞
sin ax
a
[a > 0]
coth πx dx = e−a − sinh a ln 1 − e−a
2
2
0 1+x
∞
a
sin ax
π
[a > 0]
coth x dx = sinh a ln coth
2
1
+
x
2
2
0
∞
x cos ax
π
tanh x dx = −ae−a − cosh a ln 1 − e−2a
2
1+x
2
0
∞
0
0
∞
8.
0
∞
9.
0
BI (389)(6)
BI (388)(7)
x cos ax
π
π
a
tanh x dx − ea + cosh a ln coth + 2 sinh a arctan (ea )
1 + x2
4
2
2
x cos ax
a
1
coth πx dx = − e−a − − cosh a ln 1 − e−a
2
1+x
2
2
a
x cos ax
π
coth x dx = −1 + cosh a ln coth
2
1+x
2
2
BI (388)(8)
BI (389)(15)a, ET I 33(31)a
[a > 0]
BI (389)(12)
π
x cos ax
π
a
coth x dx = −2 + e−a + cosh a ln coth + 2 sinh a arctan e−a
2
1+x
4
2
2
[a > 0]
BI (389)(5)
[a > 0]
∞
7.
BI (388)(4)
[a > 0]
6.
BI (389)(13)
∞
1
x sin ax
1
1
π
πa coth πa − 1
dx =
2
1
2 sinh 2 πa 2
2
0 ∞ cosh x
pπ
1 − cos px dx
·
= ln cosh
4.119
sinh qx
x
2q
0
4.1188
a > 0]
ET I 89(14)
BI (387)(2)a
4.123
Trigonometric and hyperbolic functions and powers
521
4.121
∞
1.
0
∞
2.
0
sin ax − sin bx dx
·
= 2 arctan
cosh βx
x
bπ
aπ
− exp
2β
2β
(a + b)π
1 + exp
2β
exp
bπ
cos ax − cos bx dx
2β
·
= ln
aπ
sinh βx
x
cosh
2β
[Re β > 0]
GW (336)(19b)
[Re β > 0]
GW (336)(19a)
cosh
4.122
γπ
sinh
cos βx sin γx dx
2δ
·
= arctan
βπ
cosh δx
x
0
cosh
2δ
∞
1 cosh 2aπ + cos βπ
cosh βx dx
2
·
= ln
sin ax
sinh x
x
4
1 + cos βπ
0
1.6
2.
4.123
∞
∞
1.
0
∞
2.
0
∞
3.
0
∞
4.
0
∞
5.
0
[Re δ > |Im β| + |Im γ|]
[|Re β| < 1]
ET I 93(46)a
BI (387)(7)
x dx
sin x
1 1
·
= arctan −
cosh ax + cos x x2 − π 2
a a
BI (390)(1)
x dx
sin x
a
1
· 2
=
− arctan
2
2
cosh ax − cos x x − π
1+a
a
BI (390)(2)
x dx
sin 2x
1 1 + 2a2
1
· 2
·
=
− arctan
2
cosh 2ax − cos 2x x − π
2a 1 + a2
a
BI (390)(4)
x dx
cosh ax sin x
−1
·
=
cosh 2ax − cos 2x x2 − π 2
2a (1 + a2 )
LI (390)(3)
dx
cos ax
πe−aγ
· 2
=
2
cosh πx + cos πβ x + γ
2γ (cos γπ + cos βπ)
∞
e−(2k+1−β)a
e−(2k+1+β)a
1
− 2
+
sinh βπ
γ 2 − (2k + 1 − β)2
γ − (2k + 1 + β)2
k=0
[0 < Re β < 1,
∞
6.
0
7.
0
∞
Re γ > 0,
a > 0]
ET I 33(27)
∞
sin ax sinh bx
Γ(p)
a (−1)k
xp−1 dx =
p arctan
p sin
cos 2ax + cosh 2bx
b
(2k + 1)p
(a2 + b2 ) 2
k=0
sin ax2
sin
sinh
1 ∂ ϑ1 (z | q)
· x dx =
cos πx + cosh πx
4
∂z
πx
2
πx
2
[p > 0]
BI (364)(8)
z=0,q=e−2a
[a > 0]
ET I 93(49)
522
4.124
Trigonometric Functions
1
1.
0
√
cos px cosh q 1 − x2
π
√
dx = J 0
p2 − q 2
2
1 − x2
∞
2.
u
4.125
∞
1.
0
dx
π
cos ax cosh β (u2 − x2 ) · √
= J0
2
2
2
u −x
4.124
MO (40)
u
ET I 34(38)
a2 − β 2
(−1)n−1 a2n−1 π
dx
a2
=
sinh (a sin x) cos (a cos x) sin x sin 2nx
1+
x
(2n − 1)! 8
2n(2n + 1)
LI (367)(14)
∞
(−1)n−1 a2(n−1) π
dx
a2
=
cosh (a sin x) cos (a cos x) sin x cos(2n − 1)x
1−
x
[2(n − 1)]! 8
2n(2n − 1)
0
2.
LI (367)(15)
∞
3.
sinh (a sin x) cos (a cos x) cos x cos 2nx
0
∞
(−1)n a2n+1 3π
dx π (−1)k a2k+1
=
+
x
2
(2k + 1)!
(2n + 1)! 8
k=n+1
(−1)n−1 a2n−1 π
+
(2n − 1)! 8
LI (367)(21)
4.126 1.
∞
sin (a cos bx) sinh (a sin bx)
0
[b > 0]
∞
2.
sin (a cos bx) cosh (a sin bx)
0
cos (a cos bx) sinh (a sin bx)
0
BI (381)(2)
dx
π
cos (a cos bc) sinh (a sin bc)
=
c2 − x2
2c
[b > 0,
∞
3.
x dx
π
= [cos (a cos bc) cosh (a sin bc) − 1]
2
−x
2
c2
c > 0]
BI (381)(1)
x dx
π
= [a cos bc − sin (a cos bc) cosh (a sin bc)]
2
−x
2
c2
[b > 0]
∞
4.
cos (a cos bx) cosh (a sin bx)
0
BI (381)(4)
dx
π
= − sin (a cos bc) sinh (a sin bc)
c2 − x2
2c
[b > 0]
BI (381)(3)
4.13 Combinations of trigonometric and hyperbolic functions and exponentials
4.131
1.
0
∞
sin ax sinhν γxe−βx dx = −
⎧
i Γ(ν + 1) ⎨ Γ
2ν+2 γ
⎩Γ
β−νγ−ai
2γ
β+νγ−ai
2γ
[Re ν > −2,
+1
−
Re γ > 0,
Γ
Γ
β−νγ+ai
2γ
β+γν+ai
2γ
⎫
⎬
+1 ⎭
|Re(γν)| < Re β]
ET I 91(30)a
4.133
Trigonometric and hyperbolic functions and exponentials
∞
2.
cos ax sinhν γxe−βx dx =
0
⎧
Γ(ν + 1) ⎨ Γ
2ν+2 γ ⎩ Γ
β−νγ−ai
2γ
β+γν−ai
2γ
[Re ν > −1,
∞
3.
e−βx
0
∞
4.
0
4.132
∞
1.
0
∞
2.
0
e−x
∞
2a
sin ax
dx =
2
2
sinh γx
a + [β + (2k − 1)γ]
k=1 β + γ + ia
1
=
ψ
−ψ
2γi
2γ
+1
−
β−νγ+ai
2γ
Γ
Γ
β+νγ+ai
2γ
∞
3.
0
∞
4.
0
∞
0
β + γ − ia
2γ
[Re β > |Re γ|]
π
aπ
1
sin ax
dx = coth
−
sinh x
2
2
a
∞
1.11
0
sinh 2πa
sin ax sinh βx
a
π
γ
dx
=
−
+
·
2πβ
eγx − 1
2 (a2 + β 2 ) 2γ cosh 2πa
γ − cos γ
β
a
a
β
i
+i +1 −ψ
−i +1
+
ψ
2γ
γ
γ
γ
γ
[Re γ > |Re β|, a > 0]
2.11
0
ET I 92(33)
sinh 2πa
a
π
sin ax cosh βx
γ
dx
=
−
+
·
eγx − 1
2 (a2 + β 2 ) 2γ cosh 2πa
−
cos 2πβ
γ
γ
a
π
sin ax cosh βx
dx =
− ·
eγx + 1
2 (a2 + β 2 ) γ
BI (265)(5)a, ET I 92(34)
βπ
sinh aπ
γ cos γ
2βπ
cosh 2aπ
γ − cos γ
ET I 92(35)
sin 2πβ
β
π
cos ax sinh βx
γ
dx
=
−
·
eγx − 1
2 (a2 + β 2 ) 2γ cosh 2aπ
−
cos 2βπ
γ
γ
LI (265)(8)
πβ
πa
β
π sin γ cosh γ
cos ax sinh βx
dx
=
−
+
2βπ
eγx + 1
2 (a2 + β 2 ) γ cosh 2aπ
γ − cos γ
sin ax sinh βx exp −
x2
4γ
dx =
cos ax cosh βx exp −
2
x
4γ
dx =
ET I 34(39)
√
πγ exp γ β 2 − a2 sin(2aβγ)
[Re γ > 0]
∞
ET I 91(28)
ET I 91(29)
[Re γ > |Re β|]
4.133
ET I 34(40)a
BI (264)(9)a
[Re γ > |Re β|]
5.
+1 ⎭
[Re γ > |Re β|]
⎫
⎬
|Re(γν)| < Re β]
Re γ > 0,
[Re γ > |Re β|]
523
ET I 92(37)
√
πγ exp γ β 2 − a2 cos(2aβγ)
[Re γ > 0]
ET I 35(41)
524
4.134
Trigonometric Functions
∞
1.
e−βx (cosh x − cos x) dx =
2
0
∞
2.
e−βx (cosh x − cos x) dx =
2
0
4.135
∞
−βx2
sin ax cosh 2γxe
1.
0
∞
2.
1
π
cosh
β
4β
[Re β > 0]
ME 24
1
π
sinh
β
4β
[Re β > 0]
ME 24
.
2
1
dx =
2
4
π2
βγ 2
exp
−
a2 + β 2
a2 + β 2
2
0
1
2
sin
4
a2
π2
βγ 2
exp − 2
2
+β
a + β2
cos
[Re β > 0]
4.136
∞
1.
0
√
4
2π
sinh2 x + sin x2 e−βx dx = √ I 14
4 β
√
∞
−βx4
2π
2
2
sinh x − sin x e
dx = √ I 14
4
β
0
2.
1
8β
1
8β
√
∞
−βx4
2π
2
2
cosh x + cos x e
dx = √ I − 14
4
β
0
√
∞
−βx4
2π
2
2
cosh x − cos x e
dx = √ I − 14
β
4
0
4.
sinh
cosh
sinh
∞
1.
0
0
3.
0
1
β
cos
sin 2x2 cosh 2x2 e−βx dx = 4
4
π
128β 2
J 14
1
β
cos
cos 2x2 sinh 2x2 e−βx dx = 4
4
−π
128β 2
J 14
1
β
sin
ME 24
ME 24
MI 32
1
π
−
β
4
[Re β > 0]
∞
ME 24
1
π
+
β
4
[Re β > 0]
∞
2.
4
π
sin 2x2 sinh 2x2 e−βx dx = J − 14
4
128β 2
ME 24
1
8β
[Re β > 0]
4.137
LI (268)(8)
1
8β
[Re β > 0]
1
8β
aγ 2
a
1
+ arctan
+ β2
2
β
a2
1
8β
[Re β > 0]
1
8β
LI (268)(7)
1
8β
cosh
[Re β > 0]
3.
aγ 2
a
1
+ arctan
2
2
a +β
2
β
[Re β > 0]
.
cos ax2 cosh 2γxe−βx dx =
4.134
MI 32
1
π
−
β
4
[Re β > 0]
MI 32
4.141
Trigonometric and hyperbolic functions, exponentials, and powers
∞
4.
0
4
π
cos 2x2 cosh 2x2 e−βx dx = J − 14
4
128β 2
1
β
525
1
π
+
β
4
sin
[Re β > 0]
4.138
∞
1.
0
0
cos
1
β
2
4
π
sin 2x cosh 2x2 − cos 2x2 sinh 2x2 e−βx dx = J1
4
32β 2 4
MI 32
1
β
sin
1
β
[Re β > 0]
∞
3.
0
1
β
[Re β > 0]
∞
2.
2
4
π
sin 2x cosh 2x2 + cos 2x2 sinh 2x2 e−βx dx = J1
4
32β 2 4
MI 32
2
4
π
cos 2x cosh 2x2 + sin 2x2 sinh 2x2 e−βx dx = J − 14
4
32β 2
MI 32
1
β
cos
1
β
[Re β > 0]
∞
4.
0
2
4
π
cos 2x cosh 2x2 − sin 2x2 sinh 2x2 e−βx dx = J − 14
4
32β 2
[Re β > 0]
MI 32
1
β
sin
1
β
MI 32
4.14 Combinations of trigonometric and hyperbolic functions, exponentials, and
powers
4.141
∞
−βx2
xe
1.
0
∞
2.
1
cosh x sin x dx =
4
xe−βx sinh x cos x dx =
2
0
∞
3.
x2 e−βx cosh x cos x dx =
2
0
4.
0
∞
1
4
x2 e−βx sinh x sin x dx =
2
1
4
1
4
π
β3
π
β3
π
β3
cos
cos
1
1
+ sin
2β
2β
[Re β > 0]
MI 32
[Re β > 0]
MI 32
1
1
− sin
2β
2β
cos
1
1
1
− sin
2β
β
2β
[Re β > 0]
π
β3
sin
MI 32
1
1
1
+ cos
2β
β
2β
[Re β > 0]
MI 32
526
4.142
Trigonometric Functions
∞
1.
xe−βx (sinh x + sin x) dx =
2
0
2.
3.
∞
1
2
4.142
π
1
cosh
β3
4β
[Re β > 0]
ME 24
π
1
[Re β > 0]
sinh
3
β
4β
0
∞
1
1
1 π
1
2 −βx2
+
sinh
x e
(cosh x + cos x) dx =
cosh
3
2
β
4β
2β
4β
0
∞
4.
xe−βx (sinh x − sin x) dx =
2
1
2
x2 e−βx (cosh x − cos x) dx =
2
0
1
2
ME 24
[Re β > 0]
π
β3
sinh
ME 24
1
1
1
+
cosh
4β
2β
4β
[Re β > 0]
4.143
∞
−βx2
xe
1.
0
∞
2.
1
(cosh x sin x + sinh x cos x) dx =
2β
xe−βx (cosh x sin x − sinh x cos x) dx =
2
0
∞
4.144
e−x sinh x2 cos ax
2
0
4.145
∞
1.
−βx2
xe
0
∞
2.
dx
=
x2
1
2β
1
π
cos
β
2β
[Re β > 0]
xe−βx sinh (2ax sin t) cos (2ax cos t) dx =
2
0
a
2
[Re β > 0]
a
√
8
[a > 0]
∞
1.8
e−βx sinh ax sin bx dx =
2
0
2.8
0
∞
e−βx cosh ax cos bx dx =
2
1
2
1
2
π
exp
β
a2 − b 2
4β
π
exp
β
a −b
4β
ET I 35(44)
[Re β > 0]
2
BI (363)(5)
2
a
π
a
sin 2t
exp − cos 2t sin t −
β3
β
β
sin
ab
2β
[Re β > 0]
2
MI 32
a2
π
a2
cos
2t
cos
t
−
sin 2t
exp
−
β3
β
β
[Re β > 0]
4.14610
MI 32
1
π
sin
β
2β
π − a2
πa
e 8 −
1−Φ
2
4
a
cosh (2ax sin t) sin (2ax cos t) dx =
2
ME 24
2
cos
ab
2β
[Re β > 0]
BI (363)(6)
4.212
Logarithmic functions
∞
3.
−βx2
xe
0
∞
4.
a
cosh ax sin ax dx =
4β
xe−βx sinh ax cos ax dx =
2
0
∞
5.8
a
4β
x2 e−βx cosh ax sin ax dx =
2
0
∞
6.8
x2 e−βx cosh ax cos ax dx =
2
0
1
4
1
4
π
β
cos
527
a2
a2
+ sin
2β
2β
[Re β > 0]
π
β
2
cos
2
a
a
− sin
2β
2β
[Re β > 0]
π
β3
sin
a2
a2
a2
+
cos
2β
β
2β
[Re β > 0]
π
β3
2
cos
2
a
a2
a
−
sin
2β
β
2β
[Re β > 0]
4.2–4.4 Logarithmic Functions
4.21 Logarithmic functions
4.211 1.
∞
dx
= −∞
1
e ln
x
u
dx
= li u
0 ln x
2.
4.212
1
1.7
0
1
2.
0
3.
1
7
1
4.
5.
1
8
1
6.
7.
1
BI (31)(4)
dx
= −ea Ei(−a)
a − ln x
[a > 0]
BI (31)(5)
[a ≥ 0]
BI (31)(14)
[a > 0]
BI (31)(16)
dx
1
= − + e−a Ei(a)
a
(a + ln x)
dx
e
1
+ ea Ei(−a)
a
2
=
2
= 1 + (1 − a)e−a Ei(a)
[a ≥ 0]
BI (31)(15)
2
= 1 + (1 + a)ea Ei(−a)
[a > 0]
BI (31)(17)
2
=
ln x dx
ln x dx
(a − ln x)
0
[a > 0]
(a + ln x)
0
dx
= e−a Ei(a)
a + ln x
(a − ln x)
0
FI III 653, FI II 606
2
0
BI (33)(9)
ln x dx
(1 + ln x)
e
−1
2
BI (33)(10)
528
Logarithmic Functions
8.
1
7
0
4.213
n−1
dx
1
1
e−a Ei(a) −
(n − k − 1)!ak−n
n =
(n − 1)!
(n − 1)!
(a + ln x)
k=1
[a ≥ 0]
1
9.
0
BI (31))(22)
n−1
dx
(−1)n a
(−1)n−1 e
=
Ei(−a)
+
(n − k − 1)!(−a)k−n
n
(n − 1)!
(n − 1)!
(a − ln x)
k=1
[a > 0,
n odd]
BI (31)(23)
m
(ln x)
dx, it is convenient to make the substitution x = e−t .
n l
[an + (ln x) ]
Results 4.212 3, 4.212 5, and 4.212 8 [for n > 1] and 4.213 6, 4.213 8 below are divergent but may
be considered to be valid if deﬁned as follows:
a
a
n−1 f (z) dz
f (z)
d
1
dz
PV
n =
(n − 1)! dz0
0 (z − z0 )
0 z − z0
where a > z0 > 0, n = 1, 2, 3, . . . and PV indicates the Cauchy principal value.
4.213
1
dx
1
[a > 0]
1.
BI (31)(6)
2 = a [ci(a) sin a − si(a) cos a]
2
0 a + (ln x)
1
dx
1 −a
2.7
Ei(a) − ea Ei(−a)
[a > 0] , (cf. 4.212 1 and 2)
2 = 2a e
2
0 a − (ln x)
In integrals of the form
BI (31)(8)
1
3.
0
ln x dx
a2 + (ln x)
1
4.7
= ci(a) cos a + si(a) sin a
2
=−
ln x dx
a2 − (ln x)
0
2
1 −a
e Ei(a) + ea Ei(−a)
2
[a > 0]
[a > 0] ,
BI (31)(7)
(cf. 4.212 1 and 2)
BI (31)(9)
1
+
5.
0
dx
2
a2 + (ln x)
,2 =
1
1
[ci(a) sin a − si(a) cos a] − 2 [ci(a) cos a + si(a) sin a]
3
2a
2a
[a > 0]
1
6.8
+
0
1
+
7.
0
dx
2
a2 − (ln x)
ln x dx
2
a2 + (ln x)
,2
,2 =
is divergent
1
1
[ci(a) sin a − si(a) cos a] − 2
2a
2a
[a > 0]
8.8
0
1
+
ln x dx
2
a2 − (ln x)
,2
LI (31)(18)
is divergent
BI (31)(19)
4.221
4.214
Logarithms of more complicated arguments
1
1.
0
dx
a4
− (ln x)
4
=−
1 a
e Ei(−a) − e−a Ei(a) − 2 ci(a) sin a + 2 si(a) cos a
4a3
[a > 0]
1
2.
ln x dx
a4
0
− (ln x)
4
=−
1
3.
2
(ln x) dx
a4 − (ln x)
0
4
=−
1
3
(ln x) dx
a4
0
− (ln x)
4
=−
1
1.
ln
0
2.
3.
4.
4.216
1.
2.∗
1
x
BI (31)(13)
μ−1
dx = Γ(μ)
[Re μ > 0]
FI II 778
[Re μ < 1]
BI (31)(1)
1
dx
π
cosec μπ
μ =
Γ(μ)
1
0
ln
x
√
1
π
1
ln dx =
x
2
0
1
√
dx
= π
1
0
ln
x
BI (31)(12)
1 a
e Ei(−a) + e−a Ei(a) + 2 ci(a) cos a + 2 si(a) sin a
4
[a > 0]
4.215
BI (31)(11)
1 a
e Ei(−a) − e−a Ei(a) + 2 ci(a) sin a − 2 si(a) cos a
4a
[a > 0]
4.7
BI (31)(10)
1 a
e Ei(−a) + e−a Ei(a) − 2 ci(a) cos a − 2 si(a) sin a
2
4a
[a > 0]
529
BI (32)(1)
BI (32)(3)
1/e
dx
<
= K 0 (1)
2
0
(ln x) − 1
√
1/e
π
dx
√
=
e
− ln x − 1
0
GW (32)(2)
4.22 Logarithms of more complicated arguments
4.221
1
1.
ln x ln(1 − x) dx = 2 −
π2
6
BI (30)(7)
ln x ln(1 + x) dx = 2 −
π2
− 2 ln 2
12
BI (30)(8)
0
2.
0
1
530
Logarithmic Functions
∞
1
3.
ln
0
4.222
ln(1 + k)
1 − ax dx
=−
ak
1 − a ln x
k
∞
ln
0
a2 + x2
dx = (a − b)π
b2 + x2
∞
2.
ln x ln
0
a2 + x2
aa
dx
=
π(b
−
a)
+
π
ln
b2 + x2
bb
∞
3.
ln x ln 1 +
0
∞
4.
2 2
ln 1 + a x
0
∞
0
0
∞
7.
∞
a2
x2
ln a2 +
0
8.∗
dx = πb (ln b − 1)
GW (322)(20)
[a > 0,
b > 0]
BI (33)(1)
[b > 0]
b2
ln 1 + 2
x
b > 0]
1 + ab
ln(1 + ab) − b
dx = 2π
a
ln 1 +
1
x2
b
x2
0
∞
1.
0
∞
2.
0
∞
3.
0
4.224
b
%
[a > 0,
[a > 0,
b > 0]
b!
(−1)b−m−1 1/a
1
e Ei −
b−m
(b
−
m)!
a
a
m=0
+
b−m
k=1
BI (33)(7)
(k − 1)!
(−a)b−m−k
&
an integer]
π2
ln 1 − e−x dx = −
6
BI (256)(11)
t2
π2
−
ln 1 + 2e−x cos t + e−2x dx =
6
2
u
π
π
−u −L
2
2
π/4
ln sin x dx = −
0
BI (33)(5)
BI (256)(10)
0
2.
BI (33)(4)
π2
ln 1 + e−x dx =
12
ln sin x dx = L
1.
b > 0]
b > 0]
1 + ab
ln(1 + ab) − b ln b
dx = 2π
a
[b > 0,
4.223
BI (33)(3)
dx = 2π [(a + b) ln(a + b) − a ln a − b ln b]
2
ln (1 + ax) xb e−x dx =
b > 0]
dx = 2π [(a + b) ln(a + b) − a ln a − b]
b2
x2
ln 1 +
BI (33)(2)
[a > 0,
ln 1 +
b2
x2
b2
ln a2 + x2 ln 1 + 2
x
∞
6.
BI (31)(3)
[a > 0,
[a > 0,
5.
[a < 1]
k=1
1.
4.222
1
π
ln 2 − G
4
2
[|t| < π]
BI (256)(18)
LO III 186(15)
BI (285)(1)
4.225
Logarithms of more complicated arguments
π/2
3.
ln sin x dx =
0
1
2
π
ln sin x dx = −
0
531
π
ln 2
2
FI II 629,643
u
ln cos x dx = − L(u)
4.
LO III 184(10)
0
π/4
ln cos x dx = −
5.
0
1
π
ln 2 + G
4
2
BI (286)(1)
π/2
π
ln 2
2
0
π/2
π
π2
2
2
(ln sin x) dx =
(ln 2) +
2
12
0
π/2
π
π2
2
2
(ln cos x) dx =
(ln 2) +
2
12
0
√
π
a + a2 − b 2
ln (a + b cos x) dx = π ln
2
0
π
ln (1 ± sin x) dx = −π ln 2 ± 4G
ln cos x dx = −
6.
7.
8.
9.8
10.
BI 306(1)
BI (305)(19)
BI (306)(14)
[a ≥ |b| > 0]
GW (322)(15)
GW (322)(16a)
0
∞ k k
b
(−1)n+1
π a
[a > 0]
11.
ln (1 + a sin x) dx = ln + 2G + 2
2 2
k n=1 2n − 1
0
k=1
π
= − ln 2 + 2G
[a = 1]
2
√
π
2
1 + 1 − a2
12.
a ≤1
ln (1 + a cos x) dx = π ln
2
0
⎧
√
2
⎪
π
⎨2π ln 1 + 1 − a
for a2 ≤ 1
2
2
12 (1)
ln (1 + a cos x) dx =
⎪
0
⎩ π a2
for a2 ≥ 1
2 ln 4
π/2
∞
2k+1
2
2a
22k (k!)
13.
ln 1 + 2a sin x + a2 dx =
(2k + 1) · (2k + 1)!! 1 + a2
0
k=0
2
a ≤1
nπ
14.11
ln a2 − 2ab cos x + b2 dx = 2nπ ln [max (|a|, |b|)]
π/2
7
b=
1−a
1+a
BI (330)(1)
BI (308)(24)
0
15.8
[ab > 0]
nπ
ln 1 − 2a cos x + a2 dx = 0
0
= nπ ln a2
4.225
π/4
ln (cos x − sin x) dx = −
1.
0
1
π
ln 2 − G
8
2
a2 ≤ 1
a2 ≥ 1
FI II 142, 163, 688
GW (322)(9b)
532
Logarithmic Functions
π/4
2.
ln (cos x + sin x) dx =
0
1
2
π/2
ln (cos x + sin x) dx = −
0
2π
3.
ln (1 + a sin x + b cos x) dx = 2π ln
1+
0
2π
4.
4.226
√
1 − a2 − b 2
2
1
π
ln 2 + G
8
2
a2 + b 2 < 1
ln 1 + a2 + b2 + 2a sin x + 2b cos x dx = 0
= 2π ln a2 + b2
0
GW (322)(9a)
BI (332)(2)
a2 + b 2 ≤ 1
a2 + b 2 ≥ 1
BI (322)(3)
4.226
π/2
1.
2
ln a2 − sin2 x dx = −2π ln 2
0
= 2π ln
a+
√
a2 − 1
= 2π (arccosh a − ln 2)
2
a2 ≤ 1
[a > 1]
FI II 644, 687
π/2
2.
0
3.
4.
u
u
π/2
1 π ln 1 + a sin2 x dx =
ln 1 + a sin2 x dx =
ln 1 + a cos2 x dx
2 0
0
√
1 π 1+ 1+a
2
=
ln 1 + a cos x dx = π ln
2 0
2
BI (308)(15), GW(322)(12)
[a ≥ −1]
α
ln 1 − sin2 α sin2 x dx = (π − 2θ) ln cot + 2u ln
2
1
π
sin α − ln 2
2
2
0
π
− 2u
+ L(θ + u) − L(θ − u) + L
2
+
π
π,
cot θ = cos α tan u; −π ≤ α ≤ π, − ≤ u ≤
LO III 287
2
2
% &
π/2
2
α
β
β
1
2
2
2
2
2 α
+ cos4 + sin
cos2
ln 1 − cos x sin α − sin β sin x dx = π ln
cos
2
2
2
2
2
0
[α > β > 0]
2
ln 1 −
5.
0
sin x
sin2 α
dx = −u ln sin2 α − L
LO III 283
π
π
−α+u +L
−α−u
2
,
+ π 2
π
− ≤ u ≤ , |sin u| ≤ |sin α|
2
2
LO III 287
6.
0
π/2
1
ln a2 cos2 x + b2 sin2 x dx =
2
0
π
a+b
ln a2 cos2 x + b2 sin2 x dx = π ln
2
[a > 0,
b > 0]
GW (322)(13)
4.227
Logarithms of more complicated arguments
π/2
7.
0
4.227
t
1 + sin
π−t
1 + sin t cos2 x
2
dx = π ln
= π ln cot
ln
2
t
1 − sin t cos x
4
cos
2
+
π,
|t| <
2
u
π
π
−u −L
2
2
ln tan x dx = L(u) + L
1.
0
π/4
2.
0
BI (286)(11)
4
ln (a tan x) dx =
0
4.
LO III 283
LO III 186(16)
π
ln tan x dx = − π2 ln tan x dx = −G
π/2
3.
533
π/4
7
π
ln a
2
n
[a > 0]
n
(ln tan x) dx = n!(−1)
0
=
1
2
π
2
∞
k=0
n+1
BI (307)(2)
(−1)k
(2k + 1)n+1
|En |
[n even]
BI (286)(21)
π/2
5.7
(ln tan x)
2n
dx = 2(2n)!
0
∞
k=0
π/2
6.
(ln tan x)
2n+1
k
π
(−1)
=
2n+1
(2k + 1)
2
2n+1
|E2n |
BI (307)(15)
dx = 0
BI (307)(14)
π3
16
BI (286)(16)
5 5
π
64
BI (286)(19)
0
π/4
7.
2
(ln tan x) dx =
0
π/4
8.
4
(ln tan x) dx =
0
π/4
9.
ln (1 + tan x) dx =
π
ln 2
8
BI (287)(1)
ln (1 + tan x) dx =
π
ln 2 + G
4
BI (308)(9)
ln (1 − tan x) dx =
π
ln 2 − G
8
BI (287)(2)
0
π/2
10.
0
π/4
11.
0
π/2
2
(ln (1 − tan x)) dx =
12.11
0
π/4
13.
14.
0
BI (308)(10)
ln (1 + cot x) dx =
π
ln 2 + G
8
BI (287)(3)
ln (cot x − 1) dx =
π
ln 2
8
BI (287)(4)
0
π
ln 2 − 2G
2
π/4
534
Logarithmic Functions
π/4
15.
ln (tan x + cot x) dx =
0
π/4
1
2
2
(ln (cot x − tan x)) dx =
16.11
0
4.228
π/2
ln (tan x + cot x) dx =
0
1
2
π/2
π
ln 2
2
2
(ln (cot x − tan x)) dx =
0
BI (287)(5), BI (308)(11)
π
ln 2
2
BI (287)(6), BI (308)(12)
π/2
17.
0
1
ln a2 + b2 tan2 x dx =
2
π
ln a2 + b2 tan2 x dx = π ln(a + b)
0
[a > 0,
4.228
π/2
1.
ln sin t sin x +
0
u
2.
ln cos x +
1 − cos2 t sin2 x dx =
cos2 x − cos2 t dx = −
0
π
ln 2 − 2 L
2
t
2
b > 0]
− 2L
GW (322)(17)
π−t
2
LO III 290
1
1
π
− t − ϕ ln cos t + L(u + ϕ) − L(u − ϕ) − L(ϕ)
2
2
2
sin u
π
cos ϕ =
0≤u≤t≤
sin t
2
LO III 290
3.
4.
π
− t ln cos t
ln cos x + cos2 x − cos2 t dx = −
2
0
&
%
u
t
sin u + sin t cos x sin2 u − sin2 x
t
2
2
sin u + 1
ln
dx = π ln tan sin u + tan
2
2
0
sin u − sin t cos x sin2 u − sin2 x
t
[t > 0, u > 0]
√
∞
π/4 √
π
(−1)k
ln cot x dx =
2
(2k + 1)3
0
k=0
π/4
∞
√ dx
(−1)k
√
√
= π
2k + 1
ln cot x
0
k=0
π/4
√
√
√
√
1
1 π/2
π
ln
tan x + cot x dx =
ln
tan x + cot x dx = ln 2 + G
2
8
2
0
0
LO III 285
LO III 283
5.
6.
7.
BI (297)(9)
BI (304)(24)
BI (287)(7), BI (308)(22)
π/4
ln2
8.
0
√
√
1
cot x − tan x dx =
2
π/2
ln2
0
√
√
π
cot x − tan x dx = ln 2 − G
4
BI (287)(8), BI (308)(23)
4.229
1
1.
ln ln
0
2.11
PV
0
1
1
x
dx = −C
dx
1
ln ln
x
FI II 807
= PV
0
∞ −u
e
du ≈ −0.154479
ln u
BI (31)(2)
4.231
Logarithms and rational functions
1
3.
ln ln
1
x
dx
ln ln
1
x
ln
0
1
4.11
0
5.7
1
ln
x
1
x
535
√
= − (C + 2 ln 2) π
BI (32)(4)
μ−1
dx = ψ(μ) Γ(μ)
[Re μ > 0]
BI (30)(10)
If the integrand contains (ln ln x1 ), it is convenient to make the substitution ln x1 = u so that
x = e−u .
1
ln (a + ln x) dx = ln a − e−a Ei(a)
[a > 0]
BI (30)(5)
0
1
ln (a − ln x) dx = ln a − ea Ei(−a)
6.
0
π/2
7.
π/4
π
ln ln tan x dx = ln
2
[a > 0]
BI (30)(6)
Γ 34 √
2π
Γ 14
BI (308)(28)
4.23 Combinations of logarithms and rational functions
4.231
1
1.
0
1
2.
0
1
3.
0
1
4.
0
π2
ln x
dx = −
1+x
12
FI II 483a
π2
ln x
dx = −
1−x
6
FI II 714
π2
x ln x
dx = 1 −
1−x
6
BI (108)(7)
π2
1+x
ln x dx = 1 −
1−x
3
BI (108)(9)
∞
5.11
0
6.
0
7.
1
∞
0
∞
9.
0
0
√ Γ n − 12
π
1
dx
a
−C−ψ n−
ln x 2
2 ln
n =
4(n − 1)!a2n−1 b
2b
2
(a + b2 x2 )
ln x dx
a
π
ln
=
2
2
+b x
2ab b
[ab > 0]
ln px
π
ln pq
dx =
q 2 + x2
2q
[p > 0,
ln x dx
π2
=
−
a2 − b2 x2
4ab
[ab > 0]
a2
0
10.
BI (139)(1)
BI (111)(1)
[a > 0,
∞
8.
[0 < a]
ln x
dx = − ln 2
(1 + x)2
7
ln x dx
ln a
=
2
(x + a)
a
∞
b > 0]
LI (139)(3)
BI (135)(6)
q > 0]
BI (135)(4)
536
Logarithmic Functions
11.
12.
13.
14.
15.
16.
a
ln x dx
π ln a G
−
=
2 + a2
x
4a
a
0
1
∞
ln x
ln x
dx = −
dx = −G
2
1
+
x
1
+ x2
0
1
1
ln x dx
π2
=
−
2
8
0 1−x
1
x ln x
π2
dx
=
−
2
48
0 1+x
1
x ln x
π2
dx = −
2
24
0 1−x
1
n
1 − x2n+2
(n + 1)π 2 n − k + 1
+
ln x
dx
=
−
2
8
(2k − 1)2
(1 − x2 )
0
ln x
0
BI (111)(2)
1 − xn+1
(n + 1)π 2 n − k + 1
+
dx
=
−
(1 − x)2
6
k2
BI (111)(3)
π2
x ln x
dx = −1 +
1+x
2
1
π2
(1 − x) ln x
dx = 1 −
1+x
6
υ
ln uυ
(u + υ)2
ln x dx
=
ln
(x + u)(x + υ)
2(υ − u)
4uυ
1.
u
BI (111)(5)
n−k+1
1 + (−1)n xn+1
(n + 1)π 2 −
dx
=
−
(−1)k
2
(1 + x)
12
k2
1
0
4.232
GW (324)(7b)
k=1
0
20.∗
BI (108)(11)
n
1
ln x
FI II 482, 614
k=1
18.
GW (324)(7b)
n
1
0
19.∗
[a > 0]
k=1
17.
4.232
∞
2.
0
2
BI (145)(32)
2
ln x dx
(ln β) − (ln γ)
=
(x + β)(x + γ)
2(β − γ)
[|arg β| < π,
|arg γ| < π]
ET II 218(24)
∞
3.
0
4.233
1.
3
2.3
3.11
4.3
2
π 2 + (ln a)
ln x dx
=
x+ax−1
2(a + 1)
[a > 0]
1
ln x dx
2 2π 2
−ψ
=
= −0.7813024129 . . .
2
9 3
3
0 1+x+x
1
1
ln x dx
1 2π 2
−
ψ
=
= −1.17195361934 . . .
2
3 3
3
0 1−x+x
1
1
x ln x dx
1 7π 2
−
ψ
=
−
= −0.15766014917 . . .
2
9 6
3
0 1+x+x
1
1
x ln x dx
1 5π 2
−ψ
=
= −0.3118211319 . . .
2
6 6
3
0 1−x+x
BI (140)(10)
1
LI (113)(1)
LI (113)(2)
LI (113)(2)
LI (113)(4)
4.236
Logarithms and rational functions
∞
5.
0
4.234
∞
1.11
2
1
2.
x ln x dx
∞
3.
∞
4.
1
5.
0
∞
0
∞
7.
0
1.
2.
3.11
4.
4.236
2
ln x dx = −
BI (142)(2)a
π
2
BI (142)(1)a
BI (112)(21)
bπ
a
ln x dx
=
ln
(a2 + b2 x2 ) (1 + x2 )
2a (b2 − a2 ) b
ln x
dx
π
·
=
x2 + a2 1 + b2 x2
2 (1 − a2 b2 )
∞
(a2
0
4.235
BI (111)(4)
[ab > 0]
x2 ln x dx
aπ
b
=
ln
+ b2 x2 ) (1 + x2 )
2b (b2 − a2 ) a
[ab > 0]
π
(1 − x)xn−2
π2
[n > 1]
dx = − 2 tan2
2n
1
−
x
4n
2n
0
∞
1 − x2 xm−1
π 2 sin m+1
π sin nπ
n
ln x
dx = − 2 2 mπ 2 m+2 1 − x2n
4n sin 2n sin
0
2n π
∞
1 − x2 xn−3
π
π2
[n > 2]
ln x
dx = − 2 tan2
2n
1
−
x
4n
n
0
1
xm−1 + xn−m−1
π2
ln x
dx
=
−
[n > m]
1 − xn
n2 sin2 m
0
nπ
1
1.
0
2.
0
b > 0]
LI (140)(12)
LI (140)(12), BI (317)(15)a
∞
ln x
BI (317)(16)a
1
ln a + b ln b
a
[a > 0,
8.
GW (324)(13c)
1
= − ln 2
4
π2
x2 ln x dx
√ =− 2
4
(1 − x ) (1 + x )
16 2 + 2
6.
0 < t < π]
BI (144)(18)a
ln x dx = 0
1 − x2
[a > 0,
G π
−
2
8
=
2
(1 + x2 )
0
1 + x2
(1 − x2 )
0
2
x2 )
(1 +
0
ln x dx
(1 + x2 )
1
ln x dx
t ln a
=
x2 + 2xa cos t + a2
a sin t
537
1
1 + (p − 1) ln x
x ln x
+
1−x
(1 − x)2
1
x ln x
+
1 − x (1 − x)2
dx =
LI (135)(12)
BI (135)(11)
BI (108)(15)
xp−1 dx = −1 + ψ (p)
[p > 0]
BI (135)(10)
π2
−1
6
BI (111)(6)a, GW (326)(13)
GW (326)(13a)
538
Logarithmic Functions
4.241
4.24 Combinations of logarithms and algebraic functions
4.241
1.
2.
3.
2n
x2n ln x
(2n − 1)!! π (−1)k−1
√
·
− ln 2
dx =
(2n)!!
2
k
1 − x2
0
k=1
1 2n+1
2n+1
(−1)k
x
ln x
(2n)!!
√
dx =
ln 2 +
(2n + 1)!!
k
1 − x2
0
k=1
1
2n
1
(2n − 1)!! π (−1)k−1
2n
2
·
−
− ln 2
x
1 − x ln x dx =
(2n + 2)!! 2
k
2n + 2
0
1
BI (118)(5)a
BI (118)(5)a
k=1
1
x2n+1
4.
1 − x2 ln x dx =
0
(2n)!!
(2n + 3)!!
ln 2 +
2n+1
k=1
1
(−1)k
−
k
2n + 3
LI (117)(4), GW (324)(53a)
BI (117)(5), GW (324)(53b)
5.
<
1
(2n − 1)!!
2n−1
π [ψ(n + 1) + C + ln 4]
ln x · (1 − x2 )
dx = −
4 · (2n)!!
0
.
6.
0
7.
8.
9.
10.
1
2 √ln x dx = − π ln 2 − 1 G
4
2
1 − x2
BI (145)(1)
1
ln x dx
π
√
= − ln 2
2
2
1
−
x
0
∞
ln x dx
√
= 1 − ln 2
2
x2 − 1
1 x
1
π π
1 − x2 ln x dx = − − ln 2
8
4
0
1 x 1 − x2 ln x dx = 13 ln 2 − 49
0
1
11.
0
4.242
BI (117)(3)
0
2.
0
2π
=−
Γ
8
x (1 − x2 )
∞
1.
b
√
ln x dx
ln x dx
1
4
1
K
=
2
2
2
2
2a
(a + x ) (x + b )
+
x2 ) (b2
BI (144)(17)
BI (117)(1), GW (324)(53c)
BI (117)(2)
2
GW (324)(54a)
√
a2 − b 2
ln ab
a
ln x dx
(a2
FI II 614, 643
−
x2 )
1
= √
K
2
2 a + b2
[a > b > 0]
b
√
2
a + b2
ln ab −
[a > 0,
π
K
2
BY (800.04)
a
√
2
a + b2
b > 0]
BY (800.02)
4.247
Logarithms and algebraic functions
∞
3.
b
b
4.
ln x dx
1
= √
2 a2 + b 2
(x2 + a2 ) (x2 − b2 )
0
%
ln x dx
(a2 − x2 ) (b2 − x2 )
=
1
K
2a
b
a
K
a
√
2
a + b2
π
ln ab + K
2
[a > 0, b > 0]
√
&
a2 − b 2
π
ln ab − K
2
a
[a > b > 0]
√
a
ln x dx
a2 − b 2
1
K
ln ab
=
2a
a
(a2 − x2 ) (x2 − b2 )
b
%
√
&
∞
b
ln x dx
a2 − b 2
1
π
K
=
ln ab + K
2a
a
2
a
(x2 − a2 ) (x2 − b2 )
a
539
b
√
2
a + b2
BY (800.06)
BY (800.01)
5.
6.
[a > b > 0]
1
4.243
0
4.244
1
1.
0
1
2.
0
1
3.
0
4.245
1.
2.
x ln x
π
√
dx = − ln 2
4
8
1−x
ln x dx
<
2
3
x (1 − x2 )
1
3
3
GW (324)(54b)
π
ln 3 + √
3 3
x ln x dx
π
<
= √
2
3
3 3
(1 − x3 )
BI (118)(7)
π
√ − ln 3
3 3
BI (118)(8)
2n
x4n+1 ln x
(2n − 1)!! π (−1)k−1
√
·
− ln 2
dx =
(2n)!!
8
k
1 − x4
0
k=1
1 4n+3
2n+1
(−1)k
x
ln x
(2n)!!
√
ln 2 +
dx =
4 · (2n + 1)!!
k
1 − x4
0
BY (800.05)
GW (324)(56b)
1
=− Γ
8
ln x dx
π
√
=− √
3
3 3
1 − x3
BY (800.03)
1
GW (324)(56a)
GW (324)(56c)
k=1
1
4.246
0
1−x
2
n−
%
&
1
n
1
(2n − 1)!! π
2 ln x dx = −
·
2 ln 2 +
(2n)!!
4
k
GW (324)(55)
k=1
4.247
1
1.6
0
2.6
0
1
1 1
,
2n 2n
π
8n2 sin
2n
1 1
,
πB
ln x dx
2n 2n
=
−
π
n
xn−1 (1 − x2 )
8 sin
2n
ln x
√
dx = −
n
1 − x2n
πB
[n > 1]
GW (324)(54c)a
GW (324)(54)
540
Logarithmic Functions
4.251
4.25 Combinations of logarithms and powers
4.251
∞
1.
0
0
1
3.10
0
1
4.
0
ln x
π
dx = πaμ−1 cot μπ ln a −
a−x
sin2 μπ
xμ−1 ln x
dx = − ψ (μ) = − ζ(2, μ)
1−x
[Re μ > 0]
BI (108)(8)
1
BI (108)(4)
2n−1
(−1)k
π2
x2n−1
dx =
+
ln x
1+x
12
k2
BI (108)(5)
k=1
∞
μ−1
π
xμ−1 ln x
dx =
β
ln β − γ μ−1 ln γ − π cot μπ β μ−1 − γ μ−1
(x + β)(x + γ)
(γ − β) sin μπ
[|arg β| < π, |arg γ| < π, 0 < Re μ < 2, μ = 1] BI (140)(9)a, ET 314(6)
∞
π
xμ−1 ln x dx
π − β μ−1 (sin μπ ln β − π cos μπ)
=
2
(x + β)(x − 1)
(β + 1) sin μπ
[|arg β| < π, 0 < Re μ < 2,
2.
0
μ = 1]
BI (140)(11)
∞
3.
0
π 2 (−1)k−1
x2n
dx = −
+
1+x
12
k2
2n
0
ET I 314(5)
GW (324)(6), ET I 314(3)
1.
0 < Re μ < 1]
k=1
0
[a > 0,
[Re μ > 0]
ln x
11
4.252
x
xμ−1 ln x
dx = β (μ)
x+1
0
6.
0 < Re μ < 1]
μ−1
1
5.11
[|arg β| < π,
BI (135)(1)
∞
2.
πβ μ−1
xμ−1 ln x
dx =
(ln β − π cot μπ)
β+x
sin μπ
∞
4.6
0
xp−1 ln x
pπ
π2
cosec2
dx
=
−
2
1−x
4
2
xμ−1 ln x
(1 − μ)aμ−2 π
dx =
2
(x + a)
sin μπ
[0 0,
Re ν > 0,
r > 0]
GW (324)(3b)a, BI (107)(5)a
2.
0
1
xp−1
π
ln x dx = − cosec pπ
p+1
(1 − x)
p
[0 < p < 1]
bi (319)(10)a
4.255
Logarithms and powers
∞
3.
u
(x − u)μ−1 ln x dx
= uμ−λ B(λ − μ, μ) [ln u + ψ(λ) − ψ(λ − μ)]
xλ
[0 < Re μ < Re λ]
∞
4.11
x
a2 + x2
ln x
0
∞
5.
p
p p
ln a
dx
= pB
,
x
2a
2 2
(x − 1)p−1 ln x dx =
1
∞
6.7
ln x
0
541
ln x
0
π
cosec πp
p
dx
n+ 12
(a + x)
=
2
(2n − 1)a
p > 0]
n− 12
ln a + 2 ln 2 − 2
[Re μ > 0,
n−1
k=1
1
2k − 1
1
1.
0
xp−1 ln x
1
dx = − 2 ψ 1 − xq
q
∞
2.
0
∞
3.
4.
1
3
0
5.
0
1
6.
0
4.255
1
1.
0
2.
0
3.
ln x dx
=
− 1 xp
1
p
q
pπ
cos
xp−1 ln x
π2
q
dx = − 2
1 + xq
q sin2 pπ
q
1 − x2 xp−2
ln x
dx = −
1 + x2p
π
2p
2
1 + x2 xp−2
ln x
dx = −
1 − x2p
π
2p
2
∞
pπ
1 − xp
π2
tan2
dx =
2
1−x
4
2
π
sin 2p
cos2
π
2p
sec2
π
2p
a = 0,
|arg a| < π]
NT 68(7)
n = 1, 2, . . .]
q > 0]
BI (142)(5)
GW (324)(5)
[0 1]
BI (140)(2)
[p > 0,
q > 0]
[0 0,
xp−1 ln x
π2
dx = −
pπ
q
1−x
q 2 sin2
q
xq
0
p
q
BI (289)(12)a
[|arg a| < π,
4.254
BI (140)(6)
[−1 0,
ET II 203(18)
GW (324)(7)
BI (135)(7)
[q > 0]
BI (108)(12)
[p > 1]
BI (108)(13)
[p > 1]
BI (108)(14)
[p < 1]
BI (140)(3)
542
Logarithmic Functions
1
4.256
ln
0
xμ−1 dx
1
1
μ m
<
,
= 2B
x n
n
n n
n−m
(1 − xn )
ψ
μ+m
n
4.256
−ψ
μ
n
[Re μ > 0]
4.257
∞
1.
0
LI (118)(12)
+
,
x
γ
ν
ν
ν
dx
π
γ
ln
+
π
(β
−
γ
)
cot
νπ
β
β
=
(x + β)(x + γ)
sin νπ(γ − β)
xν ln
[|arg β| < π,
ln
0
∞
p
x
q
q 2p
dx
=0
x
x
+ x2p
xp
2p
q + x2p
x
q
[q > 0]
BI (140)(4)a
[q > 0]
BI (140)(4)a
r
dx
=0
2 + x2
q
0
,
+
2
2
∞
+
(ln
a)
ln a
4π
dx
x
=
ln x ln
a (x − 1)(x − a)
6(a − 1)
0
ln
4.
| Re ν| < 1]
ET II 219(30)
∞
2.
3.
|arg γ|< π,
[a > 0]
(for a = 1 see 4.261 5)
BI (141)(5)
∞
5.
ln x ln
0
xp dx
π 2 [(ap + 1) ln a − 2π (ap − 1) cot pπ]
x
=
a (x − 1)(x − a)
(a − 1) sin2 pπ
2
p < 1, a > 0
BI (141)(6)
4.26–4.27 Combinations involving powers of the logarithm and other powers
4.261
1.
7
2.
3.
4.
5.
6.
7.
8.11
t π 2 − t2
dx
[0 ≤ t ≤ π]
(ln x)
=
1 + 2x cos t + x2
6 sin t
0
1
2
2
1 ∞ (ln x) dx
10π 3
(ln x) dx
√
=
=
2
2 0 x2 − x + 1
81 3
0 x −x+1
1
2
2
1 ∞ (ln x) dx
8π 3
(ln x) dx
√
=
=
2
2 0 x2 + x + 1
81 3
0 x +x+1
,
+
2
2
∞
+
(ln
a)
ln a
π
dx
2
=
[a > 0]
(ln x)
(x − 1)(x + a)
3(1 + a)
0
∞
dx
2
2
(ln x)
= π2
2
(1 − x)
3
0
1
3
dx
π
2
(ln x)
=
2
1
+
x
16
0
√
1
2
2
1 ∞
3 2 3
2 1+x
2 1+x
π
(ln x)
dx
=
(ln
x)
dx
=
1 + x4
2 0
1 + x4
64
0
√
1
8 3π 3 + 351 ζ(3)
2 1−x
(ln x)
dx =
1 − x6
486
0
1
2
BI (113)(7)
GW (324)(16c)
GW (324)(16b)
BI (141)(1)
BI (139)(4)
BI (109)(3)
BI (109)(5), BI (135)(13)
4.261
Powers and logarithms
dx
π
π2
2
=
(ln 2) +
2
12
1 − x2
0
∞
2
3
μ−1
π 2 − sin μπ
2 x
dx =
(ln x)
[0 < Re μ < 1]
1+x
sin3 μπ
0
1
∞
n
n
(−1)n+k
(−1)k
3
2 x dx
n
=2
ζ(3) + 2
(ln x)
= (−1)
1+x
(k + 1)3
2
k3
0
9.
10.
11.7
543
1
2
(ln x) √
k=n
2
(ln x)
0
1
2
(ln x)
0
1
xn dx
1
=2
= 2 ζ(3) −
3
1−x
(k + 1)
k3
n
k=n
13.11
ET I 315(10)
k=1
∞
1
12.7
BI (118)(13)
[n = 0, 1, . . .]
BI (109)(1)
[n = 0, 1, . . .]
BI (109)(2)
k=1
∞
n
k=n
k=1
x2n dx
1
1
7
ζ(3)
−
2
=
2
=
1 − x2
(2k + 1)3
4
(2k − 1)3
[n = 0, 1, . . .]
∞
14.
(ln x)
0
2
x2
BI (109)(4)
xp−1 dx
π sin(1 − p)t 2
=
π − t2 + 2π cot pπ [π cot pπ + t cot(1 − p)t]
+ 2x cos t + 1
sin t sin pπ
[0 < t < π, 0 < p < 2, p = 1]
⎧
&2 ⎫
% 2n
2n
⎨ π2 ⎬
1
2n
k
k
x
dx
(−1)
(−1)
(2n
−
1)!!
2
π
+
+
ln
2
(ln x) √
=
+
⎭
2 · (2n)!! ⎩ 12
k2
k
1 − x2
0
k=1
k=1
⎧
&2 ⎫
%2n+1
1
2n+1
⎬
⎨
2n+1
2
k
k
dx
(−1)
(−1)
(2n)!!
π
2 x
−
+
ln
2
(ln x) √
=
+
−
⎭
(2n + 1)!! ⎩ 12
k2
k
1 − x2
0
GW (324)(17)
15.
16.
k=1
GW (324)(60a)
k=1
GW (324)(60b)
1
17.7
2
2
(ln x) xμ−1 (1 − x)ν−1 dx = B(μ, ν) [ψ(μ) − ψ(ν + μ)] + ψ (μ) − ψ (μ + ν)
0
[Re μ > 0,
1
18.
2
(ln x)
0
1
19.
0
20.
1
2
(ln x)
0
0
LI (111)(8)
1 + (−1)n xn+1
n−k+1
3
dx = (n + 1) ζ(3) − 2
(−1)k−1
2
(1 + x)
2
k3
1 − x2n+2
LI (111)(7)
2
(1 − x2 )
n−k+1
7
(n + 1) ζ(3) − 2
4
(2k − 1)3
n
dx =
k=1
[n = 0, 1, . . .]
1
2
q−1
(ln x) xp−1 (1 − xr )
21.
n−k+1
k3
ET I 315(11)
k=1
7
k=1
Re ν > 0]
n
2
(ln x)
1 − xn+1
dx = 2(n + 1) ζ(3) − 2
(1 − x)2
n
dx =
p
1
,q
B
r3
r
ψ
LI (111)(9)
+
,2 p
p
p
p
+q + ψ
+q
− ψ
−ψ
r
r
r
r
[p > 0,
q > 0,
r > 0]
GW (324)(8a)
544
4.262
Logarithmic Functions
1
1.
3
(ln x)
0
1
2.
3
(ln x)
0
3.
4.
7 4
dx
=−
π
1+x
120
BI (109)(9)
π4
dx
=−
1−x
15
BI (109)(11)
,2
+
2
π 2 + (ln a)
dx
3
=
(ln x)
(x + a)(x − 1)
4(a + 1)
0
%
&
1
n−1
n
4
(−1)k
3 x dx
n+1 7π
= (−1)
−6
(ln x)
1+x
120
(k + 1)4
0
4.262
∞
[a > 0]
BI (141)(2)
[n = 1, 2, . . .]
BI (109)(10)
π4
xn dx
1
=−
+6
1−x
15
(k + 1)4
[n = 1, 2, . . .]
BI (109)(12)
x2n dx
1
π4
+6
=
−
2
1−x
16
(2k + 1)4
[n = 1, 2, . . .]
BI (109)(14)
k=0
n−1
1
5.
3
(ln x)
0
k=0
n−1
1
6.
3
(ln x)
0
k=0
3
(ln x)
0
1 + (−1)n xn+1
n−k+1
7(n + 1)π 4
+
6
dx
=
−
(−1)k−1
(1 + x)2
120
k4
BI (111)(10)
n
1
3
(ln x)
0
k=1
1
9.
3
(ln x)
0
4.263
BI (111)(11)
k=1
8.
n−k+1
1 − xn+1
(n + 1)π 4
+
6
dx
=
−
(1 − x)2
15
k4
n
1
7.
∞
1.8
0
1 − x2n+2
2
(1 − x2 )
n−k+1
(n + 1)π 4
+6
16
(2k − 1)4
n
dx = −
,+
,
+
2
2
7π 2 + 3 (ln a)
ln a π 2 + (ln a)
dx
4
=
(ln x)
(x − 1)(x + a)
15(1 + a)
[a > 0]
1
2.
4
(ln x)
0
1
3.
0
1
1.
dx
5π
=
2
1+x
64
BI (109)(17)
t π 2 − t2 7π 2 − 3t2
dx
(ln x)
=
1 + 2x cos t + x2
30 sin t
4
5
(ln x)
0
2.
1
5
(ln x)
0
BI (141)(3)
5
[|t| < π]
4.264
BI (111)(12)
k=1
BI (113)(8)
31π 6
dx
=−
1+x
252
BI (109)(20)
8π 6
dx
=−
1−x
63
BI (109)(21)
4.267
Powers and logarithms
∞
3.
0
545
,2 +
,
+
2
2
3π 2 + (ln a)
π 2 + (ln a)
dx
5
=
(ln x)
(x − 1)(x + a)
6(1 + a)
[a > 0]
1
4.265
6
(ln x)
0
4.266 1.
1
7
(ln x)
0
1
2.
7
(ln x)
0
4.267 1.
1
0
1
2.
0
1
3.8
0
4.
5.
6.11
7.
8.
9.
10.
dx
61π
=
2
1+x
256
BI (109)(25)
127π 8
dx
=−
1+x
240
BI (109)(28)
8π 8
dx
=−
1−x
15
BI (109)(29)
2
1 − x dx
= ln
1 + x ln x
π
BI (127)(3)
π
(1 − x)2 dx
= ln
2
1 + x ln x
4
BI (128)(2)
(1 − x)2
dx
·
mx
2 ln x
+x
1 + 2x cos
n
n−1
kmπ
1
mπ =
(−1)k sin
n
sin n k=1
=
BI (141)(4)
7
sin
1
mπ n
1
2 (n−1)
k=1
(−1)k sin
n+k+1 2 k+2 k Γ 2n Γ 2n
Γ
ln 2n2 k+1
n+k
Γ 2n
Γ 2n Γ n+k+2
2n
n−k+1 2 k+2 k Γ n Γ n
Γ
kmπ
n
ln 2 n−k n−k+2 k+1
n
Γ
Γ
Γ
n
n
[m + n is even]
n
[m < n]
1
ln 2
1−x
dx
·
=−
·
2 ln x
1
+
x
1
+
x
2
0
√
1
2
x
2 2
1−x
dx
·
= ln
·
2 ln x
π
0 1+x 1+x
1
∞
p
dx
ln(1 + k)
[p ≥ 1]
=
(1 − x)p
(−1)k
k
ln
x
0
k=1
1
dx
1 − xp
−p
= ln Γ(p + 1)
1−x
ln x
0
1 p−1
p
x
− xq−1
dx = ln
[p > 0,
ln
x
q
0
1 p−1
Γ q Γ p+1
dx
x
− xq−1
2
·
= ln 2p q+1
[p > 0,
ln x
1+x
Γ 2 Γ 2
0
1 p−1
1 ∞ xp−1 − x−p
pπ
x
− x−p
dx =
dx = ln tan
(1
+
x)
ln
x
2
(1
+
x)
ln
x
2
0
0
[m + n is odd]
BI (130)(3)
1
BI (130)(16)
BI (130)(17)
BI (123)(2)
GW (326)(10)
q > 0]
FI II 647
q > 0]
FI II 186
[0 0,
n+k−1
k
ak ln
p+k
q+k
p+q+1
dx
= ln
ln x
(p + 1)(q + 1)
p > 0,
p > 0,
q > 0]
q > 0,
a2 < 1
[p > −1,
q > −1,
LI (123)(5)
BI (130)(15)
p + q > −1]
GW (324)(19b)
1
14.
0
xp − xq 1 + x2n+1
·
dx = ln
1+x
x ln x
q+1
q
Γ 2
+ n Γ p+1
2
p
+n+1 Γ 2
2
p+1
q
+n+1 Γ
+n Γ
Γ
2
2
Γ
[p > 0,
1
15.
0
1
16.
0
1
17.
0
Γ(q + 1) Γ(p + r + 1)
xp − xq 1 − xr
·
dx = ln
1−x
ln x
Γ(p + 1) Γ(q + r + 1)
[p > −1, q > −1,
x
−x
dx = ln
(1 + xr ) ln x
p−1
q−1
Γ
Γ
q
p+r
Γ 2r
2r
q+r
p
Γ
2r
2r
∞
18.
0
1
20.
0
21.
0
22.
0
1
pπ
qπ
xp−1 − xq−1
dx = ln tan
cot
r
(1 + x ) ln x
2r
2r
q > 0,
[0 0,
GW (324)(21)
BI (128)(6)
0 < q < r]
0 < q < r]
q > 0]
Γ
q+2
4(2n+1)
Γ
p+4n+2
4(2n+1)
Γ
q
4(2n+1)
q+4n+4
4(2n+1)
Γ
p+2
4(2n+1)
Γ
q+4n+2
4(2n+1)
Γ
p
4(2n+1)
q > 0]
BI (143)(4)
BI (128)(11)
p+4n+4
4(2n+1)
[p > 0,
∞
r > 0]
GW (324)(23)
GW (324)(22), BI (143)(2)
q
Γ p+2
xp−1 − xq−1 1 − x2
2n Γ 2n
dx = ln q+2 p ·
1 − x2n
ln x
Γ 2n Γ 2n
Γ
xp−1 − xq−1 1 + x2
dx
=
ln
1 + x2(2n+1) ln x
Γ
BI (127)(7)
[0 < q < p]
pπ ⎞
∞ p−1
sin
x
− xq−1
r ⎠
dx = ln ⎝
r
sin qπ
0 (1 − x ) ln x
r
p
2
q + r > −1]
⎛
19.
Γ
q > 0]
p + r > −1,
[p > 0,
1 − x2p−2q xq−1 dx
qπ
= ln tan
1 + x2p
ln x
4p
q+1
2
BI (128)(7)
pπ
(p + 2)π
qπ
(q + 2)π
xp−1 − xq−1 1 + x2
dx = ln tan
· tan
· cot
· cot
·
ln x
4(2n + 1)
4(2n + 1)
4(2n + 1)
4(2n + 1)
1 + x2(2n+1)
[0 < p < 4n, 0 < q < 4n]
BI (143)(5)
4.267
Powers and logarithms
∞
23.
0
pπ
· sin (q+2)π
sin
2n
xp−1 − xq−1 1 − x2
2n
dx = ln
(p + 2)π
qπ
1 − x2n
ln x
· sin
sin
2n
2n
[0 0,
Γ(p + r) Γ(q + r)
xr−1 dx
= ln
(1 − x) ln x
Γ(p + q + r) Γ(r)
[r > 0, r + p > 0,
1
q > 0,
r + q > 0,
r > 0]
BI (123)(8)
r + p + q > 0]
(p + q + 1)(q + r + 1)(r + p + 1)
dx
= ln
ln x
(p + q + r + 1)(p + 1)(q + 1)(r + 1)
r > −1, p + q > −1, p + r > −1, q + r > −1,
FI II 815a
(1 − xp ) (1 − xq ) (1 − xr )
26.
0
[p > −1,
q > −1,
p + q + r > −1]
GW (324)(19c)
1
Γ(p + 1) Γ(q + 1) Γ(r + 1) Γ(p + q + r + 1)
dx
= ln
(1 − x) ln x
Γ(p + q + 1) Γ(p + r + 1) Γ(q + r + 1)
r > −1, p + q > −1, p + r > −1, q + r > −1, p + q + r > −1]
(1 − xp ) (1 − xq ) (1 − xr )
27.
0
[p > −1,
q > −1,
FI II 815
1
(1 − xp ) (1 − xq ) (1 − xr )
28.
0
1
29.
0
(p + q + s)(p + r + s)(q + r + s)s
xs−1 dx
= ln
ln x
(p + s)(q + s)(r + s)(p + q + r + s)
[p > 0, q > 0, r > 0,
BI (123)(10)
q+s Γ p+s
Γ r
dx
x
p
q
r
= ln s p+q+s
(1 − x ) (1 − x )
(1 − xr ) ln x
Γ r Γ
r
s−1
[p > 0,
∞
30.
(1 − xp ) (1 − xq )
0
s−1
dx
x
=2
p+q+2s
(1 − x
) ln x
1
(1 − xp ) (1 − xq ) (1 − xr )
31.
0
1
0
q > 0,
r > 0,
s > 0]
GW (324)(23a)
1
s−1
dx
x
p+q+2s
(1 − x
) ln x
0
(p + s)π
sπ
cosec
= 2 ln sin
p + q + 2s
p + q + 2s
[s > 0, s + p > 0, s + p + q > 0] GW (324)(23b)a
(1 − xp ) (1 − xq )
Γ(p + s) Γ(q + s) Γ(r + s) Γ(p + q + r + s)
xs−1 dx
= ln
(1 − x) ln x
Γ(p + q + s)4 Γ(p + r + s) Γ(q + r + s) Γ(s)
[p > 0, q > 0, r > 0, s > 0] ∗ BI (127)(11)
q+s r+s p+q+r+s Γ p+s
Γ
Γ
Γ
xs−1 dx
t
= ln p+q+s t q+r+s t p+r+s t s (1 − x ) (1 − x ) (1 − x )
t
(1 − x ) ln x
Γ
Γ
Γ t
Γ
t
t
t
[p > 0, q > 0, r > 0, s > 0, t > 0] ∗ GW (324)(23b)
p
32.
s > 0]
q
r
∗ In 4.267.31 the restrictions can be somewhat weakened by writing, for example, s > 0, p + s > 0, q + s > 0, r + s > 0,
p + q + s > 0, p + r + s > 0, q + r + s > 0, p + q + r + s > 0, in 4.267 31 and 32.
548
Logarithmic Functions
1
33.
0
1
34.
0
1
35.
0
1
36.
0
1
37.
0
38.
39.6
xp − xp+q
−q
1−x
dx
Γ(p + q + 1)
= ln
[p > −1, p + q > −1]
BI (127)(19)
ln x
Γ(p + 1)
dx
xμ − x
− x(μ − 1)
= ln Γ(μ)
[Re μ > 0]
WH, BI (127)(18)
x−1
x ln x
dx
(1 − xp ) (1 − xq )
= − ln {B(p, q)}
1−x−
1−x
x ln x
[p > 0, q > 0]
BI (130)(18)
p−1
pq−1
x
x
1
dx
1
−
+
= q ln p
−
q
q
1−x 1−x
x(1 − x) x (1 − x ) ln x
xq−1
xpq−1
p − 1 p−1 p − 1 p−1
−
x
−
x
−
1 − x 1 − xp
1 − xp
2
[p > 0]
p
q
BI (130)(20)
dx
1−p
1
=
ln(2π) + pq −
ln x
2
2
(1 − x ) (1 − x ) − (1 − x)
dx = ln B(p, q)
x(1 − x) ln x
0
1
n
n
n dx
p
=
(x − 1)
(−1)n−k ln(pk + 1)
n
−
k
ln
x
0
1
4.268
ln p
[p > 0,
q > 0]
BI (130)(22)
[p > 0,
q > 0]
GW (324)(24)
[n > 0,
pn > −1]
2
k=0
GW (324)(19d), BI (123)(12)a
1
40.6
0
1
p n
(1 − x ) dx
=
1 − x ln x
(−1)k−1 ln Γ[(n − k)p + 1]
[n > 1,
pn > −1]
[n > 0,
q > 0,
BI (127)(12)
k=0
n
dx
ln[q + (n − k)p]
=
(−1)k
k
ln x
n
n
(xp − 1) xq−1
41.
n
0
k=0
pn > −q]
BI (123)(12)
1
n
(1 − xp ) xq−1
42.6
0
dx
=
(1 − x) ln x
n
(−1)k−1 ln Γ[(n − k)p + q]
k=0
[n > 1,
q > 0,
pn > −q]
BI (127)(13)
1
n
m
(xp − 1) (xq − 1)
43.10
0
r−1
x
dx
=
ln x
[n ≥ 0,
4.268 1.
0
1
(xp − xq ) (1 − xr )
(ln x)
2
n
(−1)j
j=0
m ≥ 0,
n
j
m
(−1)k
k=0
n + m > 0,
m
ln[r + (m − k)q + (n − j)p]
k
r > 0,
pn + qm + r > 0]
BI (123)(16)
dx = (p + 1) ln(p + 1) − (q + 1) ln(q + 1)
−(p + r + 1) ln(p + r + 1) + (q + r + 1) ln(q + r + 1)
[p > −1,
q > −1,
p + r > −1,
q + r > −1]
GW (324)(26)
4.269
Powers and logarithms
1
2
(xp − xq )
2.
0
dx
2
(ln x)
= (2p + 1) ln(2p + 1) + (2q + 1) ln(2q + 1) − 2(p + q + 1) ln(p + q + 1)
1
(1 − xp ) (1 − xq ) (1 − xr )
3.
549
0
p > − 12 ,
q > − 12
GW (324)(26a)
dx
(ln x)
2
= (p + q + 1) ln(p + q + 1) + (q + r + 1) ln(q + r + 1) + (p + r + 1) ln(p + r + 1)
−(p + 1) ln(p + 1) − (q + 1) ln(q + 1) − (r + 1) ln(r + 1) − (p + q + r) ln(p + q + r)
[p > −1,
q > −1,
r > −1,
p + q > −1,
p + r > −1,
q + r > −1,
p + q + r > 0]
BI (124)(4)
4.
1
n
(pk + q)2 ln(pk + q)
k
k=0
+
q,
q > 0, p > −
n
⎛
⎞
1
n
m
dx
p n
q m r−1
j n ⎠
k m
⎝
(1 − x ) (1 − x ) x
(−1)
(−1)
2 =
j
k
(ln x)
0
j=0
k=0
n
(1 − xp ) xq−1
0
5.
n
dx
2
(ln x)
=
1
2
(−1)k
BI (124)(14)
×[(m − k)q + (n − j)p + r] ln[(m − k)q + (n − j)p + r]
[r > 0,
1
6.
mq + r > 0,
(q − r)xp−1 + (r − p)xq−1 + (p − q)xr−1
0
np + r > 0,
mq + np + r > 0]
BI (124)(8)
dx
(ln x)
2
= (q − r)p ln p + (r − p)q ln q + (p − q)r ln r
7.
0
4.269
1.
2.11
1
[p > 0,
q > 0,
r > 0]
BI (124)(9)
xp−1
xq−1
xr−1
+
+
+
(p − q)(p − r)(p − s) (q − p)(q
− r)(q − s) (r − p)(r − q)(r − s)
xs−1
dx
p2 ln p
q 2 ln q
1
+
+
=
2
(s − p)(s − q)(s − r) (ln x)
2 (p − q)(p − r)(p − s) (q − p)(q − r)(q − s)
2
2
s ln s
r ln r
+
+
(r − p)(r − q)(r − s) (s − p)(s − q)(s − r)
[p > 0, q > 0, r > 0, s > 0] BI (124)(16)
√ 1
∞
π
(−1)k
1 dx
ln
=
x 1 + x2
2
(2k + 1)3
0
k=0
1
∞
√ dx
(−1)k
√
= π
2k + 1
1
0
k=0
1 + x2
ln
x
BI (115)(33)
BI (133)(2)
550
3.
4.
5.
6.
7.
4.271
Logarithmic Functions
1
1
1 π
ln xp−1 dx =
[p > 0]
x
2 p3
0
1 p−1
x
π
[p > 0]
dx =
p
1
0
ln
x
1
n
√ sin t − xn sin[(n + 1)t] + xn+1 sin nt
sin kt
dx
√
·
π
=
2
1
−
2x
cos
t
+
x
k
1
0
k=1
ln
x
[|t| < π]
1
cos kt
√ n−1
cos t − x − xn−1 cos nt + xn cos[(n − 1)t]
dx
√
·
π
=
2
1 − 2x cos t + x
k
1
0
k=1
ln
x
[|t| < π]
v
dx
=π
[uv > 0]
x v
u
x · ln ln
u x
1
2n
(ln x)
1.
0
1
2.
22n − 1
dx
=
· (2n)! ζ(2n + 1)
1+x
22n
2n−1
(ln x)
0
1
3.
2n−1
(ln x)
0
4.
4.271
1
GW (324)(1c)
BI (133)(1)
BI (133)(5)
BI (133)(6)
BI (145)(37)
BI (110)(1)
1 − 22n−1 2n
dx
=
π |B2n |
1+x
2n
[n = 1, 2, . . .]
BI (110)(2)
1
dx
= − 22n−2 π 2n |B2n |
1−x
n
[n = 1, 2, . . .]
BI (110)(5), GW(324)(9a)
dx
= ei(p−1)π Γ(p) ζ(p)
[p > 1]
1−x
1
∞
dx
(−1)k
n
n
(ln x)
=
(−1)
n!
1 + x2
(2k + 1)n+1
0
k=0
1
dx
dx
1 ∞
π 2n+1
2n
2n
(ln x)
=
(ln x)
= 2n+2 |E2n |
2
2
1+x
2 0
1+x
2
0
∞
2n+1
(ln x)
dx = 0
[|b| < 2]
2
0 1 + bx + x
1
dx
22n+1 − 1
2n
(ln x)
=
· (2n)! ζ(2n + 1)
[n = 1, 2, . . .]
1 − x2
22n+1
0
∞
dx
2n
(ln x)
=0
1 − x2
0
1
dx
dx
1 ∞
1 − 22n 2n
2n−1
2n−1
π |B2n |
(ln x)
=
(ln x)
=
2
2
1−x
2 0
1−x
4n
0
p−1
(ln x)
GW (324)(9b)
0
5.
6.
7.
8.
9.
10.
[n = 1, 2, . . .]
BI (110)(11)
GW (324)(10)a
BI (135)(2)
BI (110)(12)
BI (312)(7)a
BI (290)(17)a, BI(312)(6)a
4.272
Powers and logarithms
1
11.
2n−1
(ln x)
0
1
12.
2n
(ln x)
2
dx =
22n − 1 2n
π |B2n |
2
[n = 1, 2, . . .]
BI (290)(19)a
[n = 1, 2, . . .]
BI (296)(17)a
∞
1
2n+1
(ln x)
0
1 + x2
(1 − x2 )
0
13.
x dx
1
= − π 2n |B2n |
2
1−x
4n
551
cos 2akπ
(cos 2aπ − x) dx
= −(2n + 1)!
2
1 − 2x cos 2aπ + x
k 2n+2
k=1
∞
14.6
0
[a is not an integer]
x
dx
d
n
ν−2 sin(ν − 1)t
(ln x) 2
= −π cosec t n a
a + 2ax cos t + x2
dν
sin νπ
[a > 0, 0 < Re ν < 2,
ν−1
LI (113)(10)
n
0 < |t| < π]
ET I 315(12)
1
15.
n
(ln x)
0
1
16.3
n
(ln x)
0
4.272
1
0
q−1
1
ln
x
1
ln
0
1
4.
ln
0
1
x
μ
1
1
6.
ln
0
7.
0
[p > 0,
q > 0]
GW (324)(10)
[|t| < π, q < 1]
LI (130)(1)
1
1
x
1
ln
x
LI (130)(5)
∞
xν−1 dx
ak sin kt
Γ(μ + 1) =
1 − 2ax cos t + x2 a2
a sin t
(ν + k − 1)μ+1
Re μ > 0,
Re ν > 0,
−π < t < π]
BI (140)(14)a
∞
pk−1 cos kλ
cos λ − px
q−1
x
dx
=
Γ(r)
1 + p2 x2 − 2px cos λ
(q + k − 1)r
k=1
[r > 0,
(ln x)
GW (324)(9)
k=1
r−1
1
x
∞
5.
q > 0]
∞
1
(1 + x) dx
t
k−1 cos k − 2 t
= sec · Γ(q)
(−1)
1 + 2x cos t + x2
2
kq
k=1
|t| < π, q < 12
[a > 0,
[p > 0,
k=1
2.
3.9
p
q
q−1
1
dx
∞
sin kt
x
=
cosec
t
Γ(q)
(−1)k−1 q
2
1 + 2x cos t + x
k
0
xp−1
1
dx = n+1 β (n)
1 + xq
q
p
q
1 ln
1.
xp−1
1
dx = − n+1 ψ (n)
1 − xq
q
p
dx
= Γ(1 + p)
x2
μ−1
xν−1 dx =
n− 12
ν−1
x
[p > −1]
1
Γ(μ)
νμ
(2n − 1)!!
dx =
(2ν)n
q > 0]
[Re μ > 0,
π
ν
[Re ν > 0]
BI (113)(11)
BI (149)(1)
Re ν > 0]
BI (107)(3)
BI (107)(2)
552
Logarithmic Functions
1
x
n−1
ln
1
x
n−1
1
x
μ−1
ln
1
8.11
ln
0
1
9.
0
1
10.
0
xν−1
1
dx = Γ 3 −
1+x
n
p n −3 − q n −3
1
1
xν−1
dx = (n − 1)! ζ(n, ν)
1−x
(x − 1)n a +
4.272
nx
x−1
xa−1 dx = Γ(μ)
[Re ν > 0]
BI (110)(4)
[Re ν > 0]
BI (110)(7)
n
(−1)k n(n − 1) . . . (n − k + 1)
k=0
(a + n − k)μ−1 k!
[Re μ > 0]
1
11.
ln
0
1
12.
ln
0
1
x
n−1
1
x
μ−1
1 − xm
dx = (n − 1)!
1−x
m
k=1
LI (110)(10)
1
kn
LI (110)(9)
∞
xν−1 dx
1
1
ν
= Γ(μ)
= μ Γ(μ) ζ μ,
2
1−x
(ν + 2k)μ
2
2
k=0
[Re μ > 0,
1
13.
0
xq − x−q
1 − x2
1
14.
ln
0
1
x
ln
r−1
1
x
p
dx = Γ(p + 1)
Re ν > 0]
1
1
−
(2k + q − 1)p+1
(2k − q − 1)p+1
p > −1, q 2 < 1
∞
k=1
∞
xp−1 dx
=
Γ(r)
s
(1 + xq )
BI (110)(13)
LI (326)(12)a
−s
1
k (p + kq)r
k=0
[p > 0,
q > 0,
r > 0,
0 < s < r + 2]
GW (324)(11)
1
15.
ln
0
1
x
n
m
(1 + xq ) xp−1 dx = n!
m
m
1
k (p + kq)n+1
k=0
[p > 0,
1
16.
ln
0
1
17.
ln
0
1
18.
0
1
x
1
x
1
ln
x
n
m
(1 − xq ) xp−1 dx = n!
m
BI (107)(6)
[p > 0,
q > 0]
BI (107)(7)
[p > 0,
q > 0,
m
(−1)k
k (p + kq)n+1
k=0
p−1
q > 0]
∞
ak
xq−1 dx
1
=
Γ(p)
1 − axq
aq p
kp
1
2− n
p−1
x
− xq−1 dx =
n
Γ
n−1
1
n
19.
1
ln
0
1
x
LI (110)(8)
1
1
q 1− n − p1− n
[q > p > 0]
a < 1]
k=1
2n−1
xp − x−p q−1
1
x
dx = 2n
q
1−x
p
∞
k=n
2pπ
q
k
|B2k |
2k · (2k − 2n)!
+
q,
p<
2
BI (133)(4)
LI (110)(16)
4.282
Rational functions of ln x and powers
x p−1
v q−1 dx
v
ln
= B(p, q) ln
u
x
x
u
u
√
1
√
q
qπ
e x dx
= √
q
e
0 x − (1 + ln x)
4.274
4.275
v
ln
4.273
1
%
1.
0
1
2.
x−
&
q−1
1
ln
x
−x
p−1
1
1 − ln x
0
q
(1 − x)
q−1
dx =
p+q−1
[p > 0,
q > 0,
uv > 0]
[q > 0]
BI (145)(36)
BI (145)(4)
Γ(q)
[Γ(p + q) − Γ(p)]
Γ(p + q)
[p > 0,
dx
= − ψ(q)
x ln x
553
q > 0]
[q > 0]
BI (107)(8)
BI (126)(5)
4.28 Combinations of rational functions of ln x and powers
4.281
1
1.
0
∞
2.
1
3.
4.
5.
6.
7.
1
1
+
dx = C
ln x 1 − x
BI (127)(15)
1
dx
= li(p)
x2 (ln p − ln x)
p
LA 281(30)
1
xp−1 dx
= ±e∓pq Ei (±pq)
[p > 0, q > 0]
0 q ± ln x
1
1
xμ−1
+
dx = − ψ(μ)
[Re μ > 0]
ln x 1 − x
0
1 p−1
x
xq−1
+
dx = ln p − ψ(q)
[p > 0, q > 0]
ln x
1−x
0
1
1
ln 2
dx
1
=
+
2
1−x
2x ln x ln x
2
0
1
1
ln 2π
1 (1 − x) (1 + q ln x) + x ln x q−1 dx
= − q − ln Γ(q) +
x
q− +
2
2
(1
−
x)
ln
x
2
2
0
[q > 0]
4.282
1
1.
0
1
2.
0
2
·
·
1 1
dx
= − C
1−x
4 2
dx
1
β
=
1 + x2
2a
2a + π
4π
dx
4−π
=
2
1
+
x
4π
+ (ln x)
1
π2
0
4.
+ (ln x)
1
1
3.
2
a2 + (ln x)
0
ln x
4π 2
1
+ (ln x)
2
·
dx
1
=
1 − x2
2
WH
BI (127)(17)
LI (130)(19)
BI (128)(15)
BI (129)(1)
+
π,
a>−
2
BI (129)(9)
BI (129)(6)
2
ln x
π2
LI (144)(11,12)
1
− ln 2
2
BI (129)(10)
554
Logarithmic Functions
1
5.
a2 + (ln x)
0
1
6.
1
8.
1
11.
0
1
13.
0
x−1
−x
ln x
1
1
0
1
3.
0
1
4.
0
1
5.
0
6.
7.
8.
,
√
dx
1 +
√
=
2
−
1
π
+
2
ln
1 + x2
8π 2
2
·
√
1
1
dx
π
+ √ ln
=− √ +
2−1
2
1−x
32 2 16 16 2
,2
2π
a
∞
π
x dx
π2 =
−
|B2k |
2
4
1−x
4a
a
BI (129)(8)
BI (129)(12)
2k−2
BI (129)(4)
2k−2
BI (129)(16)
k=1
p2 < 1
BI (132)(13)a
k=1
2.
2
·
∞
dx
xp − x−p
sin kpπ
2π =
(−1)k−1
2
2
2
x − 1 q + (ln x)
q
2q + kπ
0
2
a2 + (ln x)
BI (129)(13)
BI (129)(11)
∞
π2 dx
=− 4
|B2k |
+
,2
1−x
a
2
k=1
a2 + (ln x)
1.
dx
2−π
=
2
1−x
16
ln x
BI (129)(14)
BI (129)(7)
ln x
+
0
·
ln x
1
12.
4.283
2
π 2 + 16 (ln x)
0
dx
ln 2
=
2
1+x
4π
1
1
10.
·
ln x
[a > 0]
1
−C
2
2
π 2 + 16 (ln x)
0
+ 4 (ln x)
1
9.10
x dx
1
=
2
1−x
2
π 2 + 4 (ln x)
0
·
1
π2
0
2
+ (ln x)
1
7.
π
a ,
x dx
1+π
+
ln
+
ψ
=
1 − x2
2 2a
a
π
ln x
π2
0
2
·
ln x
4.283
%
dx
= ln 2 − 1
ln x
BI (132)(17)a
1
1
1
+
−
ln x 1 − x 2
ln 2π
dx
=
−1
ln x
2
BI (127)(20)
1
x
x
+
+
ln x 1 − x 2
dx
ln 2π
=
x ln x
2
BI (127)(23)
x
2 − (1 − x)2
(ln x)
1
&
1
1
1
−
+
1 − x2
2 ln x 2
dx = C −
1
2
GW (326)(8a)
dx
ln 2 − 1
=
ln x
2
BI (128)(14)
1
dx
1
1 1+x
ln 2π
+ ·
− ln x
=
ln
x
2
1
−
x
ln
x
2
0
1
dx
1
−x
= −C
1 − ln x
x ln x
0
&
1% q
q
x −1
2 − ln x dx = q ln q − q
x (ln x)
0
BI (127)(22)
GW (326)(11a)
[q > 0]
BI (126)(2)
4.291
Logarithmic functions and powers
10.
x+
dx
a
1
= ln + C
a ln x − 1 x ln x
q
0
p−1
1
1
1+x
x
1
+
dx = − ln Γ(p) + p −
ln
x
2(1
−
x)
ln
x
2
0
9.
1
1
p−1−
11.
1
%
−
12.
0
1
0
1
14.
0
4.284
1
%
1.
0
1
%
2.
2
(ln x)
p−
13.
1
2
+
1
2
1−
−x
ln x
p−1
x
1
q−
2
q > 0]
ln p − p +
ln 2π
2
GW (326)(9)
1
ln 2π
− p ln p + p −
2
2
dx
=
ln x
[p > 0]
BI (127)(25)
r−1
1
ln x
2p−1
x
−1
pq−1
rq−1
px
rx
+
−
p
1−x
1 − xr
[p > 0]
dx
=
ln x
GW (326)(8)
1
− p (ln p − 1)
2
[p > 0]
BI (132)(23)a
1
dx
ln 2π
= (p − r)
− q − ln Γ(q) +
ln x
2
2
[q > 0]
BI (132)(13)
[q > 0]
BI (126)(3)
xq − 1
q
−
q
x (ln x)
3
−
&
q2
2x (ln x)
2
−
11
q3
q3
ln q − q 3
dx =
6 ln x
6
36
[q > 0]
1
0
&
3 2
q2
q2
3 −
2 − 2 ln x dx = 2 ln q − 4 q
x (ln x)
x (ln x)
xq − 1
x (ln x)
4.285
BI (126)(8)
(p − 2)xp − (p − 1)xp−1
3
dx = − ψ(p) + p −
2
(1 − x)
2
x3 +
4
0
xp−1
[a > 0,
[p > 0]
&
1
1
1
−
2 ln x
1
+
1−x
0
555
BI (126)(4)
n−1
xp−1 dx
pn−1 −pq
1
e
=
Ei(pq)
−
(n − k − 1)!(pq)k−1
n
n−1
(n − 1)!
(n − 1)!q
(q + ln x)
k=1
[p > 0,
q < 0]
BI (125)(21)
n
xa (ln x) dx
, we should make the substitution x = et or x = e−t and
m l
[b ± (ln x) ]
then seek the resulting integrals in 3.351–3.356.
In integrals of the form
4.29–4.32 Combinations of logarithmic functions of more complicated arguments
and powers
4.291
1.
0
1
π2
ln(1 + x)
dx =
x
12
FI II 483
556
Logarithmic Functions
1
2.
0
π2
ln(1 − x)
dx = −
x
6
1/2
3.
0
ln 1 −
0
7.7
8.
9.
10.
11.
12.
13.
dx
1
π2
2
= (ln 2) −
x
2
12
x
2
BI (114)(18)
1+x
2
2 dx = 1 (ln 2)2 − π
1−x
2
12
0
BI (145)(2)
1 ln
5.
6.
FI II 714
1
ln(1 − x)
π2
2
dx = (ln 2) −
x
2
12
1
4.
4.291
BI (115)(1)
1
ln(1 + x)
1
2
dx = (ln 2)
1+x
2
0
a
∞
ln(1 + ax)
ln u du
π 2
ln
1
+
a
−
dx
=
2
2
1
+
x
4
0
0 1+u
1
ln(1 + x)
π
dx = ln 2
2
1+x
8
0
∞
ln(1 + x)
π
dx = ln 2 + G
1 + x2
4
0
1
ln(1 − x)
π
dx = ln 2 − G
2
1+x
8
0
∞
ln(x − 1)
π
dx = ln 2
1 + x2
8
1
1
π2
1
ln(1 + x)
2
dx =
− (ln 2)
12 2
0 x(1 + x)
∞
π2
ln(1 + x)
dx =
.
x(1 + x)
6
0
BI (114)(14)a
[a > 0]
GI II (2209)
FI II 157
BI (136)(1)
BI (114)(17)
BI (144)(4)
BI (144)(4)
BI (141)(9)a
1
14.
0
a+b
2 ln 2
ln(1 + x)
1
ln
+ 2
dx =
(ax + b)2
a(a − b)
b
b − a2
1
= 2 (1 − ln 2)
2a
[a = b,
ab > 0]
[a = b]
LI (114)(5)a
∞
15.
0
16.
1
ln(a + x)
0
17.
a
ln
ln(1 + x)
b
dx =
(ax + b)2
a(a − b)
√
dx
1
= √ arccot a ln[(1 + a)a]
2
a+x
2 a
[ab > 0]
BI (139)(5)
[a > 0]
BI (114)(20)
∞
dx
a ln a − b ln b
=
(b + x)2
b(a − b)
a
ln(1 + ax)
1
dx = arctan a ln 1 + a2
2
1
+
x
2
0
ln(a + x)
[a > 0,
b > 0,
a = b]
LI (139)(6)
0
18.
GI II (2195)
4.291
Logarithmic functions and powers
19.
20.
557
√
ln(1 + ax)
1
dx = √ arctan a ln(1 + a)
[a > 0]
2
1 + ax
2 a
0
1
1
ln(ax + b)
1
(a + b) ln(a + b) − b ln b − a ln 2
dx =
2
a−b 2
0 (1 + x)
1
∞
21.
0
ln(ax + b)
1
[a ln a − b ln b]
dx =
(1 + x)2
a−b
∞
22.
x dx
ln(a + x)
(b2 + x2 )
0
=
2
1
2
2 (a + b2 )
ln b +
[a > 0,
b > 0,
[a > 0,
b > 0]
BI (139)(8)
1
23.
b > 0]
BI (139)(9)
ln(1 + x)
23
1 + x2
1
dx = − ln 2 +
(1 + x)4
3
72
ln(1 + x)
+π ,
1 + x2
dx
1
2
ln
1
+
a
−
2
arctan
a
·
ln
a
·
=
a2 + x2 1 + a2 x2
2a (1 + a2 ) 2
0
1
24.
0
1
25.
0
26.
[a > 0]
1
dx
1
1
a+b
1−x
1
ln(a + b) − ln b − ln a
= 2
2
2
2
(ax + b) (bx + a)
a −b
a−b
ab
a
b
4 ln 2
+ 2
b − a2
a > 0, b > 0, a2 = b2
∞
0
2
0
(1 +
ln(a + x)
0
∞
29.
30.
0
dx =
2
x2 )
b2 − x2
(b2 +
ln2 (a − x)
0
b > 0]
2
∞
LI (114)(13)
ln2 (a − x)
dx =
2
x2 )
b2 − x2
2
(b2 + x2 )
x dx
(b2 +
2
x2 )
a
a
1 (1 + a)
1
π
ln(1 + a) − ·
ln 2 − ·
2
2
2 1+a
2 1+a
4 1 + a2
a2
dx =
=
LI (139)(14)
2
[a > −1]
∞
28.
1−x
1
ln(1 + ax)
LI (114)(11)
b
1−x
dx
1
ln
·
=
2
2
2
2
(ax + b) (bx + a)
ab (a − b ) a
[a > 0,
27.
2
ln(1 + x)
BI (114)(22)
LI (114)(12)
2
ln(1 + x)
a = b]
aπ a2
+ 2 ln a
2b
b
[a > 0,
BI (114)(21)
a2
1
+ b2
a ln
2
a2 + b 2
1
+ b2
bπ
b
−
a
2
a ln
ln b −
BI (114)(23)
[a > 0,
b > 0]
BI (139)(11)
[a > 0,
b > 0]
BI (139)(12)
b > 0]
BI (139)(10)
a bπ
−
b
2
aπ a2
+ 2 ln a
2b
b
[a > 0,
558
4.292
Logarithmic Functions
1
1.
0
1
2.
0
3.
a
1
ln (1 ± x)
π
√
dx = − ln 2 ± 2G
2
2
1−x
x ln (1 ± x)
π
√
dx = −1 ±
2
2
1−x
ln(1 + bx)
1+
√
dx = π ln
2
2
a −x
−a
4.
0
4.292
GW (325)(20)
GW (325)(22c)
√
1 − a2 b 2
2
1
0 ≤ |b| ≤
a
√
√
x ln(1 + ax)
1 − a2
π 1 − 1 − a2
√
+
arcsin a
dx = −1 + ·
2
2
a
1−x
√ a
π
a2 − 1
= −1 +
+
ln a + a2 − 1
2a
a
BI (145)(16, 17)a, GW (325)(21e)
[|a| ≤ 1]
[a ≥ 1]
GW (325)(22)
1
5.
0
4.293
π2
1
ln(1 + ax)
1
2
√
− (arccos a)
dx = arcsin a (π − arcsin a) =
2
8
2
x 1 − x2
[|a| ≤ 1]
1
xμ−1 ln(1 + x) dx =
1.
0
∞
2.6
1
[ln 2 − β(μ + 1)]
μ
∞
3.
4.
5.
6.11
7.
BI (106)(4)a
ET I 315(17)
−1
[β(−μ) + ln 2]
μ
[Re μ < 0]
xμ−1 ln(1 + x) dx =
π
μ sin μπ
[−1 < Re μ < 0]
0
[Re μ > −1]
xμ−1 ln(1 + x) dx =
1
BI (120)(4), GW (325)(21a)
1 (−1)k−1
2n
k
0
k=1
%
&
1
2n+1
(−1)k
1
2n
ln 4 +
x ln(1 + x) dx =
2n + 1
k
0
k=1
%
&
1
n
(−1)n · 4 π (−1)k
2 ln 2
n− 12
+
−
x
ln(1 + x) dx =
2n + 1
2n + 1
4
2k + 1
0
k=0
∞
π
[−1 < Re μ < 0]
xμ−1 ln |1 − x| dx = cot(μπ)
μ
0
GW (325)(3)a
2n
1
x2n−1 ln(1 + x) dx =
GW (325)(2b)
GW (325)(2c)
GW (325)(2f)
BI (134)(4), ET I 315(18)
1
xμ−1 ln(1 − x) dx = −
8.
0
9.7
1
∞
xμ−1 ln(x − 1) dx =
1
1
[ψ(μ + 1) − ψ(1)] = − [ψ(μ + 1) + C]
μ
μ
[Re μ > −1]
ET I 316(19)
[Re μ < 0]
ET I 316(20)
1
[π cot(μπ) + ψ(μ + 1) + C]
μ
4.294
Logarithmic functions and powers
∞
10.
xμ−1 ln(1 + γx) dx =
0
∞
0
1
12.
0
1
13.
0
[−1 < Re μ < 0,
|arg γ| < π]
BI (134)(3)
11.11
π
μγ μ sin μπ
559
π
xμ−1 ln(1 + x)
dx = −
[C + ψ(1 − μ)]
1+x
sin μπ
[−1 < Re μ < 1]
ET I 316(21)
2μ − 1
ln(1 + x)
ln 2
+
dx
=
−
(1 + x)μ+1
2μ μ
2μ μ2
BI (114)(6)
xμ−1 ln(1 − x)
dx = B(μ, ν) [ψ(ν) − ψ(μ + ν)]
(1 − x)1−ν
∞
14.
0
[Re μ > 0,
Re ν > 0]
xμ−1 ln(γ + x)
dx = γ μ−ν B(μ, ν − μ) [ψ(ν) − ψ(ν − μ) + ln γ]
(γ + x)ν
[0 < Re μ < Re ν]
4.294
1
1.
ln(1 + x)
0
1
2.
ln(1 + x)
0
ln(1 + x)
0
1
ln(1 + x)
0
1
ln(1 + x)
0
1
ln(1 − x)
0
1
1 (−1)k−1
1 − x2n
dx = 2 ln 2 ·
−
1+x
2k + 1 j=1 j
k
BI (114)(8)
j
n−1
2n
1
(−1)j (−1)k−1
1 − x2n
dx = 2 ln 2 ·
+
1−x
2k + 1 i=1 j
k
BI (114)(9)
1
ln(1 − x)
0
∞
1
j
n
(−1)j 1
1 − (−1)n xn
dx =
1−x
j
k
j=1
BI (114)(15)
j
n
11
1 − xn
dx = −
1−x
j
k
j=1
BI (114)(16)
ln2 (1 − x)xp dx =
k=1
2π
cot pπ
p+1
[−2 < p < −1]
BI (134)(13)a
(−1)k n! (ln 2)
n!
r+1
+
2
(r + 1)n+1
(n − k)!(r + 1)k+1
n
n
0
n−k
LI (106)(34)a
k=0
1
n
[ln(1 − x)] (1 − x)r dx = (−1)n
0
k=1
BI (114)(10)
[ln(1 + x)] (1 + x)r dx = (−1)n−1
9.
10.
k=1
j
n
2n+1
(−1)j 1
(−1)k−1
1 − x2n+1
dx = 2 ln 2
+
1−x
2k + 1
j
k
j=1
0
j
k=1
8.
2n
k=1
7.
BI (114)(7)
k=1
k=0
6.
j
n
2n+1
1
1
(−1)k−1
1 + x2n+1
dx = 2 ln 2
−
1+x
2k + 1
j
k
j=1
k=0
5.
BI (114)(2)
k=0
4.
[0 −1]
BI (106)(35)a
560
Logarithmic Functions
1
11.
ln
0
1
12.
2n
(ln x)
0
13.
1
1 − x2
1
6
0
4.295
n
x2q−1 dx =
n!
ζ(n + 1, q + 1)
2
∞
ln μx2 + β
0
[−1 < q < 0]
BI (311)(15)a
dx
π 2n+2
=−
|B2n+2 |
ln 1 − x2
x
2(n + 1)(2n + 1)
m
∞
Γ(m + 1)
1
ln 1 − x2 dx = −
ln
x
n(2n
+ 1)m+1
n=1
1.
4.295
dx
π
√
= √ ln
μγ + β
2
γ+x
γ
BI (309)(5)a
[m + 1 > 0,
[Re β > 0,
n + 1 > 0]
Re μ > 0,
|arg γ| < π]
ET II 218(27)
2.
3.
4.
5.
6.
7.
dx
π
ln 1 + x2 2 = − ln 2
x
2
0
∞
dx
ln 1 + x2 2 = π
x
0
∞
π
dx
2a
+ ln a
ln 1 + x2
=
2
2
(a + x)
1 + a 2a
0
1
dx
π
ln 1 + x2
= ln 2 − G
2
1
+
x
2
0
∞
dx
π
ln 1 + x2
= ln 2 + G
2
1+x
2
1
∞
2
ag + bc
dx
π
ln
ln a + b2 x2 2
=
2 x2
c
+
g
cg
g
0
8.
1
9.
10.
11.
12.
13.
14.
GW (325)(4c)
[a > 0]
BI (319)(6)a
BI (114)(24)
BI (114)(5)
[a > 0,
b > 0,
c > 0,
g > 0]
BI (136)(11-14)a
∞
ln a2 + b2 x2
0
c2
bc
dx
π
= − arctan
2
2
−g x
cg
ag
∞
ln 1 + p2 x2 − ln 1 + q 2 x2
dx = π(p − q)
x2
0
1
1 + a2 x2 dx
2
ln
= − (arctan a)
2 1 − x2
1
+
a
0
1
dx
π2
=−
ln 1 − x2
x
12
0
∞
dx
ln2 1 − x2 2 = 0
x
0
1
dx
π
ln 1 − x2
= ln 2 − G
2
1+x
4
0
∞
2
dx
π
ln x − 1
= ln 2 + G
2
1
+
x
4
1
GW (325)(2g)
[a > 0,
b > 0,
c > 0,
g > 0]
BI (136)(15)a
[p > 0,
q > 0]
FI II 645
BI (115)(2)
BI (142)(9)a
GW (325)(17)
BI (144)(6)
4.295
Logarithmic functions and powers
∞
15.
ln2 a2 − x2
0
dx
π = ln a2 + b2
b2 + x2
b
17.
18.
19.
20.
[b > 0]
BI (136)(16)
[b > 0]
BI (136)(20)
∞
b2 − x2
2bπ
ln2 a2 − x2
dx = − 2
2
a + b2
(b2 + x2 )
0
1
1 π2
1
dx
2
=
−
(ln
2)
ln 1 + x2
x (1 + x2 )
2 12 2
0
∞
π2
dx
=
ln 1 + x2
x (1 + x2 )
12
0
1
dx
ln cos2 t + x2 sin2 t
= −t2
1 − x2
0
∞
b2
dx
2 ln b
+
ln a2 + b2 x2
=
(c + gx)2
cg
a2 g 2 + b 2 c 2
0
16.
561
BI (114)(25)
BI (141)(9)
BI (114)(27)a
a
c c
a2 g a
π + 2 ln + 2 2 ln
b
g g
b c
b
[a > 0, b > 0, c > 0,
g > 0]
BI (139)(16)a
21.
1
ln a2 + b2 x2
0
22.
∞
11
0
∞
23.
0
∞
24.
0
∞
25.
dx
(c + gx)2
2a
b2
a cb2 − ga2 a2 + b2
2
c c+g
ln a + 2 2
arccot
+
ln
ln
=
−
2
c(c + g)
a g + b 2 c2 b
b
b2 (c + g)
a2
g
c
[a > 0, b > 0, c > 0, g > 0] BI (114)(28)a
∞ 2
ln 1 + p2 x2
ln p + x2
q + pr
π
ln
dx =
dx =
r2 + q 2 x2
q 2 + r2 x2
qr
r
0
[qr > 0,
ln 1 + a2 x2
dx
π
= 2 2
2
2
2
2
2
2
b +c x
d + g x
b g − c 2 d2
a > 0, b > 0,
ln 1 + a2 x2
x2 dx
π
= 2 2
2
2
2
2
2
2
b +c x
d + g x
b g − c 2 d2
a > 0, b > 0,
ln a2 + b2 x2
0
26.
0
dx
(c2
+
2
g 2 x2 )
=
π
2c3 g
p > 0]
FI II 745a, BI (318)(1)a, BI (318)(4)a
g
ad
ab
c
ln 1 +
− ln 1 +
d
g
b
c
c > 0, d > 0, g > 0, b2 g 2 = c2 d2
ln
b
ab
ad
d
ln 1 +
− ln 1 +
c
c
g
g
c > 0, d > 0, g > 0, b2 g 2 = c2 d2
bc
ag + bc
−
g
ag + bc
[a > 0,
b > 0,
c > 0,
BI (141)(10)
BI (141)(11)
g > 0]
GW (325)(18a)
∞
ln a2 + b2 x2
2
x dx
(c2
+
2
g 2 x2 )
=
π
2cg 3
ln
bc
ag + bc
+
g
ag + bc
[a > 0,
b > 0,
c > 0,
g > 0]
GW (325)(18b)
562
Logarithmic Functions
1
27.
0
1
28.
0
29.
30.6
31.
32.
33.
34.
35.
36.
π
ln 1 + ax2
1 − x2 dx =
2
ln
π
ln 1 + a − ax2
1 − x2 dx =
2
1+
4.295
√
1+a 11− 1+a
√
+
2
21+ 1+a
√
[a > 0]
√
1+ 1+a 11− 1+a
√
−
ln
2
21+ 1+a
√
[a > 0]
1
[a > 0,
37.8
0
40.
1
b > 0,
a = b]
BI (119)(1)
LI (120)(11)
BI (120)(12)
BI (120)(14)
BI (119)(27)
BI (119)(3)
BI (119)(7)
BI (145)(15)
q
p
+1 +ψ
+1
ln 1 − x2 pxp−1 − qxq−1 dx = ψ
2
2
√
1
dx
1+ 1+a
2
√
ln 1 + ax
= π ln
2
1 − x2
0
1
,
+
μ
1
+1
ln 1 + x2 xμ−1 dx =
ln 2 − β
μ
2
0
∞
π
ln 1 + x2 xμ−1 dx =
μπ
0
μ sin
2
39.
BI (117)(7)
2
dx
1 + 1 − a2
a <1
ln 1 − a2 x2 √
= π ln
2
2
1−x
0
1
dx
π 2
ln 1 − a2 x2 √
= − arccos |a| −
2
2
x 1−x
0
1
π
dx
k
= ln K(k) − K (k )
ln 1 − x2 2
2
2
k
2
(1 − x ) (1 − k x )
0
1
dx
π
1 2 ± 2k
K(k)− K (k )
= ln √
ln 1 ± kx2 BI (120)(8),
2
2
2
2
8
k
(1 − x ) (1 − k x )
0
1
ln 1 − k 2 x2
dx = ln k K(k)
(1 − x2 ) (1 − k 2 x2 )
0
.
1
1 − k 2 x2
ln 1 − k 2 x2
dx = 2 − k 2 K(k) − (2 − ln k ) E(k)
2
1−x
0
.
1
1
1 − x2
2
2
ln 1 − k 2 x2 dx = 2 1 + k − k ln k K(k) − (2 − ln k ) E(k)
2
2
1
−
k
x
k
0
√
1
a2 − b 2
dx
2π
√
√
ln 1 − x2
=√
ln
a2 − b 2 a + a2 − b 2
(a + bx) 1 − x2
−1
38.
BI (117)(6)
√
[p > −2,
q > −2]
[a ≥ −1]
[Re μ > −2]
BI (106)(15)
GW (325)(21b)
BI (106)(12)
[−2 < Re μ < 0]
BI (311)(4)a, ET I 315(15)
4.298
Logarithmic functions and powers
∞
41.
0
xμ−1 dx
ln 1 + x2
1+x
π
=
sin μπ
4.296 1.
μπ
β
ln 2 − (1 − μ) sin
2
1−μ
2−μ
μπ
β
− (2 − μ) cos
2
2
2
[−2 < Re μ < 1]
ET I 316(25)
dx
π2
t2
=
−
ln 1 + 2x cos t + x2
x
6
2
0
∞
2
dx
ln a − 2ax cos t + x2
= π ln 1 + 2a|sin t| + a2
2
1+x
−∞
∞
2π cos μt
[|t| < π, −1 < Re μ < 0]
ln 1 + 2x cos t + x2 xμ−1 dx =
μ sin μπ
0
2
∞
a − b2 cos t
x2 + 2ax cos t + a2
x dx
1 2
ln
= π − πt + π arctan 2
x2 − 2ax cos t + a2 x2 + b2
2
(a + b2 ) sin t + 2ab
0
2.
3.
4.
1
[a > 0,
4.297 1.
1
0
[a > 0,
∞
ln
0
1 − x dx
π
= ln 2
x 1 + x2
8
ln
1 + x dx
=G
1 − x 1 + x2
0
1
4.
0
∞
11
ln
0
v
6.
ln
u
∞
7.
0
ln
0
BI (145)(28)
ET I 316(27)
0 < t < π]
1 + ax
1 − ax
ln
PV
0
1
BI (115)(16)
BI (139)(23)
BI (115)(5)
BI (115)(17)
π2
dx
=
x (1 + x2 )
2
BI (141)(13)
2
dx
1 + ax
√
= π arcsin a
1 − ax x 1 − x2
u
10.8
2
1
v
v + x dx
=
ln
u+x x
2
u
v
9.
1+x
1−x
b > 0]
[ab > 0]
b ln(1 + ax) − a ln(1 + bx)
b
dx = ab ln
x2
a
1
8.
dx
ax + b
=0
bx + a (1 + x)2
ln
1
3.
b > 0,
BI (114)(34)
dx
ax + b
1
a+b
− a ln a − b ln b
ln
=
(a + b) ln
bx + a (1 + x)2
a−b
2
2.
5.
563
[uv > 0]
[a > 0,
BI (145)(33)
b > 0]
[|a| ≤ 1]
FI II 647
GW (325)(21c), BI (122)(2)
dx
u
π
= F arcsin av,
2
2
2
2
v
v
(x − u ) (v − x )
!
!
!a + y !
π2
! dy
ln !!
=
!
a − y y 1 − y2
2
[|av| < 1]
[0 < a ≤ 1]
BI (145)(35)
564
4.298 1.
Logarithmic Functions
4.298
ln
ln 2
1
1 + x2 x2n−1
1
dx =
+ 2−
β(2n + 1)
x 1+x
2n
4n
2n
BI (137)(1)
ln
ln 2
1
1 + x2 x2n
1
dx =
+ 2−
β(2n + 1)
x 1+x
2n
4n
2n
BI (137)(3)
ln
ln 2
1
1 + x2 x2n−1
1
dx =
+ 2−
β(2n + 1)
x 1−x
2n
4n
2n
BI (137)(2)
ln
ln 2
1
1 + x2 x2n
1
dx = −
− 2+
β(2n + 1)
x 1−x
2n
4n
2n
BI (137)(4)
∞
0
∞
2.
0
∞
3.
0
∞
4.
0
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
∞
1
1 + x2 x2n−1
ln 2
1
+ 2−
β(2n + 1)
dx =
2
x 1+x
2n
4n
2n
0
1
n−1
1
(−1)k
1 + x2 2n
1
nπ
x dx =
+ ln 2 −
+2
(−1)
ln
x
2n + 1
2
2n + 1
2n − 2k − 1
0
k=0
1
n−1
(−1)k
1 + x2 2n−1
1
1
n+1
n+1
x
+ (−1)
(−1)
ln
dx =
ln 2 + ln 2 −
x
2n
2n
k
0
k=1
1
1 + x2 dx
π
ln
= ln 2
2
x
1
+
x
2
0
∞
2
dx
1+x
ln
= π ln 2
x 1 + x2
0
∞
1 + x2 dx
ln
=0
x 1 − x2
0
1
1 − x2 dx
π
ln
= ln 2
2
x
1
+
x
4
0
∞
2
dx
1+x
3π
ln 2
ln
=
2
x+1 1+x
8
1
1
1 + x2 dx
3π
ln 2 − G
ln
=
2
x
+
1
1
+
x
8
0
∞
1 + x2 dx
3π
ln 2 + G
ln
=
2
x
−
1
1
+
x
8
1
1
1 + x2 dx
3π
ln 2
ln
=
2
1
−
x
1
+
x
8
0
∞
1 + x2 x dx
π2
ln
=
x2 1 + x2
12
0
∞
2
2 2
dx
ag + bc
a +b x
π
ln
[a > 0, b > 0, c > 0,
ln
=
2
2
2
2
x
c +g x
cg
c
0
ln
18.
BI (294)(8)
BI (294)(9)a
BI (115)(7)
BI (137)(8)
BI (137)(9)
BI (115)(9)
BI (144)(8)
BI (115)(18)
BI (144)(9)
BI (115)(19)
BI (138)(3)
g > 0]
BI (138)(6, 7, 9, 10)a
∞
ln
0
BI (137)(10)
ag
dx
a2 + b2 x2
1
arctan
=
x2
c2 − g 2 x2
cg
bc
[a > 0, b > 0, c > 0, g > 0]
BI (138)(8, 11)a
4.311
Logarithmic functions and powers
∞
19.
ln
0
1 + x2 x2 dx
π
= (ln 4 − 1)
2
2
2
x
4
(1 + x )
1 − x2
x2
1
ln2
20.
0
1
21.
0
∞
ln
1
23.
ln
0
1.
∞
ln
0
t2
1 + 2x cos t + x2 dx
=
−
(1 + x)2
x
2
1 + 2x cos t + x2 p−1
2π (1 − cos pt)
x
dx = −
(1 + x)2
p sin pπ
dx
1 + x2 sin t
√
= π ln cot
1 − x2 sin t 1 − x2
π−t
4
BI (115)(23), BI (134)(15)
[0 < |p| < 1,
|t| < π]
[|t| < π]
(x + 1) x + a2 dx
2
= (ln a)
[a > 0]
2
(x
+
a)
x
0
1
√
(1 − ax) 1 + ax2
dx
1
ln
= √ arctan a ln(1 + a)
2
2
1 + ax
2 a
(1 − ax2 )
0
∞
1
3.
ln
0
4.
0
11
2.
3.
4.
5.
6.8
1 − a2 x2 1 + ax2
(1 −
1
ln
4.311
2
ax2 )
(x + 1) x + a
(x + a)2
2
BI (134)(14)
[a > 0]
BI (115)(25)
√
dx
1
= √ arctan a ln(1 + a)
1 + ax2
a
[a > 0]
BI (115)(26)
2
xμ−1 dx =
π (aμ − 1)
μ sin μπ
π cosec πn
ln (1 + xn )
dx =
xn
n−1
0
∞
dx
2π
ln 1 + x3
= √ ln 3
2
1
−
x
+
x
3
0
∞
dx
π2
π
√
ln
3
−
ln 1 + x3
=
1 + x3
9
3
0
∞
x dx
π2
π
√
ln
3
+
ln 1 + x3
=
1 + x3
9
3
0
∞
1−x
2
ln 1 + x3
dx = − π 2
3
1+x
9
0
√
!
∞!
3!
!
!1 − x ! dx = − π 3
!
a3 ! x3
6a2
0
BI (134)(17)
GW (325)(21d)
ln
2.
1.
FI II 643a
[|t| < π]
0
4.299
BI (139)(21)
1 − x2 dx = π
1
1 + 2x cos t + x2 dx
=
ln
(1 + x)2
x
2
22.
565
[a > 0,
Re μ > 0]
BI (134)(16)
∞
n = 2, 3, . . .
LI (136)(8)
LI (136)(6)
LI (136)(7)
BI (136)(9)
566
4.312
Logarithmic Functions
∞
1.
ln
π2
1 + x3 dx
π
√
ln
3
+
=
x3 1 + x3
9
3
BI (138)(12)
ln
π2
1 + x3 x dx
π
= √ ln 3 −
3
3
x
1+x
9
3
BI (138)(13)
0
∞
2.
0
4.313
1.
2.
∞
dx
ln x ln 1 + a2 x2 2 = πa (1 − ln a)
x
0
∞
dx
b
ln 1 + c2 x2 ln a2 + b2 x2 2 = 2π c +
x
a
0
∞
3.
0
b
ln 1 + c2 x2 ln a2 + 2
x
2
∞
4.
[a > 0]
BI (141)(7)
[a > 0]
LI (141)(8)
π 2 + (ln a)
x ln x − x − a dx
=
(x + a)2
x
1+a
∞
0
ak p + k
xp−1 − xq−1
dx =
(−1)k+1 ln
ln x
k
q+k
k=1
∞
2.
0
(q − 1)x
1
1
+
−
(1 + x)2
x + 1 (1 + x)q
[|a| < 1,
1
3.
4.
0
ln(1 + x)dx = ln
2
0
x ln x + 1 − x
1
x (ln x)
ln 1 − x2 dx
2
x q 2 + (ln x)
=−
π
ln Γ
q
p > 0,
q > 0]
BI (123)(18)
dx
= ln Γ(q)
x ln(1 + x)
[q > 0]
BI (134)(24)
2
ln2 (1 − x)
ln(1 + ax)
b > 0]
BI (134)(22, 23)a
2
(ln a)
x ln x − x − a dx
=
2
(x + a)
x
2(a − 1)
1
1.11
a + bc > 0]
BI (134)(25)
0
4.314
[a > 0,
[a ≥ |b|]
0
∞
a + bc
dx
a
ln(a + bc) − ln a − c
= 2π
x2
b
b
BI (134)(20, 21)a
b
a2 + 2bx + x2 dx
= 2π ln a arcsin
2
2
a − 2bx + x x
a
∞
7.
ln x ln
ln(1 + x)
b
ln b − c ln c
a
[a > 0, b > 0, c > 0]
ln(b + ac) −
[a > 0,
0
6.
BI (134)(18)
1 + a2 x2 dx
bb
=
π(a
−
b)
+
π
ln
1 + b2 x2 x2
aa
∞
5.
[a > 0]
ln x ln
0
4.312
4
π
q+π
π
BI (143)(7)
BI (126)(12)
+
q
π
ln 2q + ln − 1
2q
π
[q > 0]
LI (327)(12)a
4.317
4.315
Logarithmic functions and powers
1
1.
ln(1 + x) (ln x)
n−1
ln(1 + x) (ln x)
2n
ln(1 − x) (ln x)
n−1
ln(1 − x) (ln x)
2n
0
1
2.
0
1
3.
0
1
4.
0
4.316
1
ln (1 − axr ) ln
1.
0
2.
0
4.317
1.
2.
3.
4.
5.
6.
7.
8.
9.
1
x
dx
1
= (−1)n−1 (n − 1)! 1 − n
x
2
ζ(n + 1)
BI (116)(3)
22n+1 − 1
dx
=
π 2n+2 |B2n+2 |
x
(2n + 1)(2n + 2)
BI (116)(1)
dx
= (−1)n (n − 1)! ζ(n + 1)
x
BI (116)(4)
22n
dx
=−
π 2n+2 |B2n+2 |
x
(n + 1)(2n + 1)
p
BI (116)(2)
∞
ak
1
dx
= − p+1 Γ(p + 1)
x
r
k p+2
k=1
[p > −1,
1
567
1
ln 1 − 2ax cos t + a2 x2 ln
x
p
dx
= −2 Γ(p + 1)
x
∞
k=1
a < 1,
r > 0]
ak cos kt
k p+2
√
1 + x2 + a
dx
√
ln √
= π arcsin a
[|a| < 1]
2−a
1
+ x2
1
+
x
0
√
1 √
1
1 − a2 x2 − x 1 − a2 dx
2
= (arcsin a)
ln
1−x
x
2
0
v−t
√
1
cos
dx
1 + cos t 1 − x2
2
√
= π cot t
ln
v+t
1 − cos t 1 − x2 x2 + tan2 v
0
sin
2
√
1
π2
x + 1 − x2
x dx
√
ln2
=
2
2
x − 1 − x2 1 − x
0
1 √
√
dx
π
1
= ln(4k) K(k) + K (k )
ln
1 + kx + 1 − kx 2
2
2
4
8
(1 − x ) (1 − k x )
0
1 √
√
3
1
dx
= ln(4k) K(k) + π K (k )
ln
1 + kx − 1 − kx 2
2
2
4
8
(1 − x ) (1 − k x )
0
1 dx
π
1
= ln k K(k) + K (k )
ln 1 + 1 − k 2 x2 2
4
(1 − x2 ) (1 − k 2 x2 )
0
1 dx
3
1
= ln k K(k) − π K (k )
ln 1 − 1 − k 2 x2 2 ) (1 − k 2 x2 )
2
4
(1
−
x
0
√
1
2
1 + p 1 − x2 dx
√
= π arcsin p
p <1
ln
2
1
−
x
1−p 1−x
0
BI (116)(7)
LI (116)(8)
∞
BI (142)(11)
BI (115)(32)
BI (115)(30)
BI (115)(31)
BI (121)(8)
BI (121)(9)
BI (121)(6)
BI (121)(7)
BI (115)(29)
568
Logarithmic Functions
√
dx
1 + q 1 − k2 x2
√
= π F (arcsin q, k )
ln
2
2
2
1−q 1−k x
(1 − x ) (1 − k 2 x2 )
0
2
q <1
!
√
∞ !!
!
π2
! 1 + 2 1 + x2 ! dx
√
√
ln !
=
!
! 1 − 2 1 + x2 ! 1 + x2
3
−∞
10.
11.10
4.318
4.318
1
1
1.
0
BI (122)(15)
ln (1 − xq ) dx
q
ln q
q
q
= π ln Γ
+1 −
+
ln
−1
2
2π
2
2π
2π
1 + (ln x) x
%
∞
ln (1 + xr )
2.
0
xq − xp
(p − r)xp − (q − r)xq
+
2
ln x
(ln x)
&
[q > 0]
BI (126)(11)
pπ
qπ
dx
cot
= r ln tan
xr+1
2r
2r
[p < r,
q < r]
BI (143)(9)
In integrals containing ln (a + bxr ), it is useful to make the substitution xr = t and then to seek the
resulting integral in the tables. For example,
∞
1 ∞ p −1
π
xp−1 ln (1 + xr ) dx =
t r ln(1 + t) dt =
(see 4.293 3)
pπ
r 0
0
p sin
r
4.319
∞
dx
1
−2aπx
ln 2aπ + a (ln a − 1) − ln Γ(a + 1)
ln 1 − e
= −π
1.
1 + x2
2
0
[a > 0]
∞
dx
1
ln 1 + e−2aπx
= π ln Γ(2a) − ln Γ(a) + a (1 − ln a) − 2a −
2
1
+
x
2
0
ln 2
2.
∞
3.
ln
0
[a > 0]
b
> −1,
a
a
p
a + be−px dx
= ln
ln
a + be−qx x
a+b q
BI (354)(6)
BI (354)(7)
pq > 0
FI II 635, BI (354)(1)
4.321
∞
x ln cosh x dx = 0
1.
−∞
∞
2.
ln cosh x
−∞
4.322
1.11
dx
=0
1 − x2
π
x ln sin x dx =
0
2.
0
∞
BI (358)(2)a
1
2
π
x ln cos2 x dx = −
0
ln sin2 ax
π 1 − e−2ab
ln
dx
=
b2 + x2
b
2
BI (138)(20)a
π2
ln 2
2
BI (432)(1, 2) FI II 643
[a > 0,
b > 0]
GW (338)(28b)
4.324
Logarithmic functions and powers
∞
3.
0
∞
4.
0
∞
5.11
0
∞
6.
0
ln cos2 ax
π 1 + e−2ab
ln
dx
=
b2 + x2
b
2
[a > 0,
b > 0]
GW (338)(28a)
ln sin2 ax
π2
+ aπ
dx
=
−
b2 − x2
2b
[a > 0,
b > 0]
BI (418)(1)
ln cos2 ax
dx = ∞
b2 − x2
BI (418)(2)
ln cos2 x
dx = −π
x2
FI II 686
π/4
7.7
0
ln sin xxμ−1 dx = −
0
1
μ
ln (1 − cos x) xμ−1 dx = −
0
∞
10.
ln 1 ± 2p cos βx + p2
0
%
∞
ζ(2k)
2 ln 2 + −
μ
42k−1 (μ + 2k)
&
k=1
μ
π
2
π/2
9.
μ
π
4
1
ln sin xxμ−1 dx = −
2μ
π/2
8.7
569
%
[Re μ > 0]
&
LI (425)(1)
[Re μ > 0]
&
ζ(2k)
42k−1 (μ + 2k)
LI (430)(1)
[Re μ > 0]
LI (430)(2)
∞
ζ(2k)
1
−2
μ
4k (μ + 2k)
k=1
1
μ
π
2
μ
%
2
−
μ
∞
k=1
dx
π = ln 1 ± pe−βq
q 2 + x2
q
π = ln p ± e−βq
q
p2 < 1
p2 > 1
FI II 718a
4.323
π
x ln tan2 x dx = 0
1.11
BI (432)(3)
0
∞
2.
0
ln tan2 ax
π
dx = ln tanh ab
2
2
b +x
b
∞
3.
ln
1 + tan x
1 − tan x
ln
1 + sin x
1 − sin x
0
4.324
∞
1.
0
2.
∞
ln
0
2
2
[a > 0,
b > 0]
GW (338)(28c)
π2
dx
=
x
2
GW (338)(26)
dx
= π2
x
GW (338)(25)
q2
1 + 2a cos px + a2 dx
=
ln(1
+
a)
ln
1 + 2a cos qx + a2 x
p2
1
q2
= ln 1 +
ln 2
a
p
[−1 < a ≤ 1]
[a < −1 or a ≥ 1]
GW (338)(27)
570
Logarithmic Functions
∞
3.
ln a2 sin2 px + b2 cos2 px
0
4.325
π
dx
= [ln (a sinh cp + b cosh cp) − cp]
c2 + x2
c
[a > 0, b > 0, c > 0,
p > 0]
GW (338)(29)
4.325
1
1.3
ln ln
0
1
x
∞
dx
1
ln k
2
= −C ln 2 +
= −C ln 2 + 0.159868905 · · · = − (ln 2)
(−1)k
1+x
k
2
k=2
GW (325)(25a)
1
2.
ln ln
1
x
ln ln
1
x
0
1
3.
0
1
4.
ln ln
1
x
ln ln
1
x
ln ln
1
x
0
1
5.
0
1
6.
0
1
7.
ln ln
0
1
x
∞
(−1)k −ikλ
e
(C + ln k)
k
k=1
∞
dx
dx
1
=
ln ln x
=
ψ
(1 + x)2
(1 + x)2
2
1
dx
=
x + eiλ
GW (325)(26)
1
2
+ ln 2π =
π
1
ln − C
2
2
√
2π Γ 34
dx
dx
π
=
ln ln x
= ln
1 + x2
1 + x2
2
Γ 14
1
√
∞
3
2π Γ 23
dx
dx
π
=
ln ln x
= √ ln
1 + x + x2
1 + x + x2
Γ 13
3
1
∞
dx
dx
2π 5
√
ln 2π − ln Γ
=
ln
ln
x
=
1 − x + x2
1 − x + x2
3 6
1
BI (147)(7)
∞
dx
=
1 + 2x cos t + x2
∞
ln ln x
1
dx
π
ln
=
1 + 2x cos t + x2
2 sin t
BI (148)(1)
BI (148)(2)
1
6
BI (148)(5)
1
t
+
2 2π
1
t
−
2 2π
(2π)t/π Γ
Γ
BI (147)(9)
8.
0
9.
1
1
1
ln ln xμ−1 dx = − (C + ln μ)
x
μ
∞
ln ln x
1
[Re μ > 0]
BI (147)(1)
xn−2 dx
1 + x2 + x4 + · · · + x2n−2
Γ
n−1
π
π
π
kπ
k−1
tan
ln 2π +
ln
=
(−1)
sin
2n
2n
n
n
k=1
n−1
2
π π
π
tan
ln π +
=
2n
2n
n
k=1
kπ
ln
(−1)k−1 sin
n
n+k
2n
k
Γ
2n
Γ
n−k
n
k
Γ
n
[n is even]
[n is odd]
BI (148)(4)
4.331
Logarithms and exponentials
1
10.
ln ln
0
1
x
dx
(1 + x2 )
ln
1
x
∞
ln ln x
=
1
=
571
dx
√
(1 + x2 ) ln x
∞
√ (−1)k+1
√
π
[ln(2k + 1) + 2 ln 2 + C]
2k + 1
k=0
BI (147)(4)
1
11.
ln ln
1
x
ln ln
1
x
0
1
12.
0
4.326
xμ−1 dx
= − (C + ln 4μ)
1
ln
x
ln
μ−1
1
x
xν−1 dx =
1
ln (a − ln x) xμ−1 dx =
1.
0
2.
4.327
1
e ln 2 ln 1 − 1
x
0
1.
1
,
+
2
ln a2 + (ln x)
0
π
μ
x2μ−1
1
2
dx = − [Ei(−μ)]
ln x
2
2 Γ 2a+3π
dx
π π
4π
2a+π + ln
= π ln
2
1+x
2
2
Γ 4π
1
+
, dx
2 Γ a+3π
π
2
2
4π
a+π + ln π
ln a + 4 (ln x)
= π ln
2
1+x
2
Γ 4π
0
∞
3.
0
[Re μ > 0]
BI (147)(3)
1
Γ(μ) [ψ(μ) − ln(ν)]
νμ
1
[ln a − eaμ Ei(−aμ)]
μ
2.
[Re μ > 0,
Re ν > 0]
[Re μ > 0,
a > 0]
BI (147)(2)
BI (107)(23)
[Re μ > 0]
BI (145)(5)
+
π,
a>−
2
BI (147)(10)
[a > −π]
,
+
2
2
ln a2 + (ln x) xμ−1 dx = [− cos aμ ci(aμ) − sin aμ si(aμ) + ln a]
μ
[a > 0,
Re μ > 0]
BI (147)(16)a
GW (325)(28)
If the integrand contains a logarithm whose argument also contains a logarithm, for example, if the
integrand contains ln ln x1 , it is useful to make the substitution ln x = t and then seek the transformed
integral in the tables.
4.33–4.34 Combinations of logarithms and exponentials
4.331
∞
1.
e−μx ln x dx = −
1
(C + ln μ)
μ
[Re μ > 0]
BI (256)(2)
e−μx ln x dx = −
1
Ei(−μ)
μ
[Re μ > 0]
BI (260)(5)
0
2.
1
∞
572
Logarithmic Functions
1
eμx ln x dx = −
3.
0
4.332
1.
2.
1
μ
−1
dx
x
1 μx
0
e
1
2π 5
ln x dx
=√
ln 2π − ln Γ
x
−x − 1
6
3 6
0 e +e
% &
∞
2 √
Γ 3
π
ln x dx
2π
= √ ln
x
−x
+1
Γ 13
3
0 e +e
4.332
[μ = 0]
GW (324)(81a)
∞
(cf. 4.325 6)
BI (257)(6)
(cf. 4.325 5)
BI (257)(7)a, LI (260)(3)
2
π
1
[Re μ > 0]
e−μx ln x dx = − (C + ln 4μ)
4
μ
0
∞
∞
ln x dx
C + ln 4k
kπ
1 π
√
sin
4.334
=
(−1)k
2
2
x
−x
2 3
3
+1+e
k
0 e
∞
4.333
BI (256)(8), FI II 807a
BI (357)(13)
k=1
4.335 1.
2.
3.7
1 π2
2
+ (C + ln μ)
[Re μ > 0]
μ 6
0
√ ∞
π
π2
2
2
−x2
e
(ln x) dx =
(C + 2 ln 2) +
8
2
0
∞
1
π2
3
3
(C + ln μ) − ψ (1)
e−μx (ln x) dx = −
(C + ln μ) +
μ
2
0
∞
4.336
1.7
0
∞
2.
4.337 1.
0
4.7
5.∗
FI II 808
MI 26
e
dx = −0.154479567
ln x
e−μx dx
2
∞
ET I 149(13)
∞ −x
π 2 + (ln x)
0
3.
2
PV
2.
e−μx (ln x) dx =
= ν (μ) − eμ
e−μx ln(β + x)dx =
1
ln β − eμβ Ei(−βμ)
μ
BI (260)(9)
[Re μ > 0]
MI 26
[|arg β| < π,
Re μ > 0]
BI (256)(3)
∞
1 μ
μ
e−μx ln(1 + βx)dx = − e β Ei −
ET I 148(4)
[|arg β| < π, Re μ > 0]
μ
β
0
∞
1
ln a − e−aμ Ei(aμ)
e−μx ln |a − x| dx =
[a > 0, Re μ > 0]
BI (256)(4)
μ
0
!
!
∞
! β !
! dx = 1 e−βμ Ei(βμ)
e−μx ln !!
[Re μ > 0]
MI 26
!
β−x
μ
0
%
&
∞
ζ
ζ−μ
ζ−μ−k
(−1)ζ−μ−1 1/a
ζ!
1
1
ζ −x
ln(1 + ax)x e dx =
e Ei −
(k − 1)! −
+
(ζ − μ)!
aζ−μ
a
a
0
μ=0
k=1
4.338 1.
0
∞
2
e−μx ln β 2 + x2 dx = [ln β − ci(βμ) cos(βμ) − si(βμ) sin(βμ)]
μ
[Re β > 0,
Re μ > 0]
BI (256)(6)
4.352
Logarithms, exponentials, and powers
∞
2.
0
4.341
1.
2.
3.11
2 2
ln β − eβμ Ei(−βμ) − eβμ Ei(βμ)
e−μx ln2 x2 − β 2 dx =
μ
[Im β > 0, Re μ > 0]
!
!
∞
!x + 1!
! dx = 1 e−μ (ln 2μ + γ) − eμ Ei(−2μ)
e−μx ln !!
!
x−1
μ
0
[Re μ > 0]
√
√
∞
x
+
ai
+
x
−
ai
π
√
[H0 (aμ) − Y 0 (aμ)]
dx =
e−μx ln
4μ
2a
0
[a > 0, Re μ > 0]
4.339
4.342
573
BI (256)(5)
MI 27
ET I 149(20)
1 1
+ ln 2 − 2 β(2n + 1)
e
ln (sinh x) dx =
2n n
0
∞
μ
1
1
e−μx ln (cosh x) dx =
β
−
[Re μ > 0]
μ
2
μ
0
∞
1
μ
1
μ
−μx
e
[ln (sinh x) − ln x] dx =
ln − − ψ
μ
2
μ
2
0
ET I 165(32)
[Re μ > 0]
ET I 165(33)
∞
π
4.343
−2nx
BI (256)(17)
eμ cos x ln 2μ sin2 x + C dx = −π K 0 (μ)
WA 95(16)
0
4.35–4.36 Combinations of logarithms, exponentials, and powers
4.351
1
1.
(1 − x)e−x ln x dx =
0
2.
0
1
e
∞
1.
0
∞
2.
0
3.
0
BI (352)(2)
∞ −μx
1
BI (352)(1)
1
eμx μx2 + 2x ln x dx = 2 [(1 − μ)eμ − 1]
μ
3.
4.352
1−e
e
∞
1
ln x
2
dx = eμ [Ei(−μ)]
1+x
2
xν−1 e−μx ln x dx =
1
Γ(ν) [ψ(ν) − ln μ]
μν
[Re μ > 0]
[Re μ > 0,
1
n!
1 1
xn e−μx ln x dx = n+1 1 + + + · · · + − C − ln μ
μ
2 3
n
[Re μ > 0]
√ (2n − 1)!!
1
1
1 1
xn− 2 e−μx ln x dx = π
2 1 + + + ··· +
1
n+
n
3 5
2n − 1
2 μ 2
[Re μ > 0]
NT 32(10)
Re ν > 0]
BI (353)(3), ET I 315(10)a
− C − ln 4μ
ET I 148(7)
ET I 148(10)
574
Logarithmic Functions
∞
4.
4.353
xμ−1 e−x ln x dx = Γ (μ)
[Re μ > 0]
GW (324)(83a)
(x − ν)xν−1 e−x ln x dx = Γ(ν)
[Re ν > 0]
GW (324)(84)
[Re μ > 0]
BI (357)(2)
0
4.353
∞
1.
0
∞
1
μx − n −
2
2.
0
n− 12 −μx
x
e
(2n − 1)!!
ln x dx =
(2μ)n
1
(μx + n + 1)xn eμx ln x dx = eμ
3.
0
n
(−1)k−1
k=0
π
μ
n!
n!
+ (−1)n n+1
k+1
(n − k)!μ
μ
[μ = 0]
4.354
1.
∞
6
0
GW (324)(82)
∞
(−1)k−1
xν−1 ln x
dx
=
Γ(ν)
[ψ(ν) − ln k]
ex + 1
kν
k=1
1
2
= − (ln 2)
2
[Re ν > 0]
[for ν = 1]
GW (324)(86a)
∞
2.7
∞
3.
∞
4.
0
∞
1.
0
∞
2.
0
3.
0
+ 1)
∞
(−1) (k − 1)
[ψ(ν) − ln k]
kν
k
xν−1 ln x dx = Γ(ν)
x2n−1 ln x dx =
xν−1 ln x
n Γ(ν)
n dx = (−1)
x
(n − 1)!
(e + 1)
2 −μx2
∞
k=1
(x − 2n)ex − 2n
x e
2
(ex + 1)
∞
∞
k=2
2
5.
4.355
+ 1)
dx = Γ(ν)
(x − ν)ex − ν
0
ln x
2
(ex
0
x
(ex
0
ν−1
(−1)k−1
kν
[Re ν > 1]
GW (324)(86b)
[Re ν > 0]
GW (324)(87a)
[n = 1, 2, . . .]
GW (324)(87b)
22n−1 − 1 2n
π |B2n |
2n
∞
k=n
(−1)k (k − 1)!
[ψ(ν) − ln k]
(k − n)!k ν
1
(2 − ln 4μ − C)
ln x dx =
8μ
π
μ
2
ν
1
+
x μx2 − νx − 1 e−μx +2νx ln x dx =
4μ 4μ
2
2
(n − 1)!
μx − n x2n−1 e−μx ln x dx =
4μn
[Re ν > 0]
GW (324)(86c)
[Re μ > 0]
BI (357)(1)a
π
exp
μ
ν2
μ
1+Φ
ν
√
μ
[Re μ > 0]
BI (358)(1)
[Re μ > 0]
BI (353)(4)
4.356
Logarithms, exponentials, and powers
∞
4.
0
2
(2n − 1)!!
2μx2 − 2n − 1 x2n e−μx ln x dx =
2(2μ)n
575
π
μ
[Re μ > 0]
4.356
1.
2.
BI (353)(5)
+
dx
x a ,
+
ln x
= 2 ln a K 0 (2μ)
exp −μ
[a > 0, Re μ > 0]
GW (324)(91)
a
x
x
0
∞
1
b
exp −ax −
ln x 2ax2 − (2n + 1)x − 2b xn− 2 dx
x
0
n
∞
b 2
π −2√ab (n + k)!
e
=2
k
√
a
a
k=0 (n − k)!(2k)!! 2 ab
∞
[a > 0,
∞
3.
exp −ax −
0
b
x
b > 0]
dx
ln x 2ax2 + (2n − 1)x − 2b n+ 3
x 2
n
∞
π −2√ab a 2
e
=2
b
a
k=0
2ax2 − x − 2b
√
dx = 2
ln x
x x
(n + k − 1)!
√
(n − k − 1)!(2k)!! 2 ab
k
[a > 0,
b > 0]
BI (357)(11)
[a > 0,
b > 0]
GW (324)(92c)
π −2√ab
e
a
[a > 0,
b > 0]
For n = 12 :
∞
√
ax2 − b
a
ln x
exp −ax −
dx = 2 K 0 2 ab
4.
2
x
x
0
For n = 0:
∞
b
5.
exp −ax −
x
0
BI (357)(4)
BI (357)(7), GW(324)(92a)
For n = −1:
∞
b
exp −ax −
6.
x
0
7.9
0
ln x
√ 1 + 2 ab π −2√ab
2ax2 − 3x − 2b
√
e
dx =
a
a
x
[a > 0, b > 0]
LI (357)(6), GW (324)(92b)
∞
exp −ax −
b
x
ln x a −
b
x2
√
dx = K 0 2 ab
[a > 0,
b > 0]
576
Logarithmic Functions
8.
∞
9
exp −ax −
0
3
ln x 2ax2 − (2n + 1)x − 2b xn− 2 dx
b
x
=4
=2
∞
9.9
exp −ax −
0
4.357
b
a
b
a
(2n+1)/4
n
2
√
K n+ 12 2 ab
π −2√ab e
a
n
k
√
(n − k)!(2k)!! 2 ab
[n = 0, 1, . . . , a > 0, b > 0]
dx
ln ax2 − b cos (α ln x) + αx sin (α ln x) 2
x
√
= 2 cos α ln b/a K iα 2 ab
b
x
k=0
[a > 0,
∞
10.9
exp −ax −
0
(n + k)!
b
x
ln x
b > 0,
dx
ax2 − b sin (α ln x) − αx cos (α ln x) 2
x
−∞ < α < ∞]
= 2 sin α ln
∞
b
α
b
q
xα ln x a − − 2 exp −ax −
x
x
x
0
[a > 0,
11.9
dx = 2
b
a
α/2
∞
1.
0
∞
2.
0
∞
3.
0
1 + x4
exp −
2ax2
exp −
1 + x4
2ax2
1 + x4
exp −
2ax2
√
b/a K iα 2 ab
−∞ < α < ∞]
√
K α 2 ab
[a > 0,
4.357
b > 0,
b > 0,
−∞ < α < ∞]
√
2a3 π
1 + ax2 − x4
dx = − √
ln x
2
x
2ae
√
2a3 π
x4 + ax2 − 1
√
dx =
ln x
4
x
2ae
[a > 0]
BI (357)(8)
[a > 0]
BI (357)(9)
√
(1 + a) 2a3 π
x4 + 3ax − 1
√
dx =
ln x
x6
2ae
[a > 0]
4.358
1.
∞
6
1
0
3.9
0
m
dx =
∂ m −ν
μ Γ(ν, μ)
∂ν m
[m = 0, 1, . . . ,
Re μ > 0,
Re ν > 0]
MI 26
∞
2.
xν−1 e−μx (ln x)
BI (357)(10)
∞
Γ(ν) 2
2
xν−1 e−μx (ln x) dx = ν [ψ(ν) − ln μ] + ζ(2, ν)
μ
[Re μ > 0, Re ν > 0]
Γ(ν)
3
3
xν−1 e−μx (ln x) dx = ν [ψ(ν) − ln μ] + 3 ζ(2, ν) [ψ(ν) − ln μ] − 2 ζ(3, ν)
μ
[Re μ > 0,
Re ν > 0]
MI 26
MI 26
4.364
Logarithms, exponentials, and powers
4.
∞
7
Γ(ν)
ν
xν−1 e−μx (ln x) dx =
4
577
4
[ψ(ν) − ln μ] + 6 ζ(2, ν) [ψ(ν) − ln μ]
2
−8 ζ(3, ν) [ψ(ν) − ln μ] + 3 [ζ(2, ν)] + 6 ζ(4, ν)
0
2
[Re μ > 0,
∞
5.3
∂ n −ν
μ Γ(ν)
n
∂ν
xν−1 e−μx (ln x) dx =
n
0
4.359
∞
1.
e−μx
0
[n = 0, 1, 2, . . .]
xp−1 − xq−1
1
dx = [λ(μ, p − 1) − λ(μ, q − 1)]
ln x
μ
[Re μ > 0,
1
eμx
2.11
0
4.361
∞
1.
∞
2.
(x + 1)e−μx
+
2
k=0
[p > 0,
q > 0]
∞
2.
∞
1.
0
2.
[Re μ > 0]
MI 27
2
e−μx ln(2x − 1)
1
e−μx ln(a + x)
0
[Re μ > 0]
ET I 148(8)
μ(x + a) ln(x + a) − 2
dx
x+a
∞
1
μ(x − a) ln2 (x − a) − 4
2
dx = (ln a)
=
e−μx ln2 (a − x)
4 0
x−a
[Re μ > 0, a > 0]
BI (354)(4, 5)
x(1 − x)(2 − x)e−(1−x) ln(1 − x) dx =
∞
BI (352)(5)a
1+
μ ,2
dx
=
Ei −
x
2
2
2
0
1.
BI (352)(9)
, = eμ − ν(μ)
e−μx dx
1
MI 27
MI 27
xex ln(1 − x) dx = 1 − e
q > 0]
[Re μ > 0]
0
4.364
μk p + k
ln
k!
q+k
p > 0,
1
1.
4.363
∞
dx = ν (μ) − ν (μ)
x π 2 + (ln x)
0
4.362
xp−1 − xq−1
dx =
ln x
π 2 + (ln x)
0
Re ν > 0]
e−μx ln[(x + a)(x + b)]
1−e
4e
BI (352)(4)
dx
= e(a+b)μ {Ei(−aμ) Ei(−bμ) − ln(ab) Ei[−(a + b)μ]}
x+a+b
[a > 0, b > 0, Re μ > 0] BI (354)(11)
578
Logarithmic Functions
∞
2.
1
1
+
x+a x+b
e−μx ln(x + a + b)
0
4.365
dx
= (1 + ln a ln b) ln(a + b) + e−(a+b)μ {Ei(−αμ) Ei(−bμ)}
+ (1 − ln(ab)) Ei[−(a + b)μ]
[a > 0,
∞
4.365
e−x −
0
4.366
1.
2.
3.
∞
x
(1 + x)p+1 ln(1 + x)
dx
= ln p
x
b > 0,
[p > 0]
e−μx ln 1 +
[Re μ > 0] ;
∞
BI (354)(12)
BI (354)(15)
x2 dx
2
2
= [ci(aμ)] + [si(aμ)]
[Re μ > 0]
2
a
x
0
!
!
∞
!
x2 ! dx
= Ei(aμ) Ei(−aμ)
[Re μ > 0]
e−μx ln !!1 − 2 !!
a x
0
!
!
∞
! 1 + x2 !
−μx2
! dx = 1 [cosh μ sinh(iμ) − sinh μ cosh(iμ)]
!
xe
ln !
2
1−x !
μ
0
NT 32(11)a
ME 18
(cf. 4.339)
MI 27
x2 + 2β
eβμ
√
K 0 (βμ)
[|arg β| < π, Re μ > 0]
ET I 149(19)
dx =
4μ
2β 0 2u
+
2 2 ,
x2 4u2 − x2
2
dx
π −2u2 μ π
√
e
Y
2iu
e−μx ln
=
μ
−
(C
−
ln
2)
J
0
0 2iu μ
u4
2
2
4u2 − x2
0
ET I 149(21)a
[Re μ > 0]
4.367
4.368
4.369
Re μ > 0]
∞
1.
xe−μx ln
∞
x+
xν−1 e−μx [ψ(ν) − ln x] dx =
0
2.
2
xn e−μx
ln x −
0
1
2
Γ(ν) ln μ
μν
2
ψ(n + 1) −
1
2
[Re ν > 0]
ψ (n + 1) dx
=
n!
μn+1
ET I 149(12)
2
1 1
ln μ − ψ(n + 1) + ψ (n + 1)
2
2
[Re μ > 0]
MI 26
4.37 Combinations of logarithms and hyperbolic functions
4.371
∞
1.
0
2.
0
∞
ln x
dx = π ln
cosh x
%√
&
2π Γ 34
Γ 14
π
ln x dx
=
ln
cosh x + cos t
sin t
LI (260)(1)a
π+t
2π
π−t
2π
(2π)t/π Γ
Γ
t2 < π 2
BI (257)(7)a
4.374
Logarithms and hyperbolic functions
∞
3.
0
4.372
ln x dx
=ψ
cosh2 x
∞
1.
ln x
1
1
2
579
+ ln π = ln π − 2 ln 2 − C
BI (257)(4)a
n−1
π
mπ
π
sinh mx
kmπ Γ n+k
dx =
tan
ln 2π +
ln 2n
(−1)k−1 sin
k
sinh nx
2n
2n
n
n
Γ 2n
k=1
n−1
2
π
mπ
π
kmπ Γ n−k
k−1
n
=
tan
ln π +
ln
(−1)
sin
2n
2n
n
n
Γ nk
k=1
[m + n is odd]
[m + n is even]
BI (148)(3)a
∞
2.
ln x
1
cosh mx
dx
cosh nx
n
π
(2k − 1)mπ Γ 2n+2k−1
π ln 2π
k−1
4n
ln
+
(−1)
cos
=
2n cos mπ
n
2n
Γ 2k−1
2n
4n
k=1
n−1
2n−2k+1 2
π ln π
π
(2k − 1)mπ Γ
2n
2k−1
=
ln
(−1)k−1 cos
mπ +
2n cos 2n
n
2n
Γ
2n
k=1
[m + n is odd]
[m + n is even]
BI (148)(6)a
4.373
∞
1.
0
2.
3.
∞
%
&
ln a2 + x2
2 Γ 2ab+3π
2b
π
4π
− ln
dx =
2 ln
cosh bx
b
π
Γ 2ab+π
4π
+
b > 0,
ln 1 + x2
dx
4
πx = 2 ln
cosh
π
0
2
a+5 2 ∞
2
Γ 6
sinh
π 6 Γ a+4
2
3 πx
6
dx = 2 sin ln a+1 a+2
ln a + x
sinh πx
3
Γ 6 Γ 6
0
∞
4.
ln 1 + x2
0
π
dx
2
a
−ψ
=
ln +
a
π 2a
sinh2 ax
π+a
π
[a > −1] .
[a > 0]
π
cosh 2 x
2π − 4
ln 1 + x
dx =
π
2
π
0
x
sinh
2
π
√
∞
√
√
cosh 4 x
16 8 2
2
dx
=
4
+
ln
ln 1 + x
2
−
2+1
π
π
π
0
x
sinh2
4
5.
6.
4.374
1.
0
∞
∞
2
ln cos2 t + e−2x sin2 t
dx
= −2t2
sinh x
a>−
π,
.
2b
BI (258)(11)a
BI (258)(1)a
BI (258)(12)
BI (258)(5)
BI (258)(3)
BI (258)(2)
BI (259)(10)a
580
Logarithmic Functions
∞
2.
ln a + be
0
−2x
4.375
a+b
dx
2
ln(a + b) − a ln a − b ln 2
=
(b − a)
2
cosh2 x
[a > 0,
4.375
∞
1.11
ln cosh
0
ln coth x
0
4.376
∞
1.
0
BI (259)(11)
π
dx
= ln 2
cosh x
2
BI (259)(16)
∞
√ (−1)k+1
ln x
√
√
dx = 2 π
{ln(2k + 1) + 2 ln 2 + C}
x cosh x
2k + 1
∞
ln x
0
LI (259)(14)
BI (147)(4)
k=0
2.
∞
∞
(−1)k+1
(μ + 1) cosh x − x sinh x μ
dx
=
2
Γ(μ
+
1)
x
2
(2k + 1)μ+1
cosh x
k=0
[Re μ > −1]
BI (356)(10)
[n = 1, 2, . . .]
BI (356)(9)a
(n + 1) cosh x − x sinh x n
(−1) (n) 1
β
x dx =
2
2n
2
cosh x
0
∞
2n
n sinh 2ax − ax 2n−1
1 π
ln 2x
dx = −
|B2n |
x
2
n
a
sinh
ax
0
3.
n
ln x
4.
∞
5.
ln x
∞
6.
ln x
0
∞
7.
ln
0
6
−1
ax cosh ax − (2n + 1) sinh ax 2n
2
(2n)! ζ(2n + 1)
x dx = 2
2
2n+1
(2a)
sinh ax
2n+1
0
8.
π
x dx
= G − ln 2
2 cosh x
4
∞
2.
a + b > 0]
ax cosh ax − 2n sinh ax 2n−1
22n−1 − 1
π 2n
|B2n |
dx =
x
2
2n
a
sinh ax
[n = 1, 2, . . . , a > 0]
(2n + 1) cosh ax − ax sinh ax 2n
π
x dx = −
2
2a
cosh ax
2n+1
9.6
ln x
0
BI (356)(11)
2n
|B2n | n = 1, 2, . . .
[a > 0]
∞
2ax cosh ax − (2n + 1) sinh ax 2n
1
x dx =
a
sinh3 ax
π
a
2n
BI (356)(15)
|E2n |
[a > 0]
⎧2 π
⎪
22n−1 − 1
⎪
∞
⎨
a
2a
2ax sinh ax − (2n + 1) cosh ax 2n
ln x
x dx = 1
3
⎪
cosh ax
0
⎪
⎩
a
BI (356)(14)
n=0
BI (356)(2)
|B2n |
[a > 0,
n = 1, 2, . . .]
BI (356)(6)a
4.382
Logarithms and trigonometric functions
ln x
0
x
t
2
− 6 cos2
2
2 x2 dx = π − t t
2
3 sin t
(cosh x + cos t)
x sinh x − 6 sinh2
∞
10.
581
[0 < t < π]
BI (356)(16)a
∞
cosh πx + πx sinh πx dx
ln 1 + x2
=4−π
x2
cosh2 πx
0
∞
cosh πx + πx sinh πx dx
12.
ln 1 + 4x2
= 4 ln 2
x2
cosh2 πx
0
∞
ax − n 1 − e−2ax 2n−1
1 π
4.377
ln 2x
dx =
x
2
2n
a
sinh ax
0
11.
BI (356)(12)
BI (356)(13)
2n
|B2n |
[n = 1, 2, . . .]
LI (356)(8)a
4.38–4.41 Logarithms and trigonometric functions
4.381
1.
2.
3.
4.
4.382
1.
2.10
1
1
ln x sin ax dx = − [C + ln a − ci(a)]
a
0
1
π,
1+
ln x cos ax dx = − si(a) +
a
2
0
2π
1
ln x sin nx dx = − [C + ln(2nπ) − ci(2nπ)]
n
0
2π
1+
π,
ln x cos nx dx = − si(2nπ) +
n
2
0
[a > 0]
GW (338)(2a)
[a > 0]
BI (284)(2)
GW (338)(1a)
GW (338)(1b)
!
!
!x + a!
! sin bx dx = π sin ab
!
[a < 0, b > 0]
ln !
x − a!
b
0
!
∞ !
+
,
!x + a!
! cos bx dx = 2 cos(ab) si(ab) + π − sin(ab) ci(ab)
!
ln !
!
x−a
b
2
0
∞
∞
3.
ln
0
∞
4.
ln
0
5.
x2 + x + a2
2π
exp −b
sin bx dx =
x2 − x + a2
b
a2 −
1
4
b > 0]
[a > 0,
b > 0,
ET I 18(9)
c > 0]
FI III 648a, BI (337)(5)
sin
b
2
[b > 0]
∞
ln
0
a2 + x2
π −bc
e
cos cx dx =
− e−ac
2
2
b +x
c
[a > 0,
ET I 77(11)
(x + β)2 + γ 2
2π −γb
e
sin bx dx =
sin βb
(x − β)2 + γ 2
b
[Re γ > 0,
ET I 77(12)
|Im β| ≤ Re γ,
b > 0]
ET I 77(13)
582
Logarithmic Functions
4.383 1.
∞
0
∞
2.
0
4.384
β
ln 1 + e−βx cos bx dx = 2 −
2b
π
2b sinh
πb
β
β
π
coth
ln 1 − e−βx cos bx dx = 2 −
2b
2b
πb
β
4.383
[Re β > 0,
b > 0]
ET I 18(13)
[Re β > 0,
b > 0]
ET I 18(14)
1
1.
ln (sin πx) sin 2nπx dx = 0
GW (338)(3a)
0
1
2.7
1/2
ln (sin πx) sin(2n + 1)πx dx = 2
ln (sin πx) sin(2n + 1)πx dx
%
&
n
1
1
2
−2
ln 2 −
=
(2n + 1)π
2n + 1
2k − 1
0
0
k=1
GW (338)(3b)
1
3.6
1/2
ln (sin πx) cos 2nπx dx = 2
0
ln (sin πx) cos 2nπx dx
0
= − ln 2
1
=−
2n
4.
[n = 0]
[n > 0]
GW (338)(3c)
1
ln (sin πx) cos(2n + 1)πx dx = 0
GW (338)(3d)
0
π/2
ln sin x sin x dx = ln 2 − 1
5.
BI (305)(4)
0
6.
π/2
ln sin x cos x dx = −1
⎧
⎪
π/2
⎨− π ,
for n > 0
4n
ln sin x cos 2nx dx =
π
⎪
0
⎩− ln 2, for n = 0
2
π
π cos 2mn
ln sin x cos[2m(x − n)] dx = −
2m
0
π/2
π
ln sin x sin2 x dx = (1 − ln 4)
8
0
π/2
π
ln sin x cos2 x dx = − (1 + ln 4)
8
0
π/2
1
ln sin x sin x cos2 x dx = (ln 8 − 4)
9
0
π/2
π2
ln sin x tan x dx = −
24
0
BI (305)(5)
0
7.
8.
9.
10.
11.
12.
LI (305)(6)
LI (330)(8)
BI (305)(7)
BI (305)(8)
BI (305)(9)
BI (305)(11)
4.384
Logarithms and trigonometric functions
π/2
13.
π
0
ln sin x
0
18.
p2 < 1
a2 < 1
a2 > 1
FI II 484
BI (331)(8)
ln sin bx
0
17.
dx
π
1 − a2
=
ln
1 − 2a cos x + a2
1 − a2
2
a2 − 1
π
ln
= 2
a −1
2a2
π
16.
ln (1 + p cos x)
dx = π arcsin p
cos x
π
15.
BI (305)(16, 17)
0
14.
π/2
ln sin 2x cos x dx = 2 (ln 2 − 1)
ln sin 2x sin x dx =
0
583
dx
π
1−a
=
ln
2
2
1 − 2a cos x + a
1−a
2
2b
a2 < 1
BI (331)(10)
2
dx
π
1 + a2b
a <1
=
ln
2
2
1 − 2a cos x + a
1−a
2
0
π/2
π
dx
1
dx
ln sin x
=
ln sin x
2
1
−
2a
cos
2x
+
a
2
1
−
2a
cos
2x + a2
0
0
1−a
π
ln
=
2 (1 − a2 )
2
a−1
π
ln
=
2 (a2 − 1)
2a
π
ln cos bx
ln sin bx
dx
π
1−a
=
ln
1 − 2a cos 2x + a2
1 − a2
2
ln cos bx
dx
π
1 + ab
=
ln
2
2
1 − 2a cos 2x + a
1−a
2
0
π
20.
0
π/2
21.
0
1+p
ln cos x dx
π
ln
=
2
2
1 − 2p cos 2x + p
2 (1 − p )
2
p+1
π
ln
=
2 (p2 − 1)
2p
π
ln sin x
0
ln sin bx
0
24.
a2 > 1
a2 < 1
a2 < 1
p2 < 1
p2 > 1
BI (331)(18)
BI (331)(21)
cos x dx
=
1 − 2a cos 2x + a2
a2 < 1
a2 > 1
LI (331)(9)
π
ln cos bx
0
cos x dx
=0
1 − 2a cos 2x + a2
[0 < a < 1]
π
ln sin x
0
b
aπ ln 2
cos x dx
π 1 + a2 =
ln 1 − a2 −
2
2
1 − 2a cos x + a
2a 1 − a
1 − a2
2
2
π a +1 a −1
π ln 2
ln
=
−
2a a2 − 1
a2
a (a2 − 1)
π
23.
a2 < 1
BI (321)(8)
22.
BI (321)(1), BI (331)(13)
π
19.
BI (331)(11)
π ln 2
cos2 x dx
π 1+a
ln(1 − a) −
=
1 − 2a cos 2x + a2
4a 1 − a
2(1 − a)
π a+1 a−1
π ln 2
=
ln
−
4a a − 1
a
2a(a − 1)
BI (331)(19, 22)
[0 < a < 1]
[a > 1]
BI (331)(16)
584
Logarithmic Functions
π/2
25.
ln sin x
0
cos 2x dx
1
=
1 − 2a cos 2x + a2
2
4.385
π
cos 2x dx
1 − 2a cos 2x + a2
1 + a2
π
2
ln(1
−
a)
−
a
=
ln
2
2a (1 − a2 )
2
1 + a2 a − 1
π
ln
− ln 2
=
2a (a2 − 1)
2
a
ln sin x
0
2
a <1
2
a >1
BI (321)(2), BI (331)(15), LI (321))(2)
π/2
26.
0
cos 2x dx
π
ln cos x
=
2
1 − 2a cos 2x + a
2a (1 − a2 )
π
=
2a (a2 − 1)
1 + a2
2
ln(1 + a) − a ln 2
2
1 + a2 1 + a
ln
− ln 2
2
a
a2 < 1
a2 > 1
BI (321)(9)
4.385
1.
2.
3.
√
π
a2 − b 2
dx
√
=√
ln sin x
ln
[a > 0, a > b]
2
2
a + b cos x
a −b
a + a2 − b 2
0
π/2
π/2
dx
dx
ln sin x
ln cos x
2 =
2
(a sin x ± b cos x)
(a cos x ± b sin x)
0
0
1
a bπ
=
∓a ln −
b (a2 + b2 )
b
2
[a > 0, b > 0]
π/2
π/2
b
ln sin x dx
ln cos x dx
π
ln
=
=
2 sin2 x + b2 cos2 x
2 sin2 x + a2 cos2 x
2ab
a
+
b
a
b
0
0
π
[a > 0,
π/2
ln sin x 4.
0
sin 2x dx
2 =
a sin2 x + b cos2 x
=
π/2
ln cos x 0
π/2
5.
0
2
b sin x + a cos2 x
a2 sin2 x − b2 cos2 x
ln sin x 2 dx =
a2 sin2 x + b2 cos2 x
0
π/2
ln cos x ln sin x sin x
2
dx =
π/2
0
b > 0]
BI (319)(3, 7), LI (319)(3)
a2 cos2 x − b2 sin2 x
2 dx
a2 cos2 x + b2 sin2 x
π
2b(a + b)
cos x ln cos x
π
√
dx = − ln 2
8
1 + cos2 x
1 + sin x
π/2 3
π/2
sin x ln sin x
cos3 x ln cos x
ln 2 − 1
√
dx =
dx =
2
2
4
1 + cos x
0
0
1 + sin x
0
2.
π/2
BI (317)(4, 10)
sin 2x dx
[a > 0,
1.
b > 0]
a
1
ln
2b(b − a) b
=
4.386
BI (319)(1,6)a
2
[a > 0,
BI (331)(6)
b > 0]
LI (319)(2, 8)
BI (322)(1, 6)
BI (322)(2, 7)
4.387
Logarithms and trigonometric functions
π/2
3.
0
π/2
4.
0
4.387
1.
2.
ln sin x dx
1−
k2
585
π
1
= − K(k) ln k − K (k )
2
4
sin x
BI (322)(3)
2
k
π
ln cos x dx
1
= K(k) ln − K (k )
2
2
2
k
4
1 − k sin x
π/2
BI (322)(9)
π/2
ln sin x sinμ x cosν x dx =
ln cos x cosμ x sinν x dx
0
0
μ+1 ν+1
μ+1
μ+ν+2
1
,
= B
ψ
−ψ
4
2
2
2
2
[Re μ > −1, Re ν > −1] GW (338)(6c)
√
μ
π/2
πΓ
μ+1
μ
μ−1
2
−ψ
ln sin x sin
x dx =
ψ
μ+1
2
2
0
4Γ
2
[Re μ > 0]
3.
4.
5.
6.
√
ν
π/2
πΓ
ν−1
2
ln sin x cos
x dx =
ν+1
0
4Γ
2
ν
−ψ
ψ
2
GW (338)(6a)
ν+1
2
[Re ν > 0]
π/2
2n
(2n − 1)!! π (−1)k+1
2n
− ln 2
ln sin x sin x dx =
(2n)!! 2
k
0
k=1
2n+1
π/2
(−1)k
(2n)!!
2n+1
+ ln 2
ln sin x sin
x dx =
(2n + 1)!!
k
0
k=1
% n
&
π/2
(2n − 1)!! π 1
2n
+ ln 4
ln sin x cos x dx = −
(2n)!! 4
k
0
k=1
(2n − 1)!! π
[C + ψ(n + 1) + ln 4]
=−
(2n)!! 4
GW (338)(6b)
FI II 811
BI (305)(13)
BI (305)(14)
π/2
1
(2n)!!
(2n + 1)!!
2k + 1
k=0
(2n)!!
3
=−
ψ n+
2(2n + 1)!!
2
ln sin x cos2n+1 x dx = −
7.
0
n
−ψ
1
2
GW (338)(7b)
π/2
ln cos x sin2n x dx = −
8.
0
(2n − 1)!! π
{C + 2 ln 2 + ψ(n + 1)}
2n+1 · n! 2
BI (306)(8)
586
Logarithmic Functions
π/2
9.
0
(2n − 1)!! π
ln cos x cos2n x dx = −
2n n! 2
ln cos x cos2n x dx =
0
ln 2 +
2n
(−1)k
k=1
k
%
π/2
10.
4.388
2n−1
(−1)k
2n−1 (n − 1)!
ln 2 +
(2n − 1)!!
k
BI (306)(10)
&
k=1
BI (306)(9)
4.388
1.
2.
3.
4.
5.
6.
%
&
n−1
1
(−1)k
sin2n x
1
nπ
ln 2 + (−1)
+
ln sin x 2n+2 dx =
cos
x
2n + 1 2
4
2n − 2k − 1
0
k=0
%
&
π/4
n−1
(−1)k
1
sin2n−1 x
n
− ln 2 + (−1) ln 2 +
ln sin x 2n+1 dx =
cos
x
4n
n−k
0
k=1
%
&
π/4
n
1
1
(−1)k−1
sin2n x
n+1 π
+
− ln 2 + (−1)
ln cos x 2n+2 dx =
cos
x
2n + 1
2
4
2n − 2k + 1
0
k=0
%
&
π/4
n−1
(−1)k
1
sin2n−1 x
n
− ln 2 + (−1) ln 2 +
ln cos x 2n+1 dx =
cos
x
4n
n−k
0
k=0
π/2
π
pπ
sinp−1 x
[0 < p < 2]
ln sin x p+1 dx = − cosec
cos
x
2p
2
0
π/2
2
1 π
pπ
dx
=
sec
p <1
ln sin x
p−1
4
p
−
1
2
tan
x
sin
2x
0
4.389 1.
π/4
(2n − 1)!! π
(2n)!! 4n + 2
π
ln sin x sin2n 2x cos 2x dx = −
0
π/4
ln sin x cosn 2x sin 2x dx = −
2.
0
ln cos x cosμ−1 2x tan 2x dx =
0
π/2
0
π/2
ln cos x cosμ−1 x sin x dx = −
5.3
0
ln cos x cosp x cos px dx =
−π
2
0
BI (310)(3)
1
μ2
BI (306)(11)
π
[C + ψ(p + 1) − 2 ln 2]
2p+1
π/2
ln cos x cosp−1 x sin px sin x dx =
6.
BI (310)(4)
BI (286)(2)
[Re μ > 0]
π
2
BI (288)(11)
1
β(μ)
4(1 − μ)
ln sin x sinμ−1 x cos x dx =
4.
BI (288)(10)
BI (285)(2)
[Re μ > 0]
LI (288)(2)
BI (330)(9)
1
{C + ψ(n + 2) + ln 2}
4(n + 1)
π/4
3.
BI (288)(1)
π
2p+2
[p > −1]
1
C + ψ(p) − − 2 ln 2
p
[p > 0]
BI (306)(12)
4.394
4.391
Logarithms and trigonometric functions
π/4
π/4
n
0
n
(ln sin 2x) sinp−1 2x tan
(ln cos 2x) cosp−1 2x tan x dx =
1.
587
0
[p > 0]
π/4
n
(ln sin 2x) sinp−1 2x tan
2.
0
π/4
3.
2n−1
(ln cos 2x)
tan x dx =
0
π/4
4.
2n
(ln cos 2x)
tan x dx =
0
4.392
π/4
1.
0
π
1
− x dx = β (n) (p)
4
2
BI (286)(10), BI (285)(18)
(−1)n n!
π
+ x dx =
ζ(n + 1, p)
4
2
1 − 22n−1 2n
π |B2n |
4n
BI (285)(17)
[n = 1, 2, . . .]
BI (286)(7)
22n 1
(2n)! ζ(2n + 1)
22n+1
BI (286)(8)
%
&
n−1
(−1)k−1
1
1
sin2n x
n+1 π
dx =
− ln 2 +
+2
ln (sin x cos x)
(−1)
cos2n+2 x
2n + 1
2
2n + 1
2n − 2k − 1
k=0
%
π/4
2.
ln (sin x cos x)
0
(−1)k
1
sin2n−1 x
1
n
n
dx
=
+
(−1)
(−1)
ln
2
−
ln
2
+
cos2n+1 x
2n
2n
k
n−1
&
BI (294)(8)
k=1
BI (294)(9)
4.393
π/2
1.
ln tan x sin x dx = ln 2
BI (307)(3)
ln tan x cos x dx = − ln 2
BI (307)(4)
0
π/2
2.
0
π/2
3.
ln tan x sin2 x dx = −
0
π/4
0
sin x ln cot
0
π/2
1.
0
π
4
BI (307)(5, 6)
π2
ln tan x
dx = −
cos 2x
8
π/2
5.
4.394
ln tan x cos2 x dx =
0
4.
π/2
GW (338)(10b)a
x
dx = ln 2
2
LO III 290
1−a
ln tan x dx
π
ln
=
2
2
1 − 2a cos 2x + a
2 (1 − a ) 1 + a
a−1
π
ln
=
2
2 (a − 1) a + 1
a2 < 1
a2 > 1
BI (321)(15)
2.
0
π/2
ln tan x cos 2x dx
π 1+a
1−a
=
ln
2
2
1 − 2a cos 2x + a
4a 1 − a
1+a
π a2 + 1 a − 1
ln
=
4a a2 − 1 a + 1
2
a2 < 1
a2 > 1
BI (321)(16)
588
Logarithmic Functions
π
3.
0
π
4.
0
ln tan bx dx
π
1 − ab
=
ln
1 − 2a cos 2x + a2
1 − a2 1 + ab
[0 < a < 1,
ln tan bx cos x dx
=0
1 − 2a cos 2x + a2
[0 < a < 1]
ln tan x
arcsin a
cos 2x dx
=−
(π + arcsin a)
1 − a sin 2x
4a
ln tan x
π
cos 2x dx
= − arcsin a
2
2
4a
1 − a sin 2x
π/4
5.
0
π/4
6.
0
7.
8.
4.395
a2 ≤ 1
a2 < 1
b > 0]
BI (331)(24)
BI (331)(25)
BI (291)(2,3)
BI (291)(9)
π
π
cos 2x dx
=
−
arcsinh
a
=
−
ln
a
+
1 + a2
2
4a
4a
1 + a2 sin 2x
0
2
a <1
BI (291)(10)
u sin x ln cot x
2 dx = cosec 2α π ln 2 + L(ϕ − α) − L(ϕ + α) − L π − 2α
2
2
2
2
0 1 − cos α sin x
[tan ϕ = cot α cos u; 0 < u < π]
π/4
ln tan x
LO III 290
π/4
9.
0
4.395
π/2
1.
0
π/4
2.
u
4.396
2.
BI (322)(11)
√
sin υ + 1 − cos2 u cos2 υ
ln tan x sin 4x dx
π cos2 υ
ln
=−
2
2
2 sin u sin υ
sin u (1 + sin υ)
sin u + tan2 υ sin2 2x
sin 2x − sin2 u
+
π,
π
LO III 285a
0<u< , 0<υ<
2
2
π/2
π/2
ln tan x cosq−1 x cot x sin[(q + 1)x] dx = −
3.
0
π/2
ln tan x cosq−1 x cos[(q + 1)x] dx = −
4.
0
π/4
n
(ln tan x) tanp x dx =
5.
0
LO III 290a
ln tan x dx
= − ln k K(k)
2
2
1 − k sin x
a 2
Γ
2
ln (a tan x) sinμ−1 2x dx = 2μ−2 ln a
Γ(a)
0
√
π/2
π Γ u − 12
2(μ−1)x
ln tan x cos
dx = −
C+ψ
4
Γ(μ)
0
1.
,
+
π
π
ln tan x sin 2x dx
−
t
−
−
t
ln
2
=
cosec
2t
L
2
2
1 − cos2 t sin2 2x
1
2n+1
B (n)
π
2q
p+1
2
[a > 0,
Re μ > 0]
2μ − 1
+ ln 4
2
Re μ > 12
LI (307)(8)
BI (307)(9)
π
[C + ψ(q + 1)]
2
[q > −1]
BI (307)(11)
[q > 0]
BI (307)(10)
[p > −1]
LI (286)(22)
4.397
Logarithms and trigonometric functions
6.
7.
589
1 − 22n 2n
dx
=
π |B2n |
[n = 1, 2, . . .]
cos 2x
2n
0
&
%
π/4
n
(−1)n+1 π 2 (−1)k
2n+1
+
ln tan x tan
x dx =
4
12
k2
0
π/2
(ln tan x)
2n−1
BI (312)(6)
GW (338)(8a)
k=1
4.397
ln (1 + p sin x)
π2
1
dx
=
−
arccos p2
sin x
8
2
ln (1 + p cos x)
π2
1
dx
2
=
− (arccos p)
cos x
8
2
π/2
1.
0
π/2
2.
0
π
3.
ln (1 + p cos x)
0
5.
6.
p2 < 1
p2 < 1
BI (313)(1)
BI (313)(8)
BI (331)(1)
π
− α − α ln sin α
L
cos x ln (1 + cos α cos x)
2
dx
=
1 − cos2 α cos2 x
sin α cos α
0
+
π,
0<α<
2
π
π/2
− α + (π − α) ln sin α
L
cos x ln (1 − cos α cos x)
2
dx
=
1 − cos2 α cos2 x
sin α cos α
0
+
π,
0<α<
2
π
ln 1 − 2a cos x + a2 cos nx dx
4.
dx
= π arcsin p
cos x
p2 < 1
π/2
0
=
1
2
2π
π
0
8.
0
π
LO III 291
ln 1 − 2a cos x + a2 cos nx dx
0
2
a < 1 BI (330)(11), BI (332)(5)
2
a >1
GW (338)(13a)
π
= − an
n
π
=− n
na
7.
LO III 291
1
ln 1 − 2a cos x + a2 sin nx sin x dx =
2
π
=
2
1
ln 1 − 2a cos x + a2 sin nx sin x dx =
2
2π
0
an+1
an−1
−
n + 1 n− 1 a2 > 1
2π
0
π
=−
2
ln 1 − 2a cos x + a2 sin nx sin x dx
BI (330)(10), BI (332)(4)
ln 1 − 2a cos x + a2 cos nx cos x dx
an+1
an−1
+
n+1 n−1
BI (330)(12), BI (332)(6)
590
Logarithmic Functions
π
9.
4.398
ln 1 − 2a cos 2x + a2 cos(2n − 1)x dx = 0
ln 1 − 2a cos 2x + a2 sin 2nx sin x dx = 0
a2 < 1
BI (330)(15)
0
π
10.
a2 < 1
BI (330)(13)
0
π
11.
0
π
12.
π
ln 1 − 2a cos 2x + a2 sin(2n − 1)x sin x dx =
2
an
an−1
−
n
n−1
2
a <1
BI (330)(14)
ln 1 − 2a cos 2x + a2 cos 2nx cos x dx = 0
π
ln 1 − 2a cos 2x + a2 cos(2n − 1)x cos x dx = −
2
an−1
an
+
n
n−1
2
a <1
a2 < 1
BI (330)(16)
0
π
13.
0
π/2
14.
0
aπ
ln 1 + 2a cos 2x + a2 sin2 x dx = −
4
π
π ln a2
−
=
4
4a
a2 < 1
a2 > 1
BI (330)(17)
BI (309)(22), LI (309)(22)
π/2
15.
0
π
16.
0
4.398
aπ
ln 1 + 2a cos 2x + a2 cos2 x dx =
4
π
π ln a2
+
=
4
4a
ln
0
1 − 2a cos x + a
cos mx dx = 2π
1 − 2a cos nx + a2
2
ln
0
= 2π
3.
1.
0
a2 ≤ 1,
a2 < 1
b2 < 1
m
n m/n a
a
−
m
m
n −m/n a−m
a
−
m
m
BI (331)(26)
BI (330)(18)
a2 ≤ 1
a2 ≥ 1
2
ln
a2 > 1
BI (332)(9)
π
0
4.399
2n+1
1 + 2a cos x + a2
n 2πa
sin(2n
+
1)x
dx
=
(−1)
1 − 2a cos x + a2
2n + 1
2π
2.
a2 < 1
BI (309)(23), LI (309)(23)
ln 1 − 2a cos x + a2
2π ln(1 − ab)
dx =
1 − 2b cos x + b2
1 − b2
π
1.
π/2
1 + 2a cos 2x + a
cot x dx = 0
1 + 2a cos 2nx + a2
π
ln 1 + a sin x sin2 x dx =
2
2
ln
BI (331)(5), LI(331)(5)
1+
√
1+a 11− 1+a
√
−
2
21+ 1+a
√
[a > −1]
BI (309)(14)
4.413
Logarithms and trigonometric functions
π/2
2.
0
π/2
3.
0
4.411 1.
2.
π
ln 1 + a sin2 x cos2 x dx =
2
ln
1+
√
1+a 11− 1+a
√
+
2
21+ 1+a
√
ln 1 − cos β cos x
π
1 + sin α
dx = −
ln
2
2
1 − cos α cos x
sin α sin α + sin β
2
591
[a > −1]
BI (309)(15)
2
+
π
0<β< ,
2
0<α<
π,
2
LO III 285
2
dx
1 + sin x
= λ2
λ < π2
BI (331)(2)
1 + cos λ sin x sin x
0
π/2
π/2
π/2
q
p + q sin ax dx
p + q cos ax dx
p + q tan ax dx
=
=
= π arcsin
ln
ln
ln
p
−
q
sin
ax
sin
ax
p
−
q
cos
ax
cos
ax
p
−
q
tan
ax
tan
ax
p
0
0
0
[p > q > 0]
π
ln
FI II 695a, BI (315)(5, 13,17)a
π/2
3.
0
α−β
cos
1 + cos β cos x
2π
cos x
2
ln
dx =
ln
α+β
1 − cos2 α cos2 x 1 − cos β cos x
sin 2α
sin
2
+
0<α≤β<
4.412
π/4
1.
π2
dx
π
±x
=±
4
sin 2x
8
BI (293)(1)
ln tan
dx
π2
π
±x
=±
4
tan 2x
16
BI (293)(2)
ln tan
π
22n+2 − 1
dx
2n
± x (ln tan x)
=±
π 2n+2 |B2n+2 |
4
sin 2x
4(n + 1)(2n + 1)
BI (294)(24)
ln tan
π
1 − 22n+1
dx
2n−1
± x (ln tan x)
= ± 2n+2 (2n)! ζ(2n + 1)
4
sin 2x
2
n
BI (294)(25)
ln tan
dx
(−1)n−1
π
n−1
± x (ln sin 2x)
=
(n − 1)! ζ(n + 1)
4
tan 2x
2
LI (294)(20)
π/4
2.
0
π/4
3.
0
π/4
4.
0
π/4
5.
0
4.413 1.
π/2
LO III 284
ln tan
0
π,
2
ln p2 + q 2 tan2 x
0
π
ap + bq
dx
=
ln
ab
a
a2 sin2 x + b2 cos2 x
[a > 0,
b > 0,
p > 0,
q > 0]
BI (318)(1–4)a
2.
0
π/2
ln 1 + q 2 tan2 x
p2
dx
1
2
2
2
2
sin x + r cos x s sin x + t2 cos2 x
p2 − r2
qr
qt
π
t2 − s2
ln
1
+
ln
1
+
= 2 2
+
p t − s2 r2
pr
p
st
s
[q > 0, p > 0, r > 0, s > 0, t > 0] BI (320)(18)
2
592
Logarithmic Functions
π/2
ln 1 + q 2 tan2 x
dx
sin2 x
p2 sin2 x + r2 cos2 x s2 sin2 x + t2 cos2 x
t
qr
qt
π
r
ln
1
+
ln
1
+
= 2 2
−
p t − s2 r2 s
p
p
s
[q > 0, p > 0, r > 0, s > 0, t > 0] BI (320)(20)
π/2
ln 1 + q 2 tan2 x
dx
cos2 x
2
2
2
2
2
2
p sin x + r cos x s sin x + t2 cos2 x
p
qr
π
ln 1 +
= 2 2
2
2
p t −s r
r
p
[q > 0, p > 0, r > 0, s > 0,
3.
0
4.
0
5.
0
4.414
π
ln tan rx dx
π
1 − p2r
=
ln
1 − 2p cos x + p2
1 − p2 1 + p2r
π/2
1.
0
4.414
π/2
2.
0
p2 < 1
qt
s
− ln 1 +
t
s
t > 0] BI (320)(21)
BI (331)(12)
dx
ln 1 − k 2 sin2 x = ln k K(k)
1 − k 2 sin2 x
BI (323)(1)
sin2 x dx
1 ln 1 − k 2 sin2 x = 2 k 2 − 2 + ln k K(k) + (2 − ln k ) E(k)
k
1 − k 2 sin2 x
BI (323)(3)
π/2
3.
0
,
cos x 1 +
2
2
ln 1 − k 2 sin2 x
1 − k2 sin2 x = 2 1 + k − k ln k K(k) − (2 − ln k ) E(k)
dx
k
2
BI (323)(6)
π/2
4.
0
ln 1 − k 2 sin2 x <
1 = 2 k 2 − 2 K(k) + (2 + ln k ) E(k)
3
k
1 − k 2 sin2 x
dx
BI (323)(9)
π/2
5.
0
sin2 x
ln 1 − k 2 sin2 x
dx
<
3
1 − k 2 sin2 x
=
,
1 +
2
2
(2
+
ln
k
K(k)
)
E(k)
−
1
+
k
+
k
ln
k
k2 k 2
BI (323)(10)
π/2
6.
0
,
cos2 x dx
1 +
2
1
+
k
K(k)
−
(2
+
ln
k
ln 1 − k 2 sin2 x <
=
+
ln
k
)
E(k)
3
k2
1 − k 2 sin2 x
BI (323)(16)
7.
0
π/2
2
ln 1 − k 2 sin2 x
1 − k 2 sin2 x dx = 1 + k K(k) − (2 − ln k ) E(k)
BI (324)(18)
4.416
Logarithms and trigonometric functions
π/2
8.
0
593
1
2
−2 + 11k 2 − 6k 4 + 3k ln k K(k)
ln 1 − k 2 sin2 x sin2 x 1 − k 2 sin2 x dx = 2
9k
+ 2 − 10k2 − 3 1 − 2k 2 ln k E(k)
BI (324)(20)
π/2
9.
0
1
2
ln 1 − k 2 sin2 x cos2 x 1 − k 2 sin2 x dx = 2
2 + 7k 2 − 3k 4 − 3k ln k K(k)
9k
2
2
− 2 + 8k − 3 1 + k ln k E(k)
BI (324)(21), LI (324)(21)
π/2
10.
0
sin x cos x dx
2
1−2n
[1
+
(2n
−
1)
ln
k
ln 1 − k 2 sin2 x <
=
]
k
−
1
2n+1
(2n − 1)2 k 2
1 − k 2 sin2 x
BI (324)(17)
4.415 1.
∞
ln x sin ax2 dx = −
0
2.
0
4.416
1.
0
2.
3.
4.7
∞
1
4
1
ln x cos ax dx = −
4
π/2 cos x ln
2
1+
π
π
ln 4a + C −
2a
2
[a > 0]
GW (338)(19)
π
π
ln 4a + C −
2a
2
[a > 0]
GW (338)(19)
sin2 β − cos2 β tan2 α sin2 x
1 − sin2 α cos2 x
dx
= cosec 2α {(2α + 2γ − π) ln cos β + 2 L(α) − 2 L(γ) + L(α + γ) − L(α − γ)}
sin α
π
cos γ =
; 0<α<β<
LO III 291
sin β
2
π/2 cos x ln 1 − sin2 β − cos2 β tan2 α sin2 x
dx
1 − sin2 α cos2 x
0
= cosec 2α {(π + 2α − 2γ) ln cos β + 2 L(α) + 2 L(γ) − L(α + γ) + L(α − γ)}
π
sin α
; 0<α<β<
cos γ =
LO III 291
sin β
2
π/2 ln sin x + sin2 x − sin2 β
dx
1 − cos2 α cos2 x
β
tan β
1 + sin α
π
= − cosec α arctan
ln sin β + ln
sin α
2 sin α + 1 − cos2 α cos2 β
+
π,
0 < α < π, 0 < β <
LO III 285
2
π/4
n−1
ln tan x (ln cos 2x)
tan 2x dx = 12 (−1)n (n − 1)! 1 − 2−(n+1) ζ(n + 1)
0
BI (287)(20)
594
Logarithmic Functions
4.421
4.42–4.43 Combinations of logarithms, trigonometric functions, and powers
4.421
∞
ln x sin ax
1.
0
π
dx
= − (C + ln a)
x
2
∞
2.
ln ax sin bx
0
∞
ln ax cos bx
0
,
x dx
π −bβ π + bβ −bβ e
e
=
ln
(aβ
)
−
Ei
(−bβ
)
+
e
Ei
(bβ
)
β 2 + x2
2
4
[β = β sign β; a > 0,
∞
ln ax sin bx
0
,
β dx
π
π + bβ e Ei (−bβ ) − e−bβ Ei (bβ )
= e−bβ ln (aβ ) +
2
2
β +x
2
4
[β = β sign β; a > 0,
ln ax cos bx
0
x dx
π
= {− si(bc) sin bc + cos bc [ln ac − ci(bc)]}
2
−c
2
∞
1.
ln x sin axxμ−1 dx =
0
∞
2.
0
4.423
4.424
1
ln ab
2
b > 0]
GW (338)(21a)
b > 0]
GW (338)(21b)
[a > 0,
∞
sin ax
aπ
(C + ln 2a − 1)
dx = −
2
x
2
2
(ln x) sin ax
0
C+
BI (411)(5)
BI (411)(6)
2
ln x
1.
BI (422)(6)
[a > 0,
cos ax − cos bx
π
dx = [(a − b) (C − 1) + a ln a − b ln b]
2
x
2
∞
0
c > 0]
0 < Re μ < 1]
ln x
∞
3.
b > 0,
[a > 0,
a
cos ax − cos bx
dx = ln
x
b
0
BI (422)(5)
π
μπ ,
Γ(μ)
μπ +
ψ(μ)
−
ln
a
+
cot
sin
aμ
2
2
2
ln x
0
2.
c > 0]
[a > 0, |Re μ| < 1]
+
π
μπ ,
Γ(μ)
μπ
ψ(μ) − ln a − tan
ln x cos axxμ−1 dx = μ cos
a
2
2
2
∞
1.
b > 0,
dx
π
{sin bc [ci(bc) − ln ac] − cos bc si(bc)}
=
x2 − c2
2c
[a > 0,
b > 0]
x2
[a > 0,
∞
5.
4.422
b > 0]
ET I 17(3), NT 27(11)a
4.
FI II 810a
ET I 76(5), NT 27(10)a
3.
[a > 0]
[a > 0]
GW (338)(20b)
[a > 0]
ET I 77(9), FI II 810a
π
π
dx
π3
2
= C2 +
+ πC ln a + (ln a)
x
2
24
2
4.429
Logarithms, trigonometric functions, and powers
2.
∞
6
2
(ln x) sin axxμ−1 dx =
0
4.425 1.
∞
ln(1 + x) cos ax
0
∞
2.
∞
0
1
dx
2
2
=
[si(a)] + [ci(a)]
x
2
[a > 0]
b > 0]
a
dx
= −π Ei −
ln 1 + b2 x2 sin ax
x
b
[a > 0,
b > 0]
cos ax
ET I 18(11)
GW (338)(24), ET I 77(14)
1
4.
0
4.426 1.11
1
pπ
dx
π
= 2+
coth
ln 1 − x2 cos (p ln x)
x
2p
2p
2
∞
x ln
0
LI (309)(1)a
b2 + x2
π sin ax dx = 2 (1 + ac)e−ac − (1 + ab)e−ab
2
2
c +x
a
[b ≥ 0,
∞
+
ap
b2 x2 + p2
dx
ap ,
= π Ei −
ln 2 2
sin ax
− Ei −
2
c x +p
x
c
b
0
[b > 0,
2.
ET I 18(8)
[a ≥ 0,
0
3.
b+x
b−x
Γ(μ)
μπ + μπ
− 2 ψ(μ) ln a
sin
ψ (μ) + ψ 2 (μ) + π ψ(μ) cot
aμ
2
2
μπ
1
2
− π ln a cot
+ (ln a) − π 2
2
4
[a > 0, 0 < Re μ < 1]
ET I 77(10)
dx
= −2π si(ab)
x
ln2
595
c ≥ 0,
a > 0]
c > 0,
p > 0,
GW (338)(23)
a > 0]
ET I 77(15)
∞
4.427
ln x +
0
4.428 1.
2.
3.
∞
sin ax
π
π
β 2 + x2 dx = K 0 (aβ) + ln(β) [I 0 (aβ) − L(aβ)]
2
2
2
2
β +x
[Re β > 0, a > 0]
cos bx
dx = πb ln 2 − aπ
[a > 0, b > 0]
x2
0
∞
cos bx
π
ln 4 cos2 ax 2
dx = cosh(bc) ln 1 + e−2ac
2
x +c
c
0
+
π,
a 0,
∞
4.
0
ln cos2 ax
0
1
ET I 22(29)
ET I 22(30)
ET I 82(36)
cos bx
dx = −π ln 1 + e−2a cosh b + b + e−b π ln 2 − aπ
x2 (1 + x2 )
[a > 0,
4.429
b > 0]
ET I 77(16)
π
(1 + x)x
sin (ln x) dx =
ln x
4
b > 0]
ET I 22(31)
BI (326)(2)a
596
4.431
Logarithmic Functions
∞
1.
ln (2 ± 2 cos x)
0
∞
2.
ln (2 ± 2 cos x)
0
∞
3.
0
4.431
sin bx
x dx = − π sinh(bc) ln (1 ± e−c )
x2 + c2
[b > 0,
c > 0]
ET I 22(32)
[b > 0,
c > 0]
ET I 22(32)
cos bx
π
dx = cosh(bc) ln 1 ± e−c
2
2
x +c
c
sin bx
π (−a)k
dx = −
[1 + sign(b − k)]
ln 1 + 2a cos x + a2
x
2
k
[b]
k=1
[0 < a < 1,
∞
4.
0
ET I 82(25)
b
cos bx
π ak
π −c
sinh[c(b − k)]
cosh(bc) +
ln 1 − 2a cos x + a
dx = ln 1 − ae
x2 + c2
c
c
k
2
k=1
[|a| < 1,
4.432
b > 0]
∞
1.
0
sin x
dx
=
ln 1 − k sin x 2
2
1 − k sin x x
2
2
0
∞
b > 0,
ln 1 − k 2 cos2 x √
c > 0]
ET I 22(33)
sin x
dx
= ln k K(k)
2
2
1 − k cos x x
BI ((412, 414))(4)
π/2
2.
0
sin x cos x
ln 1 − k 2 sin2 x x dx
1 − k 2 sin2 x
1 = 2 πk (1 − ln k ) + 2 − k 2 K(k) − (4 − ln k ) E(k)
k
BI (426)(3)
π/2
3.
0
BI (426)(6)
∞
4.
0
5.
0
sin x cos x
1 ln 1 − k 2 cos2 x √
x dx = 2 −π − 2 − k 2 K(k) + (4 − ln k ) E(k)
k
1 − k 2 cos2 x
sin x cos x dx
1 2
2 − k 2 − k ln k K(k) − (2 − ln k ) E(k)
= 2
ln 1 − k 2 sin2 x k
1 − k 2 sin2 x x
BI (412)(5)
∞
sin x cos x dx
1 = 2 k 2 − 2 + ln k K(k) + (2 − ln k ) E(k)
ln 1 − k 2 cos2 x √
2
2
k
1 − k cos x x
BI (414)(5)
4.432
Logarithms, trigonometric functions, and powers
∞
6.
0
597
∞
dx
sin x
sin x
dx
=
ln 1 ± k sin x ln 1 ± k cos2 x √
2
2
2
2
x
1 − k cos x x
1 − k sin x
0 ∞
dx
tan x
ln 1 ± k sin2 x =
2
2
1 − k sin x x
0 ∞
tan x
dx
=
ln 1 ± k cos2 x √
2 cos2 x x
1
−
k
0 ∞
dx
tan x
ln 1 ± k sin2 2x =
2
2
1 − k sin 2x x
0 ∞
tan x
dx
=
ln 1 ± k 2 cos2 2x √
2
2
1 − k cos 2x x
0
π
1 2 (1 ± k)
K(k) − K (k )
= ln √
2
8
k
2
BI (413)(1–6), BI (415)(1–6)
∞
7.
0
1 sin x
dx
= 2 k 2 − 2 + ln k K(k) + (2 − ln k ) E(k)
ln 1 − k 2 sin2 x 2
k
1 − k 2 sin x x
3
BI (412)(6)
ln 1 − k 2 cos2 x √
∞
sin x cos2 x dx
1 2
2 − k 2 − k ln k K(k) − (2 − ln k ) E(k)
= 2
ln 1 − k 2 sin2 x k
1 − k 2 sin2 x x
0
BI (414)(6)a
9.
0
BI (412)(7)
∞
10.
0
1 sin3 x
dx
2
2 − k 2 − k ln k K(k) − (2 − ln k ) E(k)
= 2
k
1 − k 2 cos2 x x
∞
8.
∞
11.
0
sin x cos2 x dx
1 = 2 k 2 − 2 + ln k K(k) + (2 − ln k ) E(k)
ln 1 − k 2 cos2 x √
k
1 − k 2 cos2 x x
tan x
dx
=
ln 1 − k 2 sin2 x 2
1 − k 2 sin x x
BI (414)(7)
∞
ln 1 − k 2 cos2 x √
0
tan x
dx
= ln k K(k)
2
2
1 − k cos x x
BI ((412, 414))(9)
∞
sin2 x tan x dx
1 = 2 k 2 − 2 + ln k K(k) + (2 − ln k ) E(k)
ln 1 − k 2 sin2 x k
1 − k 2 sin2 x x
∞
sin x tan x dx
1 2
2 − k 2 − k ln k K(k) − (2 − ln k ) E(k)
= 2
ln 1 − k 2 cos2 x √
k
1 − k 2 cos2 x x
12.
0
BI (412)(8)
13.
0
2
BI (414)(8)
14.
0
∞
ln 1 − k 2 sin2 x <
sin2 x
∞
dx
sin x
ln 1 − k 2 cos2 x <
3 x
0
(1 − k 2 cos2 x)
1 = 2 k 2 − 2 K(k) + (2 + ln k ) E(k)
k
dx
=
3
x
1 − k 2 sin2 x
BI ((412, 414))(13)
598
Logarithmic Functions
π/2
15.
0
4.432
1 sin x cos x
π
(1
+
ln
k
ln 1 − k 2 sin2 x <
x
dx
=
)
−
(2
+
ln
k
)
K(k)
3
k2
k
1 − k 2 sin2 x
BI (426)(9)
16.
17.
π/2
∞
1
sin x cos x
ln 1 − k 2 cos2 x <
x dx = 2 {−π + (2 + ln k ) K(k)}
BI (426)(15)
k
3
0
(1 − k 2 cos2 x)
∞
∞
dx
dx
sin x cos x
sin3 x
2
2
2
2
<
=
ln 1 − k sin x <
ln
1
−
k
cos
x
3
x
x
3
0
0
(1 − k 2 cos2 x)
1 − k 2 sin2 x
1 = 2 2 − k2 + ln k K(k) − (2 + ln k ) E(k)
k
BI (412)(14), BI(414)(15)
18.
0
ln 1 − k 2 sin2 x <
3
sin x
dx
3
x
1 − k 2 sin2 x
=
∞
dx
sin x cos x
ln 1 − k 2 cos2 x <
3 x
0
(1 − k2 cos2 x)
1
2
(2 + ln k ) E(k) − 2 − k 2 + k ln k K(k)
=
2
k2 k
∞
dx
sin x cos x
=
ln 1 − k 2 sin2 x <
3 x
1 − k 2 sin2 x
∞
dx
sin x tan x
ln 1 − k 2 sin2 x <
3 x
1 − k 2 sin2 x
=
19.
0
BI (412)(15), BI(414)(14)
∞
dx
sin2 x tan x
ln 1 − k 2 cos2 x <
3 x
0
(1 − k 2 cos2 x)
1 = 2 2 − k2 + ln k K(k) − (2 + ln k ) E(k)
k
2
BI (412)(16), BI(414)(17)
20.
0
2
∞
dx
sin x cos2 x
ln 1 − k 2 cos2 x <
x
3
0
(1 − k 2 cos2 x)
1
2
2
(2
+
ln
k
K(k)
=
)
E(k)
−
2
−
k
+
k
ln
k
k2 k 2
∞
21.
0
22.
0
∞
ln 1 − k 2 sin2 x <
BI (412)(17), BI(414)(16)
∞
dx
tan x
ln 1 − k 2 cos2 x <
x
3
0
(1 − k 2 cos2 x)
1 = 2 k 2 − 2 K(k) + (2 + ln k ) E(k)
k
dx
3 x =
2
1 − k 2 sin x
tan x
dx
=
ln 1 − k 2 sin2 x
1 − k 2 sin2 x sin x
x
BI ((412, 414))(18)
dx
1 − k 2 cos2 x sin x
ln 1 − k 2 cos2 x
x
0
2
= 2 − k K(k) − (2 − ln k ) E(k)
∞
BI ((412, 414))(1)
4.511
Inverse trigonometric functions
23.
599
ln 1 − k 2 sin2 x
1 − k 2 sin2 x sin x cos x · x dx
0
1 3
2
4
2
=
(1
−
3
ln
k
)
+
22k
+
6k
−
3k
ln
k
3πk
K(k)
27k 2
π/2
− 2 − k 2 (14 − 6 ln k ) E(k)
BI (426)(1)
ln 1 − k 2 cos2 x
1 − k 2 cos2 x sin x cos x · x dx
1 2
2
−3π − 22k + 6k 4 − 3k ln k K(k) + 2 − k 2 (14 − 6 ln k ) E(k)
=
2
27k
π/2
24.
0
∞
25.
0
∞
26.
0
dx
=
ln 1 − k 2 sin2 x
1 − k 2 sin2 x tan x
x
BI (426)(2)
dx
ln 1 − k 2 cos2 x
1 − k2 cos2 x tan x
x
0
2
= 2 − k K(k) − (2 − ln k ) E(k)
∞
((412,414))(2)
∞
sin x
tan x
dx
dx
=
ln sin2 x + k cos2 x √
ln sin2 x + k cos2 x √
2
2
2
2
1 − k cos x x
1 − k cos x x
0 ∞
2
tan x
dx
2
ln sin 2x + k cos 2x √
=
2 cos2 2x x
1
−
k
0 ⎡
√ 3⎤
k ⎥
2
1 ⎢
= ln ⎣
⎦ K(k)
2
1 + k
BI (415)(19–21)
4.44 Combinations of logarithms, trigonometric functions, and exponentials
4.441
1.
∞
7
e
−qx
0
∞
2.
p 2
1
p
2
sin px ln x dx = 2
q arctan − pC − ln p − q
p + q2
q
c
e−qx cos px ln x dx = −
0
1
p2 + q 2
[q > 0,
p > 0]
q 2
p
2
ln p + q + p arctan + qC
2
q
BI (467)(1)
[q > 0]
BI (467)(2)
π/2 −p tan x
e
4.442
0
1
ln cos x dx
1
2
2
= − [ci(p)] + [si(p)]
sin x cos x
2
2
[Re p > 0]
NT 32(11)
4.5 Inverse Trigonometric Functions
4.51 Inverse trigonometric functions
4.511
∞
arccot px arccot qx dx =
0
π
2
1
p
ln 1 +
p
q
+
q
1
ln 1 +
q
p
[p > 0,
q > 0]
BI (77)(8)
600
Inverse Trigonometric Functions
4.512
π
4.512
arctan (cos x) dx = 0
BI (345)(1)
0
4.52 Combinations of arcsines, arccosines, and powers
4.521
1
1.
0
1
2.
0
4.
5.
6.
7.
8.
4.522
1.
2.
3.
4.
5.
6.
7.
FI II 614, 623
arccos x
π
dx = ∓ ln 2 + 2G
1±x
2
BI (231)(7, 8)
√
2 1+q
x
π
√
ln
arcsin x
dx =
1 + qx2
2q 1 + 1 + q
0
1
x
π
1 + 1 − p2
arcsin x
dx
=
ln
1 − p2 x2
2p2
2 1 − p2
0
1
∞
sin[(2k + 1)λ]
dx
arccos x 2
=
2
cosec
λ
2
(2k + 1)2
sin λ − x
0
k=0
√
1
π 1+ 1+q
dx
√
=
ln
arcsin x
x (1 + qx2 )
2
1+q
0
√
1
x
π 1+q−1
arcsin x
2 dx = 4q
1+q
(1 + qx2 )
0
√
1
x
π 1+q−1
arccos x
dx =
2 )2
4q
1+q
(1
+
qx
0
3.
π
arcsin x
dx = ln 2
x
2
1
[q > −1]
p2 < 1
LI (231)(3)
BI (231)(10)
[q > −1]
BI (235)(10)
[q > −1]
BI (234)(2)
[q > −1]
BI (234)(4)
1 3
2
2
π + k K(k) − 2 1 + k E(k)
x 1 − k 2 x2 arccos x dx = 2
9k 2
0
1 1
3
3
2
2
x 1 − k 2 x2 arcsin x dx = 2 − πk − k K(k) + 2 1 + k E(k)
9k
2
0
1 1 3
2
2
2
2
2
E(k)
π + k K(k) − 2 1 + k
x k + k x arcsin x dx = 2
9k 2
0
1
,
x arcsin x
1 + π
√
dx = 2 − k + E(k)
k
2
1 − k 2 x2
0
1
,
x arccos x
1 +π
√
− E(k)
dx = 2
k 2
1 − k 2 x2
0
1
,
x arcsin x
1 +π
− E(k)
dx = 2
k 2
0
k 2 + k 2 x2
1
,
x arccos x
1 + π
dx = 2 − k + E(k)
k
2
0
k 2 + k 2 x2
BI (231)(1)
1
BI (236)(9)
BI (236)(1)
BI(236)(5)
BI (237)(1)
BI (240)(1)
BI (238)(1)
BI (241)(1)
4.531
Arctangents, arccotangents, and powers
1
8.
0
1
9.
∞
x arcsin x dx
2 sin[(2k + 1)λ]
√
=
sin λ
(2k + 1)2
(x2 − cos2 λ) 1 − x2
k=0
0
1
10.
0
4.523
1.
2.
3.
4.
5.
6.
4.524
x arcsin kx
(1 −
x2 ) (1
−
k 2 x2 )
x arccos kx
(1 −
x2 ) (1
−
k 2 x2 )
dx = −
dx =
π
ln k 2k
π
ln(1 + k)
2k
π
2n n!
1
−
2n + 1 2
(2n + 1)!!
0
1
π
(2n − 1)!!
2n−1
x
arcsin x dx =
1−
4n
2n n!
0
1
2n n!
x2n arccos x dx =
(2n + 1)(2n + 1)!!
0
1
π (2n − 1)!!
x2n−1 arccos x dx =
4n
2n n!
0
1
n
2n n!
1 − x2 arccos x dx = π
(2n + 1)!!
−1
1
n− 12
π 2 (2n − 1)!!
1 − x2
arccos x dx =
2
2n n!
−1
1
1.
2
(arcsin x)
x2
0
BI (243)(11)
BI (239)(1)
BI (242)(1)
1
x2n arcsin x dx =
601
1
dx
√
= π ln 2
1 − x2
0
dx
2
(arccos x) √
2.
1 − x2
3 = π ln 2
BI (229)(1)
BI (229)(2)
BI (229)(4)
BI (229)(5)
BI (254)(2)
BI (254)(3)
BI (243)(13)
BI (244)(9)
4.53–4.54 Combinations of arctangents, arccotangents, and powers
4.531
1
1.
0
arctan x
dx =
x
∞
0
4.
0
1
arccot x
dx = G
x
FI II 482, BI (253)(8)
BI (248)(6, 7)
π
arccot x
dx = − ln 2 + G
x(1 + x)
8
BI (235)(11)
0
1
∞
π
arccot x
dx = ± ln 2 + G
1±x
4
2.
3.
∞
arctan x
dx = −G.
1 − x2
BI (248)(2)
602
Inverse Trigonometric Functions
1
5.
arctan qx
0
6.
0
7.
0
π
1
arctan x
dx = ln 2 + G
2
x (1 + x )
8
2
∞
8.
0
∞
9.
0
10.11
11.
12.
13.
4.532
5.
4.533
BI (248)(3)
x arctan x
π
dx = − ln 2
1 − x4
8
BI (248)(4)
xp arctan x dx =
4.
x arctan x
π2
dx =
4
1+x
16
1
0
3.
BI (235)(12)
x arccot x
π
dx = ln 2
4
1−x
8
0
∞
∞
arccot x
arccot x
√
√
dx =
dx = 2G
2
x
1
+
x
1 + x2
0
0
1
√
arctan x
π
√
dx = ln 1 + 2
2
2
0 x 1−x
⎤
⎡
1
π
1 ⎣
x arctan x dx
π
⎦
<
= k2 F 4 , k − < 2 2
2
2
0
(1 + x ) 1 + k x
2 2 1+k
∞
,
+π
p
1
−β
+1
2(p + 1) 2
2
pπ
π
cosec
2(p
+
1)
2
0
1
+
,
π
p
1
+β
+1
xp arccot x dx =
2(p + 1) 2
2
0
∞
pπ
π
cosec
xp arccot x dx = −
2(p
+
1)
2
0
√
∞
2q
xp
π3
Γ(q)
dx
= 2q+2
arctan x
2p
1
+
x
x
2
p
Γ
q + 12
0
∞
1.
0
2.
0
BI (234)(10)
∞
1.
2.
q
dx
1
1+q
p
1
arccot q
=
ln
+ 2
arctan q +
2
2
2
2
2
(1 + px)
2p +q
(1 + p)
p +q
1+p
[p > −1]
1
BI (243)(7)
2
arccot qx
q
dx
1
(1 + p)2
q2 − p
arctan q
=
ln
+
(1 + px)2
2 p2 + q 2
1 + q2
(1 + p) (p2 + q 2 )
[p > −1]
1
4.532
1
xp arctan x dx =
(1 − x arccot x) dx =
π
4
π
dx
π
− arctan x
= − ln 2 + G
4
1−x
8
BI (248)(12)
BI (251)(3, 10)
FI II 694
BI (294)(14)
[p > −2]
BI (229)(7)
[−1 > p > −2]
BI (246)(1)
[p > −1]
BI (229)(8)
[−1 0]
BI (250)(10)
BI (246)(3)
BI (232)(2)
4.535
Arctangents, arccotangents, and powers
1
3.
0
π
1 + x dx
1
π
− arctan x
= ln 2 + G
2
4
1−x1+x
8
2
1
BI (235)(25)
dx
π
= − ln 2
2
1
−
x
4
0
∞
∞
dx
x dx
π2
2
2
√
+ 4G
4.534
(arctan x)
=
(arccot x) √
=−
4
x2 1 + x2
1 + x2
0
0
4.535
1
1
arctan px
dx = 2 arctan p ln 1 + p2
1.
2
2p
0 1+p x
1
1 π 1
arccot px
dx = 2
+ arccot p ln 1 + p2
[p > 0]
2.
2x
1
+
p
p
4
2
0
∞
π
arctan qx
q
ln pq − pq
[p > 0, q > 0]
3.
dx = −
2
2
2
(p + x)
1+p q
2
0
∞
arccot qx
q
π
4.
dx =
ln pq +
[p > 0, q > 0]
2
2 q2
(p
+
x)
1
+
p
2pq
0
∞
x arccot px
π 1 + pq
[p > 0, q > 0]
5.
dx = ln
2
2
q +x
2
pq
0
∞
x arccot px dx
π 1 + p2 q 2
6.
= ln
[p > 0, q > 0]
2
2
x −q
4
p2 q 2
0
∞
π
arctan px
dx = ln(1 + p)
[p ≥ 0]
7.
2)
x
(1
+
x
2
0
∞
π arctan px
dx = ln 1 + p2
[p ≥ 0]
8.
2
4
0 x (1 − x )
∞
π
dx
= 2 ln(1 + pq)
9.
arctan qx
[p > 0, q ≥ 0]
2 + x2 )
x
(p
2p
0
∞
π p2 + q 2
dx
=
ln
10.
arctan qx
[p ≥ 0]
x (1 − p2 x2 )
4
p2
0
∞
x arctan qx
πq
[p > 0, q ≥ 0]
11.
dx =
2 + x2 )2
4p(1
+ pq)
(p
0
∞
x arccot qx
π
[p > 0, q ≥ 0]
12.
2 dx = 4p2 (1 + pq)
2
2
(p + x )
0
1
arctan qx
π
√
13.
dx = ln q + 1 + q 2
2
2
0 x 1−x
⎧
π
1 + |αβ|
⎪
∞
⎨
ln
sign(α) for β = γ
x arctan(αx) dx
2 − γ2
9
β
1 + |αγ|
=
14.
πα
2
2
2
2
⎪
−∞ (x + β ) (x + γ )
⎩
for β = γ
2|β| (1 + |αβ|)
4.
x arccot x −
603
1
arctan x
x
for α, β, γ real
BI (232)(1)
BI (251)(9, 17)
BI (231)(19)
BI (231)(24)
BI (249)(1)
BI (249)(8)
BI (248)(9)
BI (248)(10)
FI II 745
BI (250)(6)
BI (250)(3)
BI (250)(6)
BI (252)(12)a
BI (252)(20)a
BI (244)(11)
604
Inverse Trigonometric Functions
15.
9
4.536
∞
∞
1.
0
0
∞
3.
0
1.
8
2.
π p
arctan px − arctan qx
dx = ln
x
2 q
[p > 0,
q > 0]
FI II 635
arctan px arctan qx
π (p + q)p+q
dx = ln
2
x
2
pp q q
[p > 0,
q > 0]
FI II 745
1
1
arctan
1 − x2
arctan tan λ 1 − k 2 x2
1
arctan tan λ 1 − k 2 x2
0
.
.
cosec
π + 4λ
8
[p > 0]
∞
arctan x2
BI (245)(10)
1 − k 2 x2
π
π
dx = E (λ, k) − cot λ 1 − 1 − k 2 sin2 λ
2
1−x
2
2
√
1
arctan tan λ 1 − k 2 x2
π
dx = F (λ, k)
2
2
2
2
(1 − x ) (1 − k x )
0
BI (245)(9)
BI (245)(12)
∞
dx
dx
=
arctan x3
2
1+x
1 + x2 0
0 ∞
∞
dx
dx
π2
=
arccot x2
=
arccot x3
=
2
2
1+x
1+x
8
0
0
∞
2
1−x
π √
2.
arctan x2 dx =
2−1
2
x
2
0
∞
4.539
xs−1 arctan ae−x dx = 2−s−1 Γ(s)a Φ −a2 , s + 1, 12
0
∞
p sin qx
x dx
π 4.541
arctan
= ln 1 + pe−q [p > −eq ]
2
1 + p cos qx 1 + x
2
0
1.
,
1 − x2
π +
2
dx = 2 E (λ, k) − k F (γ, k)
2
2
1−k x
2k
<
π
− 2 cot γ 1 − 1 − k 2 sin2 γ
2k
4.538
β = γ)
π
1 + 1 + q2
dx
1
π
arctan qx arcsin x 2 = qπ ln + ln q + 1 + q 2 − − arctan q
2
x
2
2
2
1+q
0
4.
(α, β, γ real;
(β = γ)
π − 4λ
dx
π
=
ln cos
2 cos2 λ
1
−
x
2
cos
λ
8
0
1
dx
1
π
ln
p
+
arctan p 1 − x2
=
1 + p2
1 − x2
2
0
3.
5.
sign(α)
BI (230)(7)
∞
2.
4.537
⎧
⎪
⎨
π
1 + |α/γ|
ln
x arctan (α/x) dx
2 − γ2
β
1
+ |α/β|
=
πα
2 + β 2 ) (x2 + γ 2 )
⎪
(x
−∞
⎩ 2
2β (|β| + |α|)
4.536
BI (245)(11)
BI (245)(13)
BI (252)(10, 11)
BI (252)(18, 19)
BI (244)(10)a
ET I 222(47)
BI (341)(14)a
4.573
Inverse and direct trigonometric functions
605
4.55 Combinations of inverse trigonometric functions and exponentials
4.551
1
1.9
(arcsin x) e−bx dx =
0
1
2.
π
πe−b
[I 0 (b) − L0 (b)] −
2b
2b
π
1
[L0 (b) − I 0 (b) + b L1 (b) − b I 1 (b)] +
2b2
b
x (arcsin x) e−bx dx =
0
∞
3.9
arctan
0
∞
4.9
arccot
0
1 +π
x −bx
e
dx =
a
b 2
[Re b > 0]
,
− ci(ab) sin(ab) + si(ab) cos(ab)
ET I 161(3)
[Re b > 0]
ET I 161(4)
x
1
1
q
dx
=
ln Γ(q) − q −
e2πx − 1
2
2
0
∞
4.553
0
ET I 161(2)
x −bx
1
e
dx = [ci(ab) sin(ab) − si(ab) cos(ab)]
a
b
∞ arctan
4.552
ET I 160(1)
2
arccot x − e−px
π
ln q + q −
1
ln 2π
2
[q > 0]
WH
[p > 0]
NT 66(12)
dx
= C + ln p
x
4.56 A combination of the arctangent and a hyperbolic function
∞
arctan e−x
1
dx =
2q
2
cosh
px
−∞
4.561
√
Π(x)
π 3 Γ(q)
dx
=
2q
4p Γ q + 12
−∞ cosh px
[q > 0]
∞
LI (282)(10)
4.57 Combinations of inverse and direct trigonometric functions
4.571
4.572
4.573
1.
2.
π/2
sin x dx
π
arcsin (k sin x) = − ln k 2
2k
0
1 − k 2 sin x
∞
2
arccot x − cos px dx = C + ln p
π
0
p
π
1 − e− q
2p
0
∞
p
p
1
arccot qx cos px dx =
e− q Ei
2p
q
0
3.
[p > 0,
NT 66(12)
∞
1±q
sin px dx
π
ln
=±
p
1 ± 2q cos px + q 2
2pq 1 ± qe− r
q±1
π
ln
=±
p
2pq q ± e− r
q > 0]
BI (347)(1)a
q > 0]
BI (347)(2)a
p
− e Ei −
q
p
q
[p > 0,
arccot rx
0
[p > 0]
∞
arccot qx sin px dx =
BI (344)(2)
p2 < 1,
r > 0,
p>0
q 2 > 1,
r > 0,
p>0
BI (347)(10)
606
Inverse Trigonometric Functions
∞
4.
arccot px
0
4.574 1.
∞
2a
x
arctan
0
∞
2.7
arctan
0
3.
arctan
0
∞
4.
arctan
0
4.575
π
1.
π
arctan
1.
2.
4.577
π
2ax
π
sin(bx) dx = e
x2 + c2
b
2
x2
cos(bx) dx =
π/2
1.
0
[Re a > 0,
r > 0]
b > 0]
BI (347)(9)
ET I 87(8)
π
p sin x
cos nx sin x dx =
1 − p cos x
4
b > 0]
ET I 29(7)
sinh(ab)
π −b
e sin b
b
[b > 0]
ET I 87(9)
[b > 0]
ET I 29(8)
p2 < 1
pn−1
pn+1
+
n+1 n−1
2
p <1
n+1
BI (345)(4)
BI (345)(5)
n−1
p
p
−
n+1 n−1
2
p <1
dx
π 1+p
p sin x
= ln
1 − p cos x sin x
2 1−p
0
π
dx
π p sin x
= − ln 1 − p2
arctan
1 − p cos x tan x
2
0
arctan
q > 0,
[a > 0,
√
−b a2 +c2
π
p sin x
sin nx cos x dx =
1 − p cos x
4
0
4.576
arctan
π
3.
π −ab
e
sinh(ab)
b
π n
p sin x
sin nx dx =
p
1 − p cos x
2n
0
sin(bx) dx =
arctan
0
2.
1
tan x dx
r
π
= 2 ln 1 + tanh
2
2
2r
q
p
x + r sin x
[p > 0,
cos2
1 −ab
a
cos(bx) dx =
e
Ei(ab) − eab Ei(−ab)
x
2b
∞
q2
4.574
p2 < 1
p2 < 1
BI (345)(6)
BI(346)(1)
BI(346)(3)
sin2 x dx
arctan tan λ 1 − k 2 sin2 x 1 − k 2 sin+2 x
,
π
= 2 F (λ, k) − E (λ, k) + cot λ 1 − 1 − k 2 sin2 λ
2k
BI (344)(4)
2.
0
π/2
cos2 x dx
arctan tan λ 1 − k 2 sin2 x 1 − k+2 sin2 x
,
π
2
= 2 E (λ, k) − k F (λ, k) + cot λ
1 − k 2 sin2 λ − 1
2k
BI (344)(5)
4.601
Change of variables in multiple integrals
607
4.58 A combination involving an inverse and a direct trigonometric function and a
power
∞
4.58110
dx
=
x
arctan x cos px
0
∞
arctan
0
dx
π
x
cos x
= − Ei(−p)
p
x
2
[Re(p) > 0]
ET I 29(3), NT 25(13)
4.59 Combinations of inverse trigonometric functions and logarithms
4.591 1.
1
0
1
arcsin x ln x dx = 2 − ln 2 − π
2
BI (339)(1)
arccos x ln x dx = ln 2 − 2
BI (339)(2)
1
2.
0
∞
1
4.592
arccos x
0
4.593
(2k − 1)!! ln(2k + 2)
dx
=−
ln x
2k k!
2k + 1
1
1.
arctan x ln x dx =
0
π
1
1
ln 2 − + π 2
2
4
48
1
arccot x ln x dx = −
2.
0
BI (339)(8)
k=0
1
4.594
n−1
arctan x (ln x)
BI (339)(3)
1 2 π 1
π − − ln 2
48
4
2
(ln x + n) dx =
0
−n
n!
2 − 1 ζ(n + 1)
n+1
(−2)
BI (339)(4)
BI (339)(7)
4.6 Multiple Integrals
4.60 Change of variables in multiple integrals
4.601 1.
f (x, y) dx dy =
f [ϕ(u, υ), ψ(u, υ)] |Δ| du dυ
(σ )
(σ)
where x = ϕ(u, υ), y = ψ(u, υ), and Δ =
2.
D(ϕ, ψ)
∂ϕ ∂ψ ∂ψ ∂ϕ
−
≡
is the Jacobian determinant
∂u ∂υ ∂u ∂υ
D(u, υ)
of the functions ϕ and ψ.
f (x, y, z) dx dy dz =
f [ϕ(u, υ, w), ψ(u, υ, w), χ(u, υ, w)|Δ| du dυ dw]
(V )
(V )
where x = ϕ(u, υ, w), y = ψ(u, υ, w), and z = χ(u, υ, w) and where
! ∂ϕ ∂ϕ ∂ϕ !
!
!
∂υ
∂w !
! ∂u ∂ψ
D(ϕ, ψ, χ)
∂ψ !
≡
Δ = !! ∂ψ
∂u
∂υ
∂w !
! ∂χ ∂χ ∂χ ! D(u, υ, w)
∂u
∂υ
∂w
is the Jacobian determinant of the functions ϕ, ψ, and χ.
Here, we assume, both in (4.601 1) and in (4.601 2) that
608
Multiple Integrals
(a)
the functions ϕ, ψ, and χ and also their ﬁrst partial derivatives are continuous in the region
of integration;
the Jacobian does not change sign in this region;
there exists a one-to-one correspondence between the old variables x, y,z and the new ones
u, υ,w in the region of integration;
when we change from the variables x, y,z to the variables u, υ,w, the region V (resp. σ) is
mapped into the region V (resp. σ ).
(b)
(c)
(d)
4.602
Transformation to polar coordinates:
x = r cos ϕ,
4.603
4.602
y = r sin ϕ;
D(x, y)
=r
D(r, ϕ)
Transformation to spherical coordinates:
x = r sin θ cos ϕ,
y = r sin θ sin ϕ,
z = r cos θ,
D(x, y, z)
= r2 sin θ
D(r, θ, ϕ)
4.61 Change of the order of integration and change of variables
4.611
α
1.
x
dx
f (x, y) dy =
0
0
α
2.
dx
0
α
f (x, y) dx
0
y
β
αx
β
f (x, y) dy =
0
α
dy
α
dy
f (x, y) dx
α
βy
0
4.612
R
1.
0
2.
0
√
R2 −x2
dx
f (x, y) dy =
dy
0
0
2p
√
dx
√R2 −y2
R
q/p
2px−x2
f (x, y) dy =
0
f (x, y) dx
0
q
dy
0
+ √
,
p 1+ 1−(y/q)2
+ √
,
p 1− 1−(y/q)2
f (x, y) dx
4.613
4.613
Change of the order of integration and change of variables
α
1.
dx
0
β/(β+x)
β/(β+α)
f (x, y) dy =
0
0
+
α
dx
0
1
2α
3.
δ
β+γ
dx
3α−x
α
0
R
√
2 αy
f (x, y) dy =
dy
f (x, y) dx+
03α 0 3α−y
+
dy
f (x, y) dx
α
4.
f (x, y) dx
0
αβ y/β
f (x, y) dy =
dy
f (x, y) dx
0
0δ (δ−y)/γ
+
dy
f (x, y) dx
αβ
0
, a > 0, β > 0, γ > 0
x2 /4α
0
β(1−y)/y
dy
δ−νx
βx
α=
f (x, y) dx
0
β/(β+α)
2.
α
dy
x+2R
dx √
f (x, y) dy =
R2 −x2
0
R
dy √
f (x, y) dx
0
R2 −y 2
2R R
+
dy
f (x, y) dx
R
0
3R R
+
dy
f (x, y) dx
R
2R
y−2R
609
610
Multiple Integrals
π/2
4.614
4.615
0
2R
4.616
dx
4.617
α
π/2
0
ϕ2 (x)
dx
β
f (x, y)dy =
ϕ1 (x)
+
f (r cos ϕ, r sin ϕ) r dr
0
ϕ2 (x)
f (x, y)dy −
α
dx
0
2R cos ϕ
dϕ
0
0
ϕ1 (x)
dx
β
ϕ1 (x)
dx
α
f (x, y)dy −
0
0
ϕ2 (x)
f (x, y)dy
0
f (x, y) dy
0
[ϕ1 (x) ≤ ϕ2 (x) for α ≤ x ≤ β]
1
f (x, y) dy =
dx f [x, zϕ(x)] ϕ(x)dz [y = zϕ(x)]
0
0
0
1 0 ϕ(γz)
=γ
dz
f (γz, y) dy
[x = γz]
0
0
x1 y1
x1 1
dx
f (x, y) dy =
dx
(y1 − y0 ) f [x, y0 + (y1 − y0 ) t] dt
4.619
f (r cos ϕ, r sin ϕ) r dr
0
f (x, y) dy =
0
4.618
R
dϕ
2R−x2
0
β
π/2
0
√
dx
0
0
f (x, y) dy =
0
r
2R
f (r, ϕ) dϕ
0
√
R2 −x2
dx
arccos
dr
0
R
2R
f (r, ϕ) dr =
0
2R cos ϕ
dϕ
4.614
γ
dx
x0
ϕ(x)
y0
γ
x0
0
[y = y0 + (y1 − y0 ) t]
4.62 Double and triple integrals with constant limits
4.620
1.
General formulas
π ∞
π sign p
dω
f (p cosh x + q cos ω sinh x) sinh x dx = − f sign p p2 − q 2
p2 − q 2
0
0
2
2
p >q ,
lim f (x) = 0
x→+∞
LO III 389
4.621
Double and triple integrals with constant limits
2π
2.
∞
dω
0
611
f [p cosh x + (q cos ω + r sin ω) sinh x] sinh x dx
0
2π sign p
= −
f sign p p2 − q 2 − r2
2
2
2
p −q −r
2
2
2
lim f (x) = 0
LO III 390
p >q +r ,
x→+∞
π
π
3.
0
0
2π sign p
dx dy
p − q cos x
+ r cot y = − f sign p p2 − q 2 − r2
2 f
sin x sin y
sin x sin y
p2 − q2 − r2
lim f (x) = 0
p2 > q 2 + r2 ,
x→+∞
∞
4.
dx
−∞
LO III 280
∞
f (p cosh x cosh y + q sinh x cosh y + r sinh y) cosh y dy
−∞
2π sign p
= −
f sign p p2 − q 2 − r2
p2 − q 2 − r2
lim f (x) = 0
LO III 390
p2 > q 2 + r2 ,
x→+∞
5.
π
∞
∞
dx
f (p cosh x + q cos ω sinh x) sinh2 x sin ω dω = 2
f sign p p2 − q 2 cosh x sinh2 x dx
0
0
0
lim f (x) = 0
LO III 391
x→+∞
6.
∞
dx
0
2π
π
f [p cosh x + (q cos ω + r sin ω) sin θ sinh x] sinh2 x sin θ dθ
∞
f sign p p2 − q 2 − r2 cosh x sinh2 x dx
=4
0
p2 > q 2 + r2 ,
lim f (x) = 0
LO III 390
dω
0
0
x→+∞
7.
0
∞
dx
2π
dω
0
0
π
f {p cosh x + [(q cos ω + r sin ω) sin θ + s cosh θ] sinh x} sinh2 x sin θ dθ
∞
f sign p p2 − q 2 − r2 − s2 cosh x sinh2 x dx
= 4π
0
2
2
2
2
p >q +r +s ,
lim f (x) = 0
LO III 391
x→+∞
4.621
1.
2.
3.
1 − k 2 sin2 x sin2 y
π
dx dy = √
2
2
1 − k sin y
2 1 − k2
0
0
π/2 π/2
cos y 1 − k 2 sin2 x sin2 y
dx dy = K(k)
1 − k 2 sin2 y
0
0
π/2 π/2
sin α sin y dx dy
πα
=
2
2
2
2
0
0
1 − sin α sin x sin y
π/2 π/2
sin y
LO I 252(90)
LO I 252(91)
LO I 253
612
Multiple Integrals
4.622
π π
π
0
0
0
π π
√ 2
2
dx dy dz
= 4π K 2
1 − cos x cos y cos z
1.
4.622
MO 137
√
π
dx dy dz
= 3π K 2 sin
3
−
cos
y
cos
z
−
cos
x
cos
z
−
cos
x
cos
y
12
0
0
0
π π π
+
+
√
√
√ ,
√
dx dy dz
= 4π 18 + 12 2 − 10 3 − 7 6 K 2 2 − 3
0
0
0 3 − cos x − cos y − cos z
2.
3.
π
MO 137
√
√ ,
3− 2
MO 137
∞ ∞
4.6233
0 π 0
2π
π
ϕ a2 x2 + b2 y 2 dx dy =
2ab
∞
ϕ x2 x dx
0
4.624
f (α cos θ + β sin θ cos ψ + γ sin θ sin ψ) sin θ dθ dψ
π
1
= 2π
f (R cos p) sin p dp = 2π
f (Rt) dt
−1
+0
,
R = α2 + β 2 + γ 2
a b
−3/2
4.6258 pl (a, b) =
dx
dy x2 + y 2 + 1
P l 1/ x2 + y 2 + 1
0
0
0
0
Then, for even and odd subscripts:
(−1)
ab
1
√
• p2l (a, b) =
l(2l + 1)22l a2 + b2 + 1
k=0
×(2l + 2k + 1)
k
j=0
2j
j
2
2l+2k
l+k
2k
k
(2k + 1)
l−k−1 2k
l−1
1
22j (a2 + b2 + 1)j
l+k
l−k−1
1
k−j+1
(a2 + 1)
+
1
k−j+1
(b2 + 1)
(−1)l+k l
l+k+1
2l + 2k + 1
1
• p2l+1 (a, b) = 2l+1
k
k
l+k
2
(2l + 1)
22k
k=0
b
a
a
b
1
1
√
√
arctan−1 √
+
arctan−1 √
×
k
k
2+1
2+1
2+1
2+1
2
2
b
b
a
a
(b + 1)
(a + 1)
⎫
k
⎬
2j−1
2
1
1
1
+ab
+
·
j
k−j+1
k−j+1 ⎭
2j
(a2 + b2 + 1)
(a2 + 1)
(b2 + 1)
j=1 j
j
l
4.63–4.64 Multiple integrals
x
1
(x − t)n−1 f (t) dt,
(n − 1)! p
p
p
p
where f (t) is continuous on the interval [p, q] and p ≤ x ≤ q.
4.631
x
dtn−1
tn−1
dtn−2 . . .
t1
f (t) dt =
FI II 692
4.635
Multiple integrals
4.632
···
1.
dx1 dx2 · · · dxn =
x1 ≥0,x2 ≥0,...,xn ≥0
x1 +x2 +···+xn ≤h
613
hn
n!
[the volume of an n-dimensional simplex]
√
πn
···
dx1 dx2 · · · dxn =
Rn
n
+1
Γ
x21 +x22 +···+x2n ≤R2
2
2.
FI III 472
[the volume of an n-dimensional sphere]
FI III 473
dx1 dx2 · · · dxn
π (n+1)/2
[n > 1]
n+1
1 − − − · · · − x2n
Γ
x21 +x22 +···+x2n ≤1
2
half-area of the surface of an (n + 1)-dimensional sphere x21 + x22 + · · · + x2n+1 = 1
8
4.634
x1p1 −1 xp22 −1 · · · xnpn −1 dx1 dx2 . . . dxn
···
···
4.633
x1
q1
x21
=
x22
x1 ≥0,x2 ≥0,...,xn ≥0
α2
x
n αn ≤1
+ q2
+···+( x
qn )
α1
2
p1
p2
pn
Γ
...Γ
α1
α2
αn
p1
p2
pn
Γ
+
+ ··· +
+1
α1
α2
αn
qi > 0, i = 1, 2, . . . , n] FI III 477
q p1 q p2 . . . qnpn
= 1 2
α1 α2 . . . αn
[αi > 0,
4.635
1.
···
8
x1
q1
FI III 474
f
x1 ≥0,x2 ≥0,...,xn ≥0
α1
α2
x
n αn ≥1
+ q2
+···+( x
qn )
x1
q1
α1
+
x2
q2
pi > 0,
α2
+ ··· +
xn
qn
Γ
αn 2
×x1p1 −1 xp22 −1 · · · xnpn −1 dx1 dx2 · · · dxn
q p1 q p2 . . . qnpn
= 1 2
α1 α2 · · · αn
Γ
p1
p2
pn
Γ
...Γ
α1
α2
αn
p1
p2
pn
Γ
+
+ ··· +
α1
α2
αn
∞
p1
p2
pn
f (x)x α1 + α2 +···+ αn −1 dx
1
under the assumption that the integral on the right converges absolutely.
FI III 487
614
Multiple Integrals
2.
···
8
x1
q1
f
x1 ≥0,x2 ≥0,··· ,xn ≥0
α2
x
n αn ≤1
+ q2
+···+( x
qn )
x1
q1
α1
+
x2
q2
4.636
α2
+ ··· +
αn xn
qn
α1
2
×x1p1 −1 x2p2 −1 · · · xnpn −1 dx1 dx2 · · · dxn
q p1 q p2 . . . qnpn
= 1 2
α1 α2 . . . αn
Γ
p1
p2
pn
Γ
···Γ
α1
α2
αn
p1
p2
pn
Γ
+
+ ··· +
α1
α2
αn
1
p1
p2
pn
f (x)x α1 + α2 +···+ αn −1 dx
0
under the assumptions that the one-dimensional integral on the right converges absolutely and
that the numbers qi , αi , and pi are positive.
FI III 479
In particular,
···
3.
xp11 −1 xp22 −1 . . . xnpn −1 e−q(x1 +x2 +···+xn ) dx1 dx2 . . . dxn
x1 ≥0,x2 ≥0,...,xn ≥0
x1 +x2 +···+xn ≤1
=
···
4.8
x1 ≥0,x2 ≥0,··· ,xn ≥0
α
α
n
x1 1 +x2 2 +···+xα
n ≤1
Γ (p1 ) Γ (p2 ) . . . Γ (pn ) 1 p1 +p2 +···+pn −1 −qx
x
e
dx
Γ (p1 + p2 + · · · + pn ) 0
[n > 0, p1 > 0, p2 > 0, . . . , pn > 0]
xp11 −1 x2p2 −1 . . . xnpn −1
α2
αn μ dx1 dx2 . . . dxn
1
(1 − xα
1 − x2 − · · · − xn )
Γ(1 − μ)
=
α1 α2 . . . αn
[p1 > 0,
4.636
1.
8
···
x1 ≥0,x2 ≥0,...,xn ≥0
α
α
α
x1 1 +x2 2 +···+xn n ≥1
Γ
p1
α1
Γ
p2
α2
...Γ
pn
αn
p1
p2
pn
+
+ ··· +
α1
α2
αn
p2 > 0, . . . , pn > 0, μ < 1] FI III 480
Γ 1−μ+
xp11 −1 xp22 −1 . . . xnpn −1
α2
αn μ dx1 dx2 . . . dxn
1
(xα
1 + x2 + · · · + xn )
p2
pn
p1
Γ
...Γ
1
α1
α2
αn
p1
p2
pn
p2
pn
p1
−
− ··· −
Γ
+
+ ··· +
μ−
α1
α2
αn
α1
α2
αn
p1
p2
pn
p2 > 0, . . . , pn > 0; μ >
+
+ ··· +
FI III 488
α1
α2
αn
Γ
=
α1 α2 . . . αn
p1 > 0,
4.638
Multiple integrals
···
2.8
x1 ≥0,x2 ≥0,··· ,xn ≥0
α
α
n
x1 1 +x2 2 +···+xα
n ≤1
3.
8
···
x1 ≥0,x2 ≥0,...,xn ≥0
α
α
α
x1 1 +x2 2 +···+xn n ≤1
615
xp11 −1 xp22 −1 · · · xnpn −1
α2
αn μ dx1 dx2 . . . dxn
1
(xα
1 + x2 + · · · + xn )
p1
p2
pn
Γ
Γ
...Γ
1
α1
α2
αn
=
p1
p1
p2
pn
p2
pn
α1 α2 . . . αn
+
+ ··· +
−μ
Γ
+
+ ··· +
α1
α2
α
α
α
α
n
1
2
n
p1
p2
pn
μ<
+
+ ··· +
FI III 480
α1
α2
αn
.
α2
αn
1
1 − xα
1 − x2 − · · · − xn
x1p1 −1 xp22 −1 . . . xnpn −1
dx1 dx2 . . . dxn
α1
α2
n
1 + x1 + x2 + · · · + xα
n
√ Γ
π
=
2
p1
α1
p1
...Γ
α2
α1 α2 . . . αn
Γ
p1
p2
pn
where m =
+
+ ··· +
.
α1
α2
αn
8
4.637
f (x1 + x2 + · · · + xn )
···
x1 ≥0,
x2 ≥0,...,xn ≥0
x1 +x2 +···+xn ≤1
=
pn
αn
⎧
⎪
⎪
⎨
m
Γ
1
2
m
+1
Γ(m) ⎪
⎪
⎩Γ
2
Γ
−
Γ
m+1
2
m+2
2
⎫
⎪
⎪
⎬
⎪
⎪
⎭
,
FI III 480
xp11 −1 x2p2 −1 . . . xnpn −1 dx1 dx2 . . . dxn
(q1 x1 + q2 x2 + · · · + qn xn + r)
Γ (p1 ) Γ (p2 ) . . . Γ (pn )
Γ (p1 p2 + . . . pn )
1
f (x)
0
p1 +p2 +···+pn
xp1 p2 +···pn −1
p
p dx,
(q1 x + r) (q2 x + r) 2 . . . (qn x + r) n
[q1 ≥ 0, q2 ≥ 0, . . . , qn ≥ 0; r > 0]
p1
where f (x) is continuous on the interval (0, 1).
4.638 ∞ p1 −1 p2 −1
∞
∞
x1 x2
· · · xnpn −1 e−(q1 x1 +q2 x2 +···+qn xn )
...
dx1 dx2 . . . dxn
1.
s
(r0 + r1 x1 + r2 x2 + · · · + rn xn )
0
0
0
∞
er0 x xs−1 dx
Γ (p1 ) Γ (p2 ) . . . Γ (pn )
=
p1
p2
pn
Γ(s)
0 (q1 r1 x) (q1 r2 x) . . . (qn rn x)
where pi , qi , ri , and s are positive. This result is also valid for r0 = 0, provided p1 +p2 +· · ·+pn >
s.
∞ ∞
∞
x1p1 −1 xp22 −1 . . . xnpn −1
2.
...
s dx1 dx2 . . . dxn
0
0
0 (r0 + r1 x1 + r2 x2 + · · · + rn xn )
Γ (p1 ) Γ (p2 ) . . . Γ (pn ) Γ (sp1 p2 − · · · − pn )
=
r1p1 r2p2 · · · rnpn r0s−p1 −p2 −···−pn Γ(s)
[pi > 0, ri > 0, s > 0]
∞
∞ ∞
p1 −1 p2 −1
pn −1
x1 x2
. . . xn
...
3.8
q1
q2
qn s dx1 dx2 . . . dxn
0
0
0 [1 + (r1 x1 ) + (r2 x2 ) + · · · + (rn xn ) ]
p1
p2
pn
p1
p2
pn
Γ s−
−
− ··· −
Γ
Γ
...Γ
q1
q2
qn
q1
q2
qn
=
q1 q2 . . . qn r1p1 q1 r2p2 q2 . . . rnpn qn
Γ(s)
[pi > 0, qi > 0, ri > 0, s > 0]
616
4.639
Multiple Integrals
···
1.
(p1 x1 + p2 x2 + · · · + pn xn )
x21 +x22 +···+x2n ≤1
2m
dx1 dx2 . . . dxn
√
2
m
πn
(2m − 1)!!
p1 + p22 + · · · + p2n
=
n
2m
+m+1
Γ
2
FI III 482
···
2.
4.639
(p1 x1 + p2 x2 + · · · + pn xn )
2m+1
dx1 dx2 . . . dxn = 0
FI III 483
x21 +x22 +···+x2n ≤1
4.641
···
1.11
ep1 x1 +p2 x2 +···+pn xn dx1 dx2 . . . dxn
x21 +x22 +···+x2n ≤1
=
∞
√ πn
k=0
1
k! Γ
n
+k+1
2
p21 + p22 + · · · + p2n
4
k
FI III 483
···
2.
ep1 x1 p2 x2 +···p2n x2n dx1 dx2 . . . dx2n =
x21 +x22 +···+x22n ≤1
(2π)n I n
p21 + p22 + · · · + p22n
n/2
(p21 + p22 + · · · + p22n )
FI III 483a
√ <
2 π n R n−1
2
2
2
4.642
···
f
x1 + x2 + · · · + xn dx1 dx2 . . . dxn =
x
f (x) dx,
n
0
Γ
2
2
2
2
x1 +x2 +···+xn ≤R
2
where f (x) is a function that is continuous on the interval (0, R).
1 1
1
p −1
p −1
p −1
4.643
...
f (x1 x2 · · · xn ) (1 − x1 ) 1 (1 − x2 ) 2 . . . (1 − xn ) n
0
0
FI III 485
0
×xp21 xp31 +p2 · · · xpn1 +p2 +···+pn−1 dx1 dx2 . . . dxn
Γ (p1 ) Γ (p2 ) . . . Γ (pn ) 1
=
f (x)(1 − x)p1 +p2 +···+pn −1 dx
Γ (p1 + p2 + · · · + pn ) 0
under the assumption that the integral on the right converges absolutely.
n−1
? @A B
dx1 dx2 . . . dxn−1
···
4.644
f (p1 x1 + p2 x2 + · · · + pn xn )
|xn |
x21 +x22 +···+x2n =1
dx1 dx2 · · · dxn−1
f (p1 x1 + p2 x2 + · · · + pn xn ) <
=2
···
1 − x21 − x22 − · · · − x2n−1
x21 +x22 +···+x2n−1 ≤1
√
π
<
2 π n−1
=
f
p21 + p22 + · · · + p2n cos x sinn−2 x dx
n−1
0
Γ
2
where f (x) is continuous on the interval − p21 + p22 + · · · + p2n , p21 + p22 + · · · + p2n .
FI III 488
[n ≥ 3]
FI III 489
4.648
Multiple integrals
617
4.645 Suppose that two functions f (x1 , x2 , . . . , xn ) and g (x1 , x2 , . . . , xn ) are continuous in a closed,
bounded region D and that the smallest and greatest values of the function g in D are m and M ,
respectively. Let ϕ(u) denote a function that is continuous for m ≤ u ≤M. We denote by ψ(u) the
integral
1.
ψ(u) =
···
f (x1 , x2 , . . . , xn ) dx1 dx2 . . . dxn ,
m≤g(x1 ,x2 ,...,xn )≤u
2.
over that portion of the region D on which the inequality m ≤ g (x1 , x2 , . . . , xn ) ≤ u is satisﬁed.
Then
···
f (x1 , x2 , . . . , xn ) ϕ [g (x1 , x2 , . . . , xn )] dx1 dx2 . . . dxn
m≤g(x1 ,x2 ,...,xn )≤M
= (S)
M
ϕ(u)d ψ(u) = (R)
m
M
ϕ(u)
m
d ψ(u)
du
du
where the middle integral must be understood in the sense of Stieltjes. If the derivative dψ
du exists
and is continuous, the Riemann integral on the right exists.
M
" +∞
M may be +∞ in formulas 4.645 2, in which case m should be understood to mean lim
.
M →+∞
···
4.6468
x1 ≥0,x2 ≥0,...,xn ≥0
x1 +x2 +···+xn ≤1
xp11 −1 xp22 −1 . . . xnpn −1
r dx1 dx2 . . . dxn
(q1 x1 + q2 x2 + · · · + qn xn )
∞
xr−1 dx
Γ (p1 ) Γ (p2 ) . . . Γ (pn )
=
p1
Γ (p1 + p2 + · · · + pn − r + 1) Γ(r) 0 (1 + q1 x) (1 + q2 x)p2 · · · (1 + qn x)pn
p2 > 0, . . . , pn > 0, q1 > 0, q2 > 0, . . . , qn > 0, p1 + p2 + · · · + pn > r > 0]
= [p1 > 0,
···
4.647
exp
0≤x21 +x22 +···+x2n ≤1
p1 x1 + p2 x2 + · · · + pn xn
x21 + x22 + · · · + x2n
=
∞ ∞
4.6488
0
0
m
∞
···
0
FI III 493
dx1 dx2 . . . dxn
√
2 πn
n
n (p21 + p22 + · · · + p2n ) 4
exp − x1 + x2 + · · · + xn +
λn+1
x1 x2 . . . xn
1
− 12
I n2 −1
<
p21 + p22 + · · · + p2n
FI III 495
−1
2
−1
n
×xc1n+1 x2n+1 . . . xnn+1
n
1
=√
(2π) 2 e−(n+1)λ
n+1
−1
dx1 dx2 · · · dxn
FI III 496
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5 Indeﬁnite Integrals of Special
Functions
5.1 Elliptic Integrals and Functions
Notation: k =
√
1 − k 2 (cf. 8.1).
5.11 Complete elliptic integrals
5.111
K(k)k2p+3 dk =
1.
2.
1
(2p + 3)2
4(p + 1)2
+
,
2
K(k)k2p+1 dk + k 2p+2 E(k) − (2p + 3) K(k)k
⎧
⎨
1
E(k)k2p+3 dk = 2
4(p + 1)2 E(k)k2p+1 dk
4p + 16p + 15 ⎩
− E(k)k2p+2
+
BY (610.04)
⎫
⎬
2
2
(2p + 3)k − 2 − k 2p+2 k K(k)
⎭
,
BY (611.04)
5.112
1.
⎡
⎤
∞
2
πk ⎣
[(2j)!] k 2j ⎦
K(k) dk =
1+
4
4j
2
j=1 (2j + 1)2 (j!)
⎡
2.6
E(k) dk =
3.
∞
2
BY (610.00)
⎤
2j
[(2j)!] k
πk ⎣
⎦
1−
2 − 1) 24j (j!)4
2
(4j
j=1
K(k)k dk = E(k) − k K(k)
2
BY (611.00)
BY (610.01)
4.
5.
,
1 +
2
1 + k 2 E(k) − k K(k)
3
,
1 +
2
4 + k 2 E(k) − k 4 + 3k 2 K(k)
K(k)k3 dk =
9
E(k)k dk =
619
BY (611.01)
BY (610.02)
620
Elliptic Integrals and Functions
5.113
6.
7.
8.
,
1 +
2
4 + k 2 + 9k 4 E(k) − k 4 + 3k 2 K(k)
BY 611.02)
45
,
1 +
2
64 + 16k2 + 9k 4 E(k) − k 64 + 48k 2 + 45k 4 K(k)
K(k)k5 dk =
BY (610.03)
225
,
1 +
2
64 + 16k 2 + 9k 4 + 225k 6 E(k) − k 64 + 48k 2 + 45k 4 K(k)
E(k)k5 dk =
1575
E(k)k3 dk =
BY (611.03)
9.
10.
11.
12.
13.
5.113
E(k)
K(k)
dk = −
2
k
k
+
,
E(k)
1 2
k
dk
=
K(k)
−
2
E(k)
k2
k
E(k)
dk = k K(k)
k 2
,
E(k)
1 + 2
2
2
k
dk
=
−
2
E(k)
+
k
K(k)
k4
9k 3
k E(k)
dk = K(k) − E(k)
k 2
dk
= − E(k)
k
+
, dk
2
2
E(k) − k K(k)
= 2 E(k) − k K(k)
k
dk
2
= −k K(k)
1 + k 2 K(k) − E(k)
k
,
dk
1+
2
E(k) − k K(k)
[K(k) − E(k)] 2 =
k
k
+
, dk
1
2
E(k) − k K(k) 2 2 = [K(k) − E(k)]
k
k k
+
, dk
E(k)
2
1 + k2 E(k) − k K(k)
4 =
kk
k 2
[K(k) − E(k)]
1.
2.
3.
4.
5.
6.
5.114
5.115
1.
2.
3.
k K(k) dk
E(k) −
k 2
1
2 = 2
k
K(k)
− E(k)
K(k)
BY (612.05)
BY (612.02)
BY (612.01)
BY (612.03)
BY (612.04)
BY (612.06)
BY (612.09)
BY (612.12)
BY (612.07)
BY (612.13)
BY (612.11)
π 2
π 2
, r , k k dk = k 2 − r2 Π
, r , k − K(k) + E(k)
2
2
+
,
π 2
π 2
K(k) − Π
, r , k k dk = k2 K(k) − k 2 − r2 Π
,r ,k
2
2
2
π 2
E(k)
π 2
2
,
r
,r ,k
Π
+
Π
,
k
k
dk
=
k
−
r
2
2
2
k
Π
BY (612.14)
BY (612.15)
BY (612.16)
5.124
Elliptic integrals
621
5.12 Elliptic integrals
2
+
F (x, k) dx
[F (x, k)]
π,
=
0<x≤
2
2
0
1 − k 2 sin2 x
x
2
[E (x, k)]
5.12211
E (x, k) 1 − k 2 sin2 x dx =
2
0
5.123
x
1
F (x, k) sin x dx = − cos x F (x, k) + arcsin (k sin x)
1.
k
0
.
x
1 − k 2 sin2 x 1
1
2.
F (x, k) cos x dx = sin x F (x, k) + arccosh
− arccosh
k
k
k 2
0
x
5.121
5.124
BY (630.32)
BY (630.11)
1
k
BY (630.21)
,
1+
k sin x 1 − k 2 sin2 x+arcsin (k sin x)
2k
0
⎡
x
1 ⎣
k cos x 1 − k 2 sin2 x
E (x, k) cos x dx = sin x E (x, k) +
2k
0
⎤
.
2
2
1
1
−
k
sin
x
2
2
⎦
− k arccosh
− k + k arccosh
k
k 2
x
E (x, k) sin x dx = − cos x E (x, k)+
1.
2.
3.∗
a
0
x E(x) dx
π
=
2√ 2
2
2
2
2
4
(k + k x ) a − x
λ = arcsin
4.
BY (630.01)
∗
0
∗
5.
0
a
x E(x) dx
π
=
2√ 2
2
2
2
4
(k − x ) a − x
π/2
BY (630.12)
BY (630.22)
√
a 1 − a2
2
(k 2 + k 2 a2 )
√
k 2
a
+ k 2 a2
+
a2 E (λ, k)
3/2
k 2 (k 2 + k 2 a2 )
k = 1 − k2
+
1−a
2
F (λ, k)
3/2
(k 2 + k 2 a2 )
[0 < a < 1,
0 < k < 1]
√
F (φ, k)
a2 E (φ, k)
a 1 − a2
+ √
+
3/2
k 2 (k 2 − a2 ) k 2 k 2 − a2
k 2 (k 2 − a2 )
a
[0 < a < k < 1]
φ = arcsin
k
E (x, k ) sin x cos x dx
1 − k 2 cosh2 v sin2 x
1 − k 2 sin2 x
tanh v
1
π tanh v π
− [F (φ, k) − E (φ, k)]
= 2
E(k ) arctanh
−
k sinh v cosh v
k
2
2
tanh v
2
φ = arcsin
[0 < tanh v < k < 1]
k = 1−k
k
622
6.∗
Elliptic Integrals and Functions
π/2
0
E (x, k) sin x cos x dx
2
2 ψ sin2 x
1 − k cos
1 − k 2 sin2 x
1
= 2
k sin ψ cos ψ
E(k) arctan
tan ψ
k
β = arctan
7.
∗
∗
π/2
π/2
5.124
π
tan ψ
π
1 − 1 − k 2 cos2 ψ
− E (β, k) + 2
2
2
2 1 − k cos ψ
+
π,
tan ψ
0 < k < 1, 0 < ψ <
k = 1 − k2
k
2
E (x, k ) sin x cos x dx
0
1 + k 2 sinh2 μ sin2 x
1 − k 2 sin2 x
<
−1
π
2
2
= 2
E(k ) arctanh (k tanh μ) −
F (φ, k) − E (φ, k) + tanh μ 1 + k sinh μ
k sinh μ cosh μ
2
<
π
− coth μ 1 − 1 + k2 sinh2 μ
2
[0 < k < 1, 0 < tanh μ < 1]
φ = arcsin(tanh μ) k = 1 − k 2
8.
0
F (x, k ) sin x cos x dx
1 + k 2 sinh2 μ sin2 x
1 − k 2 sin2 x
+
,
−1
π
F
(φ,
k)
K(k
)
arctanh
(k
tanh
μ)
−
k 2 sinh μ cosh
2
μ
2
[0 < k < 1, 0 < tanh μ < 1]
φ = arcsin(tanh μ) k = 1 − k
=
9.∗
π/2
0
F (x, k ) sin x cos x dx
1 − k 2 cosh2 ν sin2 x
1 − k 2 sin2 x
tanh ν
1
π
F
(φ,
k)
)
arctanh
K(k
−
k 2 sinh ν cosh ν
k
2
tanh ν
[0 < k < 1, 0 < tanh ν < 1]
k = 1 − k2
k
=
φ = arcsin
10.∗
π/2
0
F (x, k) sin x cos x dx
2
1 − k cos2 ψ sin2 x
1 − k 2 sin2 x
tan ψ
1
π
F
(β,
k)
)
arctanh
−
K(k
k 2 sin ψ cos ψ
k
2
tan ψ
[0 < k < 1, 0 < ψ < 1]
k = 1 − k2
k
=
β = arctan
11.∗
b
ln
a
+x
−x
x2 dx
(x2
−
a2 ) (b2
−
x2 )
=
π 2 2
− ( − a2 ) (2 − b2 ) + πβ [F (φ, k) − E (φ, k)]
β
a
[0 < a < b < ]
φ = arcsin
k=
b
5.131
5.125
Jacobian elliptic functions
x
1.
Π x, α2 , k sin x dx
⎡.
⎤
2
2
k
−
α
1
= − cos x Π x, α2 , k + √
arctan ⎣
sin x⎦
1 − k 2 sin2 x
k 2 − α2
⎡.
⎤
2
2
α −k
1
= − cos x Π x, α2 , k + √
arctanh ⎣
sin x⎦
1 − k 2 sin2 x
α2 − k 2
0
2.
623
2
α < k2
2
α > k2
BY (630.13)
x
Π x, α2 , k cos x dx = sin x Π x, α2 , k − f − f0
0
% &
2 1 − α2 α2 − k 2 + 1 − α2 sin2 x 2k 2 − α2 − α2 k 2
f= arctan
2 (1 − α2 ) (α2 − k 2 )
2α2 (1 − α2 ) (α2 − k 2 ) cos x 1 − k2 sin2 x for 1 − α2 α2 − k 2 > 0;
% 2
2 α2 − 1 α2 − k 2 + 1 − α2 sin x α2 + α2 k 2 − 2k 2
1
= ln
1 − α2 sin2 x
2 (α2 − 1) (α2 − k 2 )
&
2α2 (α2 − 1) (α2 − k 2 ) cos x 1 − k 2 sin2 x
+
1 − α2 sin2 x
for 1 − α2 α2 − k 2 < 0,
f0 is the value of f at x = 0 BY (630.23)
where
1
Integration with respect to the modulus
2
5.126
F (x, k)k dk = E (x, k) − k F (x, k) +
1 − k 2 sin2 x − 1 cot x
BY (613.01)
,
+
1 2
1 + k 2 E (x, k) − k F (x, k) +
5.127
E (x, k)k dk =
1 − k 2 sin2 x − 1 cot x
BY (613.02)
3
5.128
Π x, r2 , k k dk = k 2 − r2 Π x, r2 , k − F (x, k) + E (x, k) +
1 − k 2 sin2 x − 1 cot x
BY (613.03)
5.13 Jacobian elliptic functions
5.131
1.
⎡
1
⎣ sn m+1 u cn u dn u + (m + 2) 1 + k2
sn m u du =
sn m+2 u du
m+1
⎤
− (m + 3)k2 sn m+4 u du⎦
SI 259, PE(567)
624
Elliptic Integrals and Functions
⎡
cn m u du =
2.
1
⎣ − cn m+1 u sn u dn u
(m + 1)k 2
+ (m + 2) 1 − 2k
2
cn m+2 u du + (m + 3)k 2
⎤
cn m+4 u du⎦
PE (568)
⎡
dn m u du =
3.
5.132
1
⎣k 2 dn m+1 u sn u cn u
(m + 1)k 2
+ (m + 2) 2 − k
2
⎤
dn m+2 u du − (m + 3) dn m+4 u du⎦
PE (569)
By using formulas 5.131, we can reduce the integrals (for m = 1)
dn m u du to the integrals 5.132, 5.133 and 5.134.
5.132
sn u
du
= ln
1.
sn u
cn u + dn u
dn u − cn u
= ln
sn u
1
du
k sn u + dn u
= ln
2.
cn u
k
cn u
1
du
k sn u − cn u
= arctan 3.
dn u k
k sn u + cn u
1
cn u
= arccos
k
dn u
1
cn u + ik sn u
= ln
ik
dn u
1
k sn u
= arcsin
k
dn u
"
5.133
1.
2.
"
sn m u du,
"
cn m u du, and
H 87(164)
SI 266(4)
SI 266(5)
H 88(166)
JA
SI 266(6)
JA
1
ln (dn u − k cn u)
k
dn u − k 2 cn u
1
= arccosh
k
1 − k2
dn u − cn u
1
= arcsinh k
k
1 − k2
1
= − ln (dn u + k cn u)
k
1
cn u du = arccos (dn u) ;
k
i
= ln (dn u − ik sn u) ;
k
1
= arcsin (k sn u)
k
sn u du =
H 87(161)
JA
;
JA
SI 365(1)
H 87(162)
SI 265(2)a, ZH 87(162)
JA
5.137
Jacobian elliptic functions
625
3.
dn u du = arcsin (sn u) ;
= am u = i ln (cn u − i sn u)
5.134
1.
2.
3.
H 87(163)
SI 266(3), ZH 87(163)
1
[u − E (am u, k)]
k2
,
1 +
2
cn 2 u du = 2 E (am u, k) − k u
k
dn 2 u du = E (am u, k)
sn 2 u du =
5.135 1
dn u + k sn u
du = ln
1.
cn u
k
cn u
1
dn u + k = ln
2k
dn u − k i
sn u
ik − k cn u
du = ln
2.
dn u
kk
dn u
1
k cn u
= arccot
kk
k
1 − dn u
cn u
du = ln
3.
sn u
sn u
1 1 − dn u
= ln
2 1 + dn u
1 1 − k sn u
cn u
du = − ln
4.
dn u
k
dn u
1 + k sn u
1
ln
=
2k 1 − k sn u
1 1 + sn u
dn u
du = ln
5.
cn u
2 1 − sn u
1 + sn u
= ln
cn u
1 1 − cn u
dn u
du = ln
6.
sn u
2 1 + cn u
PE (564)
PE (565)
PE (566)
SI 266(7)
H 88(167)
SI 266(8)
SI 266(10)
H 88(168)
SI 266(9)
H 88(172)
JA
H 87(170)
5.136 1
1.
sn u cn u du = − 2 dn u
k
2.
sn u dn u du = − cn u
3.
cn u dn u du = sn u
5.137
1.
1 dn u
sn u
du = 2
2
cn u
k cn u
H 88(173)
626
Elliptic Integrals and Functions
2.
3.
4.
5.
6.
5.138
1 cn u
sn u
du = − 2
dn 2 u
k dn u
dn u
cn u
du = −
sn 2 u
sn u
sn u
cn u
du =
dn 2 u
dn u
cn u
dn u
du = −
sn 2 u
sn u
sn u
dn u
du =
cn 2 u
cn u
H 88(175)
H 88(174)
H 88(177)
H 88(176)
H 88(178)
5.138 sn u
cn u
du = ln
1.
sn u dn u
dn u
1
sn u
dn u
du = 2 ln
2.
cn u dn u
cn u
k
sn u
dn u
du = ln
3.
sn u cn u
cn u
H 88(183)
H 88(182)
H 88(184)
5.139 cn u dn u
du = ln sn u
1.11
sn u
1
sn u dn u
du = ln
2.
cn u
cn u
1
sn u cn u
du = − 2 ln dn u
3.
dn u
k
H 88(179)
H 88(180)
H 88(181)
5.14 Weierstrass elliptic functions
The invariants g1 and g2 used below are deﬁned in 8.161.
5.141 ℘(u) du = − ζ(u)
1.
2.
3.
4.8
1 1
℘ (u) + g2 u
6
12
1
3
1
℘ (u) − g2 ζ(u) + g3 u
℘3 (u) du =
120
20
10
1
du
σ(u − v)
= 2u ζ(v) + ln
℘(u) − ℘(v)
℘ (v)
σ(u + v)
℘2 (u) du =
5.
H 120(192)
H 120(193)
[℘(v) = e1 , e2 , e3 ]
(see 8.162)
H 120(194)
au αδ − βγ
α℘(u) + β
σ(u + v)
du =
+ 2 − 2u ζ(v)
ln
γ℘(u) + δ
γ
γ ℘ (v)
σ(u − v)
where v = ℘−1
−δ
γ
H 120(195)
5.221
The exponential integral function and powers
627
5.2 The Exponential Integral Function
5.21 The exponential integral function
∞
5.211
Ei(−βx) Ei(−γx) dx =
x
1
1
+
β
γ
Ei[−(β + γ)x]
−x Ei(−βx) Ei(−γx) −
e−γx
e−βx
Ei(−γx) −
Ei(−βx)
β
γ
[Re(β + γ) > 0]
NT 53(2)
5.22 Combinations of the exponential integral function and powers
5.221
∞
1.
x
k=0
[a > 0,
∞
2.
x
3.∗
4.∗
n−1
1
Ei[−a(x + b)]
(−1)n Ei[−a(x + b)] e−ab (−1)n−k−1 ∞ e−ax
+
dx = n −
dx
k+1
xn+1
x
bn
n
n
bn−k
x x
Ei[−a(x + b)]
dx =
x2
1 1
+
x b
Ei[−a(x + b)] −
b > 0]
NT 52(3)
e−ab Ei(−ax)
b
[a > 0, b > 0]
NT 52(4)
1
xe−ax
x2
Ei(−ax) + 2 e−ax +
[a > 0]
2
2a
2a
∞
n!e−ax (ax)k
xn+1
Ei(−ax) +
xn Ei(−ax) dx =
n+1
(n + 1)an+1
k!
x Ei(−ax) dx =
k=0
5.∗
6.∗
7.∗
8.∗
[a > 0]
x Ei(−ax)e−bx dx =
[a > 0,
2
Ei(−ax)e−ax − Ei(−2ax)
Ei2 (−ax) dx = x Ei2 (−ax) +
a
x Ei2 (−ax) dx =
x2 2
Ei (−ax) +
2
Ei(−ax) dx = u Ei(−au) +
0
1
x
+
a2
a
Ei(−ax)e−ax −
1
1
Ei(−2ax) + 2 e−2ax
a2
a
[a > 0]
u
0
9.∗
b > 0]
[a > 0]
1
1
x
1
e−(a+b)x
Ei [−(a + b)x] − 2 Ei(−ax)e−bx − Ei(−ax)e−bx −
2
b
b
b
b(a + b)
∞
x Ei −
x
x
Ei −
a
b
dx =
e
−au
−1
[a > 0]
a
a2 + b 2
2
ln(a + b) −
b2
ab
a2
ln a −
ln b −
2
2
2
[a > 0,
b > 0]
628
10.
The Sine Integral and the Cosine Integral
∗
∞
0
x
x
x Ei −
Ei −
a
b
2
5.231
ab 2
2 3
3
3
3
2
a + b ln(a + b) − a ln a − b ln b −
a − ab + b
dx =
3
a+b
[a > 0,
b > 0]
5.23 Combinations of the exponential integral and the exponential
5.231
x
ex Ei(−x) dx = − ln x − C + ex Ei(−x)
1.
ET II 308(11)
0
1.
x
e−βx Ei(−αx) dx = −
0
1
β
e−βx Ei(−αx) + ln 1 +
β
α
− Ei[−(α + β)x]
ET II 308(12)
5.3 The Sine Integral and the Cosine Integral
5.31
1.
cos αx ci(βx) dx =
sin αx ci(βx) dx = −
2.
5.32
1.
cos αx si(βx) dx =
1.
cos αx ci(βx) ci(αx + βx) + ci(αx − βx)
+
α
2α
NT 49(2)
NT 49(3)
cos αx si(βx) si(αx + βx) − si(αx − βx)
+
α
2α
NT 49(4)
1
(si(αx + βx) + si(αx − βx))
ci(αx) ci(βx) dx = x ci(αx) ci(βx) +
2α
1
1
1
+
(si(αx + βx) + si(βx − αx)) − sin αx ci(βx) − sin βx ci(αx)
2β
α
β
NT 53(5)
1
(si(αx + βx) + si(αx − βx))
2β
1
1
1
−
(si(αx + βx) + si(βx + αx)) + cos αx si(βx) + cos βx si(αx)
2α
α
β
si(αx) si(βx) dx = x si(αx) si(βx) −
2.
3.
NT 49(1)
sin αx si(βx) ci(αx + βx) − ci(αx − βx)
+
α
2α
sin αx si(βx) dx = −
2.
5.33
sin αx ci(βx) si(αx + βx) + si(αx − βx)
−
α
2α
NT 54(6)
1
cos αx ci(βx)
α
1
1
1
− sin βx si(αx) −
+
ci(αx + βx) −
β
2α 2β
si(αx) ci(βx) dx = x si(αx) ci(βx) +
1
1
−
2α 2β
ci(αx − βx)
NT 54(10)
5.54
Combinations of the exponential integral and the exponential
5.34
∞
si[a(x + b)]
1.
x
1 1
+
x b
dx
=
x2
si[a(x + b)] −
cos ab si(ax) + sin ab ci(ax)
b
[a > 0,
∞
2.
ci[a(x + b)]
x
1 1
+
x b
dx
=
x2
ci[a(x + b)] +
629
b > 0]
NT 52(6)
sin ab si(ax) − cos ab ci(ax)
b
[a > 0,
b > 0]
NT 52(5)
5.4 The Probability Integral and Fresnel Integrals
5.41
11
5.42
5.43
e−α x
Φ(αx) dx = x Φ(αx) + √
α π
cos2 αx2
√
S (αx) dx = x S (αx) +
α 2π
sin2 αx2
C (αx) dx = x C (αx) − √
α 2π
2
2
NT 12(20)a
NT 12(22)a
NT 12(21)a
5.5 Bessel Functions
Notation: Z and Z denote any of J, N , H (1) , H (2) . In formulae 5.52–5.56, Z p (x) and Zp (x) are arbitrary
Bessel functions of the ﬁrst, second, or third kinds.
∞
J p+2k+1 (x)
JA, MO 30
5.51
J p (x) dx = 2
k=0
5.52
1.
2.11
5.5310
xp+1 Z p (x) dx = xp+1 Z p+1 (x)
WA 132(1)
x−p Z p+1 (x) dx = −x−p Z p (x)
WA 132(2)
p2 − q 2
α2 − β 2 x −
Z p (αx) Zq (βx) dx
x
= αx Z p+1 (αx) Zq (βx) − βx Z p (αx) Zq+1 (βx) − (p − q) Z p (αx) Zq (βx)
= βx Z p (αx) Zq−1 (βx) − αx Z p−1 (αx) Zq (βx) + (p − q) Z p (αx) Zq (βx)
JA, MO 30, WA 134(7)
5.54
1.
10
αx Z p+1 (αx) Zp (βx) − βx Z p (αx) Zp+1 (βx)
α2 − β 2
βx Z p (αx) Zp−1 (βx) − αx Z p−1 (αx) Zp (βx)
=
α2 − β 2
x Z p (αx) Zp (βx) dx =
WA 134(8)
2.
2
x [Z p (αx)] dx =
x2 2
[Z p (αx)] − Z p−1 (αx) Z p+1 (αx)
2
WA 135(11)
630
3.
Bessel Functions
∗
x Z p (ax) Zp (ax) dx =
10
5.55
5.55
x4 2 Z p (ax) Zp (ax) − Z p−1 (ax) Zp+1 (ax) − Z p+1 (ax) Zp−1 (ax)
4
Z p (αx) Zq+1 (αx) − Z p+1 (αx) Zq (αx) Z p (αx) Zq (αx)
1
Z p (αx) Zq (αx) dx = αx
+
x
p2 − q 2
p+q
Z p−1 (αx) Zq (αx) − Z p (αx) Zq−1 (αx) Z p (αx) Zq (αx)
= αx
−
p2 − q 2
p+q
WA 135(13)
5.56
1.
Z 1 (x) dx = − Z 0 (x)
JA
x Z 0 (x) dx = x Z 1 (x)
JA
2.
6–7 Deﬁnite Integrals of Special
Functions
6.1 Elliptic Integrals and Functions
Notation: k =
√
1 − k 2 (cf. 8.1).
6.11 Forms containing F (x, k)
π/2
6.111
F (x, k) cot x dx =
0
6.112
π/2
1.
0
F (x, k)
2
π
sin x cos x
1
√ −
K(k) ln
K(k )
dx =
4k
1 − k sin2 x
(1 − k) k 16k
F (x, k)
sin x cos x
1
dx = − 2 ln k K(k)
2k
1 − k 2 sin2 x
0
π/2
3.
0
6.113
π/2
1.
F (x, k )
0
π/2
2.
F (x, k)
0
BI (350)(1)
√
(1 + k) k
π
sin x cos x
1
K(k) ln
+
K(k )
F (x, k)
dx =
4k
2
16k
1 + k sin2 x
π/2
2.
π
1
K(k ) + ln k K(k)
4
2
2
sin x cos x dx
1
√ K(k )
ln
=
4(1 − k) (1 + k) k
cos2 x + k sin2 x
BI (350)(6)
BI (350)(7)
BI (350)(2)a, BY(802.12)a
BI (350)(5)
sin x cos x
dx
·
1 − k 2 sin2 t sin2 x
1 − k 2 sin2 x
=−
,
+
1
π
F
(t,
k)
tan
t)
−
K(k)
arctan
(k
k 2 sin t cos t
2
BI (350)(12)
6.114
6.115
dx
1
K(k) K
1 − tan2 u cot2 v
=
2 cos u sin v
u
sin2 x − sin2 u sin2 v − sin2 x
2
BI (351)(9)
k = 1 − cot2 u · cot2 v
√
1
(1 + k) k
π
x dx
1
K(k) ln
+
K(k )
F (arcsin x, k)
=
2
1
+
kx
4k
2
16k
0
(cf. 6.112 2)
BI (466)(1)
v
F (x, k) <
631
632
Elliptic Integrals and Functions
6.121
This and similar formulas can be obtained from formulas 6.111–6.113 by means of the substitution
x = arcsin t.
6.12 Forms containing E (x, k)
1 sin x cos x
2
1 + k K(k) − (2 + ln k ) E(k)
dx = 2
BI (350)(4)
2
2
2k
1 − k sin x
0 π/2
dx
1
6.122
E (x, k) BI (350)(10), BY (630.02)
= {E(k) K(k) − ln k }
2
2
0
1 − k 2 sin x
π/2
sin x cos x
dx
6.123
E (x, k)
·
2
2
2
1 − k sin t sin x
0
1 − k 2 sin2 x
+
,
π
1
π
E(k) arctan (k tan t) − E (t, k) + cot t 1 − 1 − k 2 sin2 t
=− 2
k sin t cos t
2
2
BI (350)(13)
⎛.
⎞
v
2
dx
1
⎝ 1 − tg u ⎠
E(k)
K
6.124
E (x, k) <
=
2 cos u sin v
tg 2 v
u
sin2 x − sin2 u sin2 v − sin2 x
⎞
⎛.
k 2 sin v
sin2 2u ⎠
+
K⎝ 1−
2 cos u
sin2 2v
2
k = 1 − cot2 u cot2 v
BI (351)(10)
π/2
6.121
E (x, k)
6.13 Integration of elliptic integrals with respect to the modulus
x
1 − cos x
= tan
sin
x
2
0 1
sin2 x + 1 − cos x
6.132
E (x, k)k dk =
3 sin x
0 1
2 1 + r sin x
x
− r2 Π x, r2 , 0
6.133
Π x, r , k k dk = tan − r ln
2
1 − r sin x
0
1
6.131
F (x, k)k dk =
BY (616.03)
BY (616.04)
BY (616.05)
6.14–6.15 Complete elliptic integrals
6.141
1
1.
K(k) dk = 2G
FI II 755
π2
4
BY (615.03)
0
1
2.
K(k ) dk =
0
1
dk
= π ln 2 − 2G
k
0
√ 1
2
dk
1 4
2
7
Γ
6.143
K(k) = K
=
k
2
16π
0
1
π2
dk
=
6.144
K(k)
1+k
8
0
6.142
K(k) −
π
2
BY (615.05)
1
4
BY (615.08)
BY (615.09)
6.161
The theta function
,
dk
1 +
2
=
24 (ln 2) − π 2
k
12
0
1
1
2
n
2
6.146 n
k K(k) dk = (n − 1)
k n−2 K(k) dk + 1
0
0
1
1
6.147
n k n K(k ) dk = (n − 1) k n−2 E(k) dk
1
K(k ) − ln
6.145
4
k
0
6.148
E(k) dk =
1
+G
2
E(k ) dk =
π2
8
0
1
2.
0
3.
∗
1
0
6.149
1
E(k) −
0
1
2.
π
2
(E(k ) − 1)
0
3.∗
BY (615.12)
[n > 1]
(see 6.152)
BY (615.11)
BY (615.02)
BY (615.04)
E(k)
dk = 1
1+k
1.
BY (615.13)
0
1
1.
633
dk
π
= π ln 2 − 2G + 1 −
k
2
BY (615.06)
dk
= 2 ln 2 − 1
k
BY (615.07)
1
E(k)
dk = 1
1+k
1
dx E(x)
π
a − x2 K(x) − √
+ x2
3
2
x
4
1−x
0
⎡
0
4.∗
6.152
6.1536
6.154
⎤
π
ln
4
1
4
√
e
√ ⎢
2
dk
1⎢
2
4K
+
E(k) = ⎢
⎢
k
8⎣
2
⎥
⎥
π2
√ ⎥
⎥
0
2 ⎦
K2
2
1
1
(n + 2) k n E(k ) dk = (n + 1) k n K(k ) dk
0
0
a
1+a
1
K(k)k dk
π
√
= √
ln
2
2
2
2
4
1−a
k
a −k
1−a
0 π/2
E (p sin x)
π
sin x dx = 2 sin2 x
1
−
p
2
1
− p2
0
6.151
=−
BY (615.10)
[n > 1]
(see 6.147)
[0 < a < 1]
p2 > 1
BY (615.14)
LO I 252
FI II 489
6.16 The theta function
6.161 1.
∞
0
2.
0
∞
s
xs−1 ϑ2 0 | ix2 dx = 2s 1 − 2−s π − 2 Γ 12 s ζ(s)
s
xs−1 ϑ3 0 | ix2 − 1 dx = π − 2 Γ 12 s ζ(s)
[Re s > 2]
ET I 339(20)
[Re s > 2]
ET I 339(21)
634
Elliptic Integrals and Functions
∞
3.
0
6.162
1
xs−1 1 − ϑ4 0 | ix2 dx = 1 − 21−s π − 2 s Γ 12 s ζ(s)
[Re s > 2]
∞
4.
0
ET I 339(22)
1
xs−1 ϑ4 0 | ix2 + ϑ2 0 | ix2 − ϑ3 0 | ix2 dx = − (2s − 1) 21−s − 1 π − 2 s Γ 12 s ζ(s)
ET I 339(24)
6.162
∞
1.11
∞
2.
e−ax ϑ1
!
bπ !! iπx
2l ! l2
e−ax ϑ2
!
(l + b)π !! iπx
! l2
2l
e−ax ϑ3
!
(l + b)π !! iπx
! l2
2l
0
∞
3.11
0
∞
4.11
√ √ l
dx = √ cosh b a cosech l a
a
bπ
2l
0
!
! iπx
!
! l2
e−ax ϑ4
0
[Re a > 0,
∞
1.
0
[Re a > 0,
∞
6.16411
[Re a > 0,
|b| ≤ l]
ET I 224(3)a
|b| ≤ l]
ET I 224(4)a
√ √ l
dx = √ cosh b a cosech l a
a
√
√
1 √
√ √ e−(a−μ)x ϑ3 (π μx |iπx ) dx = √ coth a + μ + coth a − μ
2 a
ϑ3 (iπkx | iπx) e−(k
2
+l2 )x
dx =
ET I 224(7)a
sinh 2l
l (cosh 2l − cos 2k)
1
ϑ4 0 | ie2x + ϑ2 0 | ie2x − ϑ3 0 | ie2x e 2 x cos(ax) dx
0
=
6.165
ET I 224(2)a
√ √ l
dx = − √ sinh b a sech l a
a
0
|b| ≤ l]
[Re a > 0]
∞
2.10
ET I 224(1)a
√ √ l
dx = − √ sinh b a sech l a
a
[Re a > 0,
6.16310
|b| ≤ l]
∞
e
1
2x
1
1
2 2 +ia − 1
2
ϑ3 0 | ie2x − 1 cos(ax) dx
0
=
1 − 2 2 −ia π − 4 − 2 ia Γ
1
[a > 0]
1
1
1
4
+ 12 ia ζ 12 + ia
ET I 61(11)
+
− 1 ia− 1 1
,
2
2
1
1
1
2
4 Γ
1
+
a
π
ζ
ia
+
+
ia
+
4
2
4
2
1 + 4a2
[a > 0]
ET I 61(12)
6.165
Generalized elliptic integrals
635
6.1710 Generalized elliptic integrals
1.
Set
π
Ωj (k) ≡
−(j+ 12 )
1 − k 2 cos φ
dφ,
0
j! (4m + 2j)!
π
αm (j) =
(64)m (2j)! (2m + j)!
then
Ωj (k) =
∞
αm (j)k4m =
m=0
× erf λ −
2
√
π
×
2
√
π
erf λ −
1
m!
2
,
π
λ=
2
(2j + 1)k 2
,
1 − k2
⎡
π
−j
⎣ erf λ + 1 (2j + 1)−1 1 + 1
1 − k2
2
(2j + 1)k
2
2k 2
2
1
1
13
2
λe−λ
− (2j + 1)−2 16 + 2 + 4
1 + λ2
3
12
k
k
⎤
2
4
2
λe−λ
+ . . .⎦
1 + λ2 + λ4
3
15
while for large λ
lim Ωj (k) =
j→∞
2.
−j
π
k2 1 − k2
(2j + 1)
1
1
1
4
13
−1
−2
× 1 + (2j + 1)
+
1 + 2 − (2j + 1)
1+
+ ...
2
2k
3
16k 2
16k 4
Set
Rμ (k, α, δ) =
π
cos2α−1 (θ/2) sin2δ−2α−1 (θ/2) dθ
,
μ+ 1
[1 − k 2 cos θ] 2
0 < k < 1, Re δ > Re α > 0, Re μ > −1/2,
(−1)ν 2ν μ + 12 ν Γ(α) Γ (δ − α + ν)
Mν (μ, α, δ) =
,
ν!
Γ(δ + ν)
with (λ)ν = Γ(λ + ν)/ Γ(λ), and
ν
2 μ + 12 ν Γ(α + ν) Γ (δ − α)
,
Wν (μ, α, δ) =
ν!
Γ(δ + ν)
0
then:
• for small k:
Rμ (k, α, δ) = 1 − k
1
2 −(μ+ 2 )
−(μ+ 12 )
= 1 + k2
∞
ν=0
∞
ν=0
ν
k2 / 1 − k2
Mν (μ, α, δ)
ν
k2 / 1 + k2
W ν (μ, α, δ),
636
The Exponential Integral Function and Functions Generated by It
6.211
• for k 2 close to 1:
Rμ (k, α, δ)
2 α−δ δ−α−μ− 12
2k
1 − k2
= Γ(δ − α) Γ μ + α − δ + 12 Γ μ + 12
+ μ+ 12 ,
× Γ δ − α − μ − 12 Γ(α) Γ δ − μ − 12 2k 2
Re μ + α − δ + 12 not an integer
,
+
1
= 2μ+ 2 k 2μ+1 Γ μ + 12 Γ(1 − α)
×
∞
n=0
α−δ+μ−n+ 1
2
Γ (δ − α + n) Γ(1 − α + n) Γ α − δ + μ − n + 12 n! 2k 2 / 1 − k 2
α − δ + μ + 12 = m, with m a non-negative integer
6.2–6.3 The Exponential Integral Function and Functions Generated
by It
6.21 The logarithm integral
1
li(x) dx = − ln 2
6.211
BI (79)(5)
0
6.212
1
1.
li
0
1
2.
0
1
x
li(x)
0
dx
1
= ln(1 − q)
xq+1
q
∞
4.
li(x)
1
6.213
1
li
1.
0
∞
2.
li
1
3.
1
li
0
4.
5.
BI (255)(1)
1
li(x)xp−1 dx = − ln(p + 1)
p
1
3.
x dx = 0
∞
dx
1
= − ln(q − 1)
xq+1
q
1
x
1
x
1
x
1
x
sin (a ln x) dx =
π
1
a ln a −
2
1+a
2
sin (a ln x) dx = −
cos (a ln x) dx = −
1
1 + a2
π
+ a ln a
2
π
1
ln a + a
1 + a2
2
π
1
ln a − a
2
1
+
a
2
1
1
ln 1 + a2
dx
=
li(x) sin (a ln x)
x
2a
0
li
cos (a ln x) dx =
[p > −1]
BI (255)(2)
[q < 1]
BI (255)(3)
[q > 1]
BI (255)(4)
[a > 0]
BI (475)(1)
[a > 0]
BI (475)(9)
[a > 0]
BI (475)(2)
[a > 0]
BI (475)(10)
[a > 0]
BI(479)(1), ET I 98(20)a
6.216
The logarithm integral
1
6.
li(x) cos (a ln x)
arctan a
dx
=−
x
a
li(x) sin (a ln x)
dx
1
π
=
a ln a +
x2
1 + a2
2
0
1
7.
0
∞
8.
dx
1
=
2
x
1 + a2
li(x) sin (a ln x)
1
1
9.
0
∞
10.
1
1
p−1
11.
π
− a ln a
2
π
dx
1
ln a + a
=−
2
2
x
1+a
2
li(x) cos (a ln x)
BI (479)(2)
π
dx
1
ln a − a
=
x2
1 + a2
2
li(x) cos (a ln x)
li(x) sin (a ln x) x
0
1
dx = 2
a + p2
637
[a > 0]
BI (479)(3)
[a > 0]
BI (479)(13)
[a > 0]
BI (479)(4)
[a > 0]
BI (479)(14)
a a
ln (1 + p)2 + a2 − p arctan
2
1+p
[p > 0]
1
li(x) cos (a ln x) xp−1 dx = −
12.
0
6.214
1
1.
li
0
1
x
∞
2.
li
1
6.215
1
1.
1
2.
6.216
1
0
2.
0
1
li(x) ln
li(x) ln
1
x
π
√
arcsinh p = −2
p
[p > 0]
BI (477)(2)
[0 0]
BI (444)(3)
[1 > p > 0]
BI (444)(4)
ax
1
= − Γ(p)
x
p
[0 < p ≤ 1]
BI (444)(1)
dx
π Γ(p)
=−
2
x
sin pπ
[0 < p < 1]
BI (444)(2)
ln
1
x
π
Γ(p)
sin pπ
dx = −2
1
x
xp+1
1.
dx = −
dx
.
li(x)
0
p−1
xp−1
ln
p a
+ ln (1 + p)2 + a
1+p 2
dx = −π cot pπ · Γ(p)
(ln x)
li(x) .
0
1
x
a arctan
BI (477)(1)
p−1
1
x
ln
1
a2 + p2
2
= −2
π
√
arcsin p
p
638
The Exponential Integral Function and Functions Generated by It
6.221
6.22–6.23 The exponential integral function
6.221
6.222
1 − eαp
α
0
∞
1 1
+
Ei(−px) Ei(−qx) dx =
p
q
0
p
Ei(αx) dx = p Ei(αp) +
∞
6.223
0
NT 11(7)
ln(p + q) −
Γ(μ)
Ei(−βx)xμ−1 dx = − μ
μβ
ln q ln p
−
p
q
[p > 0,
q > 0]
[Re β ≥ 0,
FI II 653, NT 53(3)
Re μ > 0]
NT 55(7), ET I 325(10)
6.224
∞
1.
Ei(−βx)e−μx dx = −
0
μ
1
ln 1 +
μ
β
= −1/β
[Re(β + μ) ≥ 0,
μ > 0]
[μ = 0]
FI II 652, NT 48(8)
∞
2.
Ei(ax)e−μx dx = −
0
μ
1
ln
−1
μ
a
[a > 0,
Re μ > 0,
μ > a]
ET I 178(23)a, BI (283)(3)
6.225
∞
1.
Ei −x
2
e
−μx2
0
∞
2.
0
6.226
∞
1.
2 px2
π
√
arcsin p
Ei −x e dx = −
p
Ei −
0
∞
2.
Ei
0
∞
3.
0
∞
4.
0
6.227
1.
0
∞
π
π
√
√
arcsinh μ = −
ln
dx = −
μ+ 1+μ
μ
μ
1
4x
a2
4x
Ei −
e−μx dx = −
2
√
K 0 ( μ)
μ
2
√
K 0 (a μ)
μ
π
√
−μx2
Ei (− μ)
dx =
e
μ
e−μx dx = −
1
4x2
1
Ei − 2
4x
e
−μx2 +
Ei(−x)e−μx x dx =
[Re μ > 0]
BI (283)(5), ET I 178(25)a
[1 > p > 0]
NT 59(9)a
[Re μ > 0]
MI 34
[a > 0,
MI 34
Re μ > 0]
[Re μ > 0]
MI 34
1
π
√
√
√
√
2
4x dx =
[cos μ ci μ − sin μ si μ]
μ
1
1
− 2 ln(1 + μ)
μ(μ + 1) μ
[Re μ > 0]
MI 34
[Re μ > 0]
MI 34
6.241
The sine integral and cosine integral functions
∞
2.
0
e−ax Ei(ax) eax Ei(−ax)
−
dx = 0
x−b
x+b
= π 2 e−ab
[a > 0,
b < 0]
[a > 0,
b > 0]
639
ET II 253(1)a
6.228
∞
1.
Ei(−x)ex xν−1 dx = −
0
∞
2.
π Γ(ν)
sin νπ
Ei(−βx)e−μx xν−1 dx = −
0
∞
[0 < Re ν < 1]
Γ(ν)
μ
2 F 1 1, ν; ν + 1;
ν(β + μ)ν
β+μ
[|arg β| < π, Re(β + μ) > 0,
ET II 308(13)
Re ν > 0]
√
dx
√
√
√
√
= 2 π (cos μ si μ − sin μ ci μ)
2
x
0
[Re μ > 0]
∞
−x −μx
1
[a < 1, Re μ > 0]
Ei(−a) − Ei −e
e
dx = γ(μ, a)
μ
− ln a
6.229
6.231
Ei −
1
4x2
exp −μx2 +
ET II 308(14)
1
4x2
MI 34
MI 34
6.232
∞
1.
ln 1 +
Ei(−ax) sin bx dx = −
0
∞
2.
0
6.233
∞
1.
2b
b
1
Ei(−ax) cos bx dx = − arctan
b
a
Ei(−x)e−μx sin βx dx = −
0
b2
a2
1
β 2 + μ2
[a > 0,
b > 0]
BI (473)(1)a
[a > 0,
b > 0]
BI (473)(2)a
β β
ln (1 + μ)2 + β 2 − μ arctan
2
1+μ
[Re μ > |Im β|]
∞
2.
Ei(−x)e−μx cos βx dx = −
0
β2
1
+ μ2
μ β
ln (1 + μ)2 + β 2 + β arctan
2
1+μ
[Re μ > |Im β|]
BI (473)(7)a
BI (473)(8)a
∞
Ei(−x) ln x dx = C + 1
6.234
NT 56(10)
0
6.24–6.26 The sine integral and cosine integral functions
6.241
∞
1.
si(px) si(qx) dx =
π
2p
[p ≥ q]
BI II 653, NT 54(8)
ci(px) ci(qx) dx =
π
2p
[p ≥ q]
FI II 653, NT 54(7)
0
2.
0
∞
640
The Exponential Integral Function and Functions Generated by It
∞
3.
0
p+q
1
ln
si(px) ci(qx) dx =
4q
p−q
1
= ln 2
q
2
2
2
p − q2
1
ln
+
4p
q4
6.242
[p = q]
[p = q]
FI II 653, NT 54(10, 12)
∞
6.242
0
6.243
1
ci(ax)
2
2
dx = −
[si(aβ)] + [ci(aβ)]
β+x
2
2.
6.244
∞
1.8
2.
x dx
π
= − ci(pq)
q 2 − x2
2
[p > 0,
q > 0]
BI (255)(6)
[p > 0,
q > 0]
BI (255)(7)
[p > 0,
q > 0]
BI (255)(8)
[a > 0,
0 < Re μ < 1]
∞
ci(px)
0
∞
1.
q2
dx
π
Ei(−pq)
=
2
+x
2q
dx
π
si(pq)
=
q 2 − x2
2q
si(ax)xμ−1 dx = −
0
ET II 253(2)
si(px)
0
2.
[a > 0]
BI (255)(6)
ci(px)
6.246
ET II 253(3)
q > 0]
∞
1.
b > 0]
[p > 0,
0
[a > 0,
x dx
π
= Ei(−pq)
q 2 + x2
2
∞
8
6.245
ET II 224(1)
si(px)
0
|arg β| < π]
∞
si (a|x|)
sign x dx = π ci (a|b|)
−∞ x − b
∞
ci (a|x|)
dx = −π sign b · si (a|b|)
−∞ x − b
1.
[a > 0,
Γ(μ)
μπ
sin
μ
μa
2
NT 56(9), ET I 325(12)a
∞
2.
ci(ax)xμ−1 dx = −
0
Γ(μ)
μπ
cos
μaμ
2
[a > 0,
0 < Re μ < 1]
NT 56(8), ET I 325(13)a
6.247
∞
μ
1
arctan
μ
β
0
.
∞
1
μ2
−μx
ci(βx)e
dx = − ln 1 + 2
μ
β
0
1.
2.
6.248
1.
8
si(βx)e−μx dx = −
∞
si(x)e
0
−μx2
π
x dx =
Φ
4μ
1
√
2 μ
[Re μ > 0]
NT 49(12), ET I 177(18)
[Re μ > 0]
NT 49(11), ET I 178(19)a
[Re μ > 0]
MI 34
−1
6.253
The sine integral and cosine integral functions
2.
6.249
∞
1
π
Ei −
μ
4μ
0
∞+
2 π , −μx
π
si x +
e
S
dx =
2
μ
0
ci(x)e
6.251 1.
∞
0
6.252 1.
∞
0
[Re μ > 0]
μ2
4
1
−
2
2
+ C
2
√
kei (2 μ)
μ
1
x
e−μx dx =
ci
1
x
e−μx dx = −
∞
2.
1
dx =
4
si
0
−μx2
641
2
√
ker (2 μ)
μ
2 ME 26
[Re μ > 0]
MI 34
[Re μ > 0]
MI 34
=0
1
−
2
[Re μ > 0]
π
2p
π
=−
4p
sin px si(qx) dx = −
μ2
4
MI 34
p2 > q 2
p2 = q 2
p2 < q 2
FI II 652, NT 50(8)
∞
2.6
cos px si(qx) dx = −
0
=
1
ln
4p
1
q
p+q
p−q
2
p2 = q 2
p = 0,
[p = 0]
FI II 652, NT 50(10)
∞
3.
sin px ci(qx) dx = −
0
1
ln
4p
p2
−1
q2
2
=0
[p = 0]
π
2p
π
=−
4p
0
cos px ci(qx) dx = −
=0
6.253
0
FI II 652, NT 50(9)
∞
4.
p2 = q 2
p = 0,
∞
m
m+1
p2 > q
2
p2 = q 2
p2 < q 2
FI II 654, NT 50(7)
π r +r
si(ax) sin bx
dx = −
2
2
1 − 2r cos x + r
4b(1
− r) (1 −mr ) m+1 π 2 + 2r − r − r
=−
4b(1 − r) (1 − r2 )
πrm+1
=−
2
2b(1
− r) (1 − r )
π 1 + r − rm+1
=−
2b(1 − r) (1 − r2 )
[b = a − m]
[b = a + m]
[a − m − 1 < b < a − m]
[a + m < b < a + m + 1]
ET I 97(10)
642
The Exponential Integral Function and Functions Generated by It
6.254 1.∗
∞
0
1
dx
=
ci(x) sin x
L2
x
2
1
2
2
− L2
1
−
2
where L2 (x) is the Euler dilogarithm deﬁned as L2 (z) = −
2.11
log(1 − t)
dt and this in turn can
t
0
be expressed as L2 (z) = Φ(z, 2, 1) in terms of the Lerch function deﬁned in 9.550, with z real.
∞+
π a
π,
dx
= ln H(a − b)
si(ax) +
cos bx ·
2
x
2
b
0
[a > 0, b > 0, H(x) is the Heaviside step function] ET I 41(11)
6.255 1.
∞
[cos ax ci (a|x|) + sin (a|x|) si (a|x|)]
−∞
z
dx
= −π [sign b cos ab si (a|b|) − sin ab ci (a|b|)]
x−b
[a > 0]
∞
2.
−∞
[sin ax ci (a|x|) − sign x cos ax si (a|x|)]
6.256 1.
∞
0
∞
2
π
si (x) + ci2 (x) cos ax dx = ln(1 + a)
a
2
[si(x) cos x − ci(x) sin x] dx =
0
3.∗
∞
si2 (x) cos(ax) dx =
0
4.∗
6.257
∞
6.258 1.
∞
0
2.
0
ET II 253(5)
[a > 0]
π
2
π
log(1 + a)
2a
π
log(1 + a)
2a
0
∞
√
π
a
sin bx dx = − J 0 2 ab
si
x
2b
0
ci2 (x) cos(ax) dx =
ET II 253(4)
dx
= −π [sin (a|b|) si (a|b|) + cos ab ci (a|b|)]
x−b
[a > 0]
2.∗
6.254
[0 ≤ a ≤ 2]
[0 ≤ a ≤ 2]
[b > 0]
+
π,
dx
si(ax) +
sin bx 2
2
x + c2
π −bc
e
=
[Ei(bc) − Ei(−ac)] + ebc [Ei(−ac) − Ei(−bc)]
4c
π
= e−bc [Ei(ac) − Ei(−ac)]
4c
ET I 42(18)
[0 0]
[0 < a ≤ b,
c > 0]
BI (460)(1)
∞
+
π,
x dx
si(ax) +
cos bx 2
2
x + c2
π −bc
e
=−
[Ei(bc) − Ei(−ac)] + ebc [Ei(−bc) − Ei(−ac)]
4
π
= e−bc [Ei(−ac) − Ei(ac)]
4
[0 0]
[0 < a ≤ b,
c > 0]
BI (460)(2, 5)
6.262
6.259
The sine integral and cosine integral functions
∞
si(ax) sin bx
1.
0
dx
π
Ei(−ac) sinh(bc)
=
x2 + c2
2c
π
= e−cb [Ei(−bc) + Ei(bc) − Ei(−ac) − Ei(ac)]
4c
π
+ Ei(−bc) sinh(bc)
2c
ci(ax) sin bx
0
∞
ci(ax) cos bx
0
4.∗
x dx
π
= − sinh(bc) Ei(−ac)
2
+c
2
π
π
= − sinh(bc) Ei(−bc) + e−bc [Ei(−bc) + Ei(bc)
2
4
− Ei(−ac) − Ei(ac)]
x2
dx
x2 + c2
π
cosh bc Ei(−ac)
=
2c
π −bc
=
e
[Ei(ac) + Ei(−ac) − Ei(bc)] + ebc Ei(−bc)
4c
5.∗
[0 < a ≤ b,
c > 0]
[0 0]
[0 < a ≤ b,
c > 0]
[0 0]
[0 < a ≤ b,
c > 0]
BI (460)(4), ET I 41(15)
∞
[ci(x) sin x − Si(x) cos x] sin x
0
c > 0]
BI (460)(3)a, ET I 97(15)a
3.
[0 < b ≤ a,
ET I 96(8)
∞
2.
643
2
x dx
1
Ei(a)e−a − Ei(−a)ea
=
2
+x
8
a2
[a real]
∞
[ci(x) sin x − Si(x) cos x]
2
0
2
π x dx
π 3 e−|a|
sinh(a) −
Ei(a)e−a − Ei(−a)ea
=
2
+x
8a
8|a|
a2
[a real]
6.261
∞
1.
−px
si(bx) cos axe
0
a p2 + (a + b)2
1
2bp
ln
dx = −
+ p arctan 2
2 (a2 + p2 ) 2 p2 + (a − b)2
b − a2 − p2
[a > 0,
∞
2.
si(βx) cos axe−μx dx = −
0
1.
∞
−μx
ci(bx) sin axe
0
p > 0]
ET I 40(8)
Re μ > |Im β|]
ET I 40(9)
μ − ai
μ + ai
arctan
β
β
−
2(μ + ai)
2(μ − ai)
arctan
[a > 0,
6.262
b > 0,
1
dx =
2
2 (a + μ2 )
2
2
μ + b2 − a2 + 4a2 μ2
2aμ
a
μ arctan 2
− ln
μ + b 2 − a2
2
b4
[a > 0, b > 0, Re μ > 0]
ET I 98(16)a
644
The Exponential Integral Function and Functions Generated by It
∞
2.
ci(bx) cos axe−px dx =
0
3.
⎧
⎨p
−1
ln
2 (a2 + p2 ) ⎩ 2
,
+
2
b2 + p2 − a2 + 4a2 p2
b4
[a > 0, b > 0,
2
(μ − ai)2
(μ + ai)
∞
ln 1 +
− ln 1 +
β2
β2
−
ci(βx) cos axe−μx dx =
4(μ + ai)
4(μ − ai)
0
[a > 0,
6.263
⎫
⎬
+ a arctan
2ap
b2 + p2 − a2 ⎭
Re p > 0]
Re μ > |Im β|]
ET I 41(16)
ET I 41(17)
6.263
π
− μ ln μ
[ci(x) cos x + si(x) sin x] e
dx = 2
1 + μ2
0
π
∞
− μ + ln μ
−μx
[si(x) cos x − ci(x) sin x] e
dx = 2
1 + μ2
0
∞
ln 1 + μ2
[sin x − x ci(x)] e−μx dx =
2μ2
0
1.
2.
3.
6.264
∞
−μx
−
[Re μ > 0]
ME 26a, ET I 178(21)a
[Re μ > 0]
ME 26a, ET I 178(20)a
[Re μ > 0]
ME 26
∞
si(x) ln x dx = C + 1
1.
NT 46(10)
0
∞
2.
ci(x) ln x dx =
0
π
2
NT 56(11)
6.27 The hyperbolic sine integral and hyperbolic cosine integral functions
6.271
∞
μ+1
1
1
ln
= arccoth μ
2μ μ − 1
μ
0
∞
1
ln μ2 − 1
2.11
chi(x)e−μx dx = −
2μ
0
∞
2
1
1 π
Ei
6.27211
chi(x)e−px dx =
4
p
4p
0
6.273
∞
ln μ
[cosh x shi(x) − sinh x chi(x)] e−μx dx = 2
1.11
μ −1
0
∞
μ ln μ
2.11
[cosh x chi(x) + sinh x shi(x)] e−μx dx =
1
− μ2
0
1.
shi(x)e−μx dx =
[Re μ > 1]
MI 34
[Re μ > 1]
MI 34
[p > 0]
MI 35
[Re μ > 0]
MI 35
[Re μ > 2]
MI 35
6.284
The probability integral
∞
11
6.274
[cosh x shi(x) − sinh x chi(x)] e
−μx2
0
1
π 4μ
1
e Ei −
μ
4μ
[Re μ > 0]
MI 35
ln μ2 − 1
−μx
[x chi(x) − sinh x] e
dx = −
[Re μ > 1]
2μ2
0 ∞
2
1
1 π
1
[cosh x chi(x) + sinh x shi(x)] e−μx x dx =
exp
Ei −
3
8
μ
4μ
4μ
0
[Re μ > 0]
6.275
6.276
6.277
1
dx =
4
645
∞
∞
1.
[chi(x) + ci(x)] e
−μx
0
∞
2.
ln μ4 − 1
dx = −
2μ
[chi(x) − ci(x)] e−μx dx =
0
μ2 + 1
1
ln 2
2μ μ − 1
MI 35
MI 35
[Re μ > 1]
MI 34
[Re μ > 1]
MI 35
6.28–6.31 The probability integral
6.281
1.
∞
6
[1 − Φ(px)] x
2q−1
0
Γ q + 12
√
dx =
2 πqp2q
[Re q > 0,
Re p > 0]
NT 56(12), ET II 306(1)a
∞
2.6
0
b
2b
1 − Φ atα ± α
dt = √
t
π
b
a
1−α
2α
+
,
K 1+α (2ab) ± K 1−α (2ab) e±2ab
2α
2α
[a > 0,
6.282
∞
1.
Φ(qt)e−pt dt =
0
∞
2.
0
1
Φ x+
2
1
1−Φ
p
1
2
−Φ
p
2q
exp
p2
4q 2
b > 0,
+
Re p > 0,
α = 0]
|arg q| <
π,
4
MO 175, EH II 148(11)
e−μx+ 4
1
(μ + 1)2
1
exp
dx =
1−Φ
(μ + 1)(μ + 2)
4
μ+1
2
ME 27
6.283
1.
2.
√
√ α
1
√
eβx 1 − Φ αx dx =
−1
β
α−β
0
∞
√
√ −pt
q
1
√
Φ qt e
dt =
p
p
+q
0
∞
[Re α > 0,
Re β < Re α]
[Re p > 0,
Re(q + p) > 0]
ET II 307(5)
EH II 148(12)
∞
6.284
0
1−Φ
q
√
2 x
e−px dx =
1 −q√p
e
p
+
Re p > 0,
|arg q| <
π,
4
EF 147(235), EH II 148(13)
646
6.285
The Exponential Integral Function and Functions Generated by It
∞
1.
[1 − Φ(x)] e−μ
2
x2
dx =
0
∞
2.
Φ(iat)e−a
t −st
2 2
dt =
0
arctan μ
√
πμ
−1
√ exp
2ai π
6.285
[Re μ > 0]
s2
4a2
Ei −
s2
4a2 +
Re s > 0,
MI 37
|arg a| <
π,
4
EH II 148(14)a
6.286
∞
1.
Γ
2
[1 − Φ(βx)] eμ
2
x
xν−1 dx =
0
∞
%
√
1−Φ
2.
0
2x
2
ν+1
2
√
πνβ ν
2F 1
ν ν+1 ν
μ2
,
; + 1; 2
2
2
2
β
Re2 β > Re μ2 ,
Re ν > 0
ET II 306(2)
&
e
x2
2
xν−1 dx = 2 2 −1 sec
ν
ν
νπ
Γ
2
2
[0 < Re ν < 1]
6.287
∞
1.
Φ(βx)e−μx x dx =
2
0
∞
2.
β
2μ μ + β 2
[1 − Φ(βx)] e−μx x dx =
2
0
1
2μ
β
1− μ + β2
Re μ > − Re β 2 ,
ET I 325(9)
Re μ > 0
ME 27a, ET I 176(4)
Re μ > − Re β 2 ,
Re μ > 0
NT 49(14), ET I 177(9)
1
r
1
A
B
r
Q(rA)
Q(rB)
dr
=
−
3.∗
I=
exp
α
arctan
+
β
arctan
B=A
2
σ2
4 2π
αB
βA
−∞ σ
1
1
1
B=A
= − α arctan
4 π
α ∞
2
x
σ 2 A2
σ2 B 2
1
1
Q(x) = √
, α=
e−t /2 dt =
, β=
,
1 − erf √
2
2
2
1+σ A
1 + σ2 B 2
2π x
2
∞
2
ai
6.288
a > 0, Re μ > Re a2
Φ(iax)e−μx x dx = MI 37a
2
2μ μ − a
0
6.289
∞
+
2
2
2
β
π,
2
2
Re
1.
Φ(βx)e(β −μ )x x dx =
μ
>
Re
β
,
|arg
μ|
<
2μ (μ2 − β 2 )
4
0
2.
0
∞
ET I 176(5)
∞
[1 − Φ(βx)] e(β
2
−μ2 )x2
x dx =
1
2μ(μ + β)
+
Re2 μ > Re β 2 ,
arg μ <
π,
4
ET I 177(10)
6.297
The probability integral
∞
3.
√
2
Φ
b − ax e−(a+μ)x x dx =
0
6.291
6.292
∞
∞
Φ(x)e−μx
2
0
∞
1.
1−Φ
0
∞
2.
1−Φ
0
∞
b > a]
ME 27
−(μx+x2 )
|arg μ| <
6.293
6.295
[Re μ > −a > 0,
μ
μ2
i
1
+ Ei −
x dx = √
Φ(ix)e
[Re μ > 0]
4
4
π μ
0 ∞
2 2
arctan μ
1
1
[1 − Φ(x)] e−μ x x2 dx = √
−
μ3
μ2 (μ2++ 1)
2 π
0
6.294
√
b−a
√
2(μ + a) μ + b
647
1−Φ
β
x
1
x
1
x
√
dx
μ+1+1
1
= ln √
= arccoth μ + 1
x
2
μ+1−1
[Re μ > 0]
e−μ
2
x2
x dx =
1
exp(−2βμ)
2μ2
MI 37
π,
4
MI 37
MI 37a
+
π
|arg β| < ,
4
|arg μ| <
π,
4
ET I 177(11)
e−μ
2
x2
dx
= − Ei(−2μ)
x
+
π,
|arg μ| <
4
MI 37
1
x2
1
dx = √ [sin 2μ ci(2μ) − cos 2μ si(2μ)]
πμ
0
+
π,
|arg μ| <
4
∞
1
1
1
π
[H1 (2μ) − Y 1 (2μ)] − 2
2.
1−Φ
exp −μ2 x2 + 2 x dx =
x
x
2μ
μ
0
+
π,
|arg μ| <
4
∞
1
dx
π
1
= [H0 (2μ) − Y 0 (2μ)]
3.
1−Φ
exp −μ2 x2 + 2
x
x
x
2
0
+
π,
|arg μ| <
4
∞
2
2 2
a2
2
a
1 −aμ√2
ax · e− 2x2 e−μ x x dx =
6.296
e
x + a2 1 − Φ √
−
π
2μ4
2x
0
,
+
π
|arg μ| < , a > 0
4
6.297
∞
2
2
1
β
√
exp [−2 (βγ + β μ)]
1.
1 − Φ γx +
e(γ −μ)x x dx = √ √
x
2 μ μ+γ
0
1.
2.
0
∞
1−Φ
exp −μ2 x2 +
2
b + 2ax
2x
[Re β > 0,
exp − μ2 − a2 x2 + ab x dx =
Re μ > 0]
MI 37
MI 37
MI 37
MI 38a
ET I 177(12)a
−bμ
e
2μ(μ + a)
[a > 0,
b > 0,
Re μ > 0]
MI 38
648
The Exponential Integral Function and Functions Generated by It
1−Φ
∞
3.
0
b − 2ax2
2x
e
−ab
+ 1−Φ
b + 2ax2
2x
e
ab
e−μx x dx =
2
6.298
1
exp −b a2 + μ
μ
MI 38
[a > 0, b > 0, Re μ > 0]
∞
2
2
b − 2ax2
b + 2ax2
1
√
exp (−b μ)
2 cosh ab − e−ab Φ
− eab Φ
e−(μ−a )x x dx =
2
2x
2x
μ
−
a
0
MI 38
[a > 0, b > 0, Re μ > 0]
∞
+
,
1
2
exp 12 a2 K ν a2
cosh(2νt) exp (a cosh t) [1 − Φ (a cosh t)] dt =
2 cos(νπ)
0
Re a > 0, − 12 < Re ν < 12
6.298
6.299
ET II 308(10)
∞
b2
1
[1 − Φ(ax)] sin bx dx =
[a > 0, b > 0]
1 − e− 4a2
b 0
√
√ ∞
b + a2 + a 2b
a 2b
1
2
√ + 2 arctan
ln
Φ(ax) sin bx dx = √
b − a2
4 2πb
b + a2 − a 2b
0
[a > 0, b > 0]
6.311
6.312
6.313
α ⎞ 12
,− 12
+
1
√ 1
α2 + β 2 2 − α
sin(βx) 1 − Φ αx dx = − ⎝ 2 2 2 ⎠
β
α +β
∞
⎛
√ cos(βx) 1 − Φ αx dx = ⎝
0
[Re α > |Im β|]
2.
0
∞
1.
sin(bx) 1 − Φ
0
a
x
ET II 307(6)
α ⎞
+
,− 12
1
2
2 2
2
⎠
α
+
β
+
α
α2 + β 2
1
2
[Re α > |Im β|]
6.314
ET I 96(3)
⎛
∞
1.
ET I 96(4)
ET II 307(7)
,
+
,
+
1
1
dx = b−1 exp −(2ab) 2 cos (2ab) 2
[Re a > 0, b > 0]
∞
+
,
+
,
1
1
a
cos(bx) 1 − Φ
dx = −b−1 exp −(2ab) 2 sin (2ab) 2
x
0
ET II 307(8)
2.
[Re a > 0,
6.315
∞
1.
ν−1
x
0
2.
0
∞
Γ 1 + 12 ν β
sin(βx) [1 − Φ(αx)] dx = √
2F 2
π(ν + 1)αν+1
1
xν−1 cos(βx) [1 − Φ(αx)] dx =
Γ 2 + 12 ν
√
πναν
2F 2
ET II 307(9)
ν+1 ν
3 ν+3
β2
, + 1; ,
;− 2
2
2
2
2
4α
[Re α > 0,
b > 0]
Re ν > −1]
ET II 307(3)
ν ν+1 1 ν
β2
,
; , + 1; − 2
2
2
2 2
4α
[Re α > 0,
Re ν > 0]
ET II 307(4)
6.323
Fresnel integrals
∞
3.
0
∞
4.
1
b2
[1 − Φ(ax)] cos bx · x dx = 2 exp − 2
2a
4a
[Φ(ax) − Φ(bx)] cos px
0
∞
5.
0
6.316
1
dx
p
=
Ei − 2
x
2
4b
√ 1
1
x− 2 Φ a x sin bx dx = √
2 2πb
∞
1 2
x
2
e
1−Φ
x
√
2
0
2
1
b2
− 2 1 − exp − 2
b
4a
− Ei
[a > 0,
p2
4a2
b > 0]
ET I 40(5)
[a > 0, b > 0, p > 0]
&
% √ &
√
b + a 2b + a2
a 2b
√
ln
+ 2 arctan
2
b − a2
b − a 2b + a
%
649
sin bx dx =
π b2
e2
2
1−Φ
[a > 0,
b
√
2
[b > 0]
b > 0]
ET I 96(3)
ET I 96(5)
√
i π − b22
e 4a
e−a x Φ(iax) sin bx dx =
[b > 0]
a 2
0 ∞
2
2
2
1 − e−p − √ (1 − Φ(p))
[1 − Φ(x)] si(2px) dx =
πp
π
0
[p > 0]
6.3176
6.318
∞
2
ET I 40(6)
2
ET I 96(2)
NT 61(13)a
6.32 Fresnel integrals
6.321
∞
1.
0
∞
2.
0
6.322
∞
1.
1
− S (px) x2q−1 dx =
2
1
− C (px) x2q−1 dx =
2
S (t)e−pt dt =
1
p
C (t)e−pt dt =
1
p
0
∞
2.
0
√ 2q + 1
π
2 Γ q + 12 sin
4
√
2q
4 πqp
√ 2q + 1
π
2 Γ q + 12 cos
4
√
2q
4 πqp
p2 1
−C
4 2
p2 1
−S
cos
4 2
cos
0 < Re q < 32 ,
p>0
0 < Re q < 32 ,
p>0
p
p2 1
−S
4 2
2
2
p
p
p 1
−C
− sin
2
4 2
2
p
2
NT 56(14)a
NT 56(13)a
+ sin
MO 173a
MO 172a
6.323
1.
0
∞
√
S
t e−pt dx =
p2 + 1 − p
2p p2 + 1
1
2
EF 122(58)a
650
The Gamma Function and Functions Generated by It
∞
2.
√
C
t e−pt dt =
0
6.324
1.
2.
6.325
p2 + 1 + p
2p p2 + 1
1
2
EF 122(58)a
1
1 + sin p2 − cos p2
− S (x) sin 2px dx =
2
4p
0
∞
1
1 − sin p2 − cos p2
− C (x) sin 2px dx =
2
4p
0
∞
∞
1.
0
∞
2.
6.324
√
π −5
2 2
S (x) sin b x dx =
b
=0
[p > 0]
NT 61(11)a
0 < b2 < 1
2
b >1
ET I 98(21)a
√
0
NT 61(12)a
2 2
C (x) cos b2 x2 dx =
[p > 0]
π −5
2 2
b
0 < b2 < 1
2
b >1
=0
ET I 42(22)
6.326
∞
∞
1.
0
2.
0
1
π
− S (x) si(2px) dx =
2
8
1
π
− C (x) si(2px) dx =
2
8
1/2
(S (p) + C (p) − 1) −
1 + sin p2 − cos p2
4p
[p > 0]
1/2
(S (p) − C (p)) −
NT 61(15)a
1 − sin p − cos p
4p
2
2
[p > 0]
NT 61(14)a
6.4 The Gamma Function and Functions Generated by It
6.41 The gamma function
∞
11
6.411
−∞
Γ(α + x) Γ(β − x) dx = −iπ21−α−β Γ(α + β)
[Re(α + β) < 1 and either Im α < 0 < Im β or Im β < 0 < Im α ]
ET II 297(1)
= iπ2
1−α−β
Γ(α + β)
[Re(α + β) < 1,
Im α < 0,
Im β < 0]
ET II 297(2)
=0
[Re(α + β) < 1,
Im α > 0,
Im β > 0]
ET II 297(3)
6.415
The gamma function
i∞
6.412
−i∞
Γ(α + s) Γ(β + s) Γ(γ − s) Γ(δ − s) ds = 2πi
651
Γ(α + γ) Γ(α + δ) Γ(β + γ) Γ(β + δ)
Γ(α + β + γ + δ)
[Re α, Re β, Re γ, Re δ > 0]
ET II 302(32)
6.413
∞
1.
√
2
|Γ(a + ix) Γ(b + ix)| dx =
0
2.
√
!2
1
1
!
Γ
b
−
a
−
π
Γ(a)
Γ
a
+
Γ(a
+
ix)
2
2
!
!
! Γ(b + ix) ! dx =
2 Γ(b) Γ b − 12 Γ(b − a)
1.
2.
3.
5.
6.
0<a<b−
[Im α = 0,
1
2
ET II 302(28)
ET II 297(4)
∞
α+β−2
ET II 297(5)
ET II 299(18)
∞
±2π 2 i Γ(α + β − γ − δ − 1)
Γ(γ + x) Γ(δ + x)
dx =
sin[π(γ − δ)] Γ(α − γ) Γ(α − δ) Γ(β − γ) Γ(β − δ)
−∞ Γ(α + x) Γ(β + x)
[Re(α + β − γ − δ) > 1, Im γ < 0, Im δ < 0. In the numerator, we take the plus sign if
Im γ > Im δ and the minus sign if Im γ < Im δ.]
ET II 300(19)
1
∞
π exp ± 2 π(δ − γ)i
Γ(α − β − γ + x + 1) dx
=
Γ(β + γ − 1) Γ 12 (α + β) Γ 12 (γ − δ + 1)
−∞ Γ(α + x) Γ(β − x) Γ(γ + x)
[Re(β + γ) > 1, δ = α − β − γ + 1, Im δ = 0. The sign is plus in the argument if the
ET II 300(20)
exponential for Im δ > 0 and minus for Im δ < 0.]
∞
Γ(α + β + γ + δ − 3)
dx
=
Γ(α
+
x)
Γ(β
−
x)
Γ(γ
+
x)
Γ(δ
−
x)
Γ(α
+
β
−
1)
Γ(β
+ γ − 1) Γ(γ + δ − 1) Γ(δ + α − 1)
−∞
[Re(α + β + γ + δ) > 3]
6.415
ET II 302(27)
Re(α − β) < −1]
2
dx
=
[Re(α + β) > 1]
Γ(α
+
x)
Γ(β
−
x)
Γ(α
+ β − 1)
−∞
∞
Γ(γ + x) Γ(δ + x)
dx = 0
Γ(α
+ x) Γ(β + x)
−∞
[Re(α + β − γ − δ) > 1, Im γ, Im δ > 0]
4.
b > 0]
∞
Γ(α + x)
dx = 0
−∞ Γ(β + x)
[a > 0,
∞ !!
0
6.414
π Γ(a) Γ a + 12 Γ(b) Γ b + 12 Γ(a + b)
2 Γ a + b + 12
−∞
1.
−∞
R(x) dx
Γ(α + x) Γ(β − x) Γ(γ + x) Γ(δ − x)
ET II 300(21)
1
Γ(α + β + γ + δ − 3)
R(t) dt
=
Γ(α + β − 1) Γ(β + γ − 1) Γ(γ + δ − 1) Γ(δ + α − 1) 0
[Re(α + β + γ + δ) > 3, R(x + 1) = R(x)] ET II 301(24)
652
The Gamma Function and Functions Generated by It
2.
R(x) dx
=
Γ(α
+
x)
Γ(β
−
x) Γ(γ + x) Γ(δ − x)
−∞
[α + δ = β + γ,
1
R(t) cos
∞
0
1
2 π(2t
6.421
+ α − β) dt
α+β
γ+δ
Γ
Γ(α + δ − 1)
2
2
Re(α + β + γ + δ) > 2, R(x + 1) = −R(x)]
Γ
ET II 301(25)
6.42 Combinations of the gamma function, the exponential, and powers
6.421
∞
1.
−∞
Γ(α + x) Γ(β − x) exp [2(πn + θ)xi] dx = 2πi Γ(α + β)(2 cos θ)−α−β exp[(β − α)iθ]
%
Re(α + β) < 1,
2.
3.
4.
6.422
−
π
π
<θ< ,
2
2
× [ηn (β) exp(2nπβi) − ηn (−α) exp(−2nπαi)]
&
0
if 12 − n Im ξ > 0
n an integer, ηn (ξ) =
sign 12 − n if 12 − n Im ξ < 0
ET II 298(7)
∞
eπicx dx
=0
−∞ Γ(α + x) Γ(β − x) Γ(γ + kx) Γ(δ − kx)
[Re(α + β + γ + δ) > 2, c and k are real,
∞
i∞
|c| > |k| + 1]
ET II 301(26)
Γ(α + x)
exp[(2πn + π − 2θ)xi] dx
−∞ Γ(β + x)
(2 cos θ)β−α−1
= 2πi sign n + 12
exp[−(2πn + π − θ)αi + θi(β − 1)]
Γ(β − α)
+
,
π
π
Re(β − α) > 0, − < θ < , n is an integer,
n + 12 Im α < 0
ET II 298(8)
2
2
∞
Γ(α + x)
exp[(2πn + π − 2θ)xi] dx = 0
−∞ Γ(β + x)
+
,
π
π
Re(β − α) > 0, − < θ < , n is an integer,
n + 12 Im α > 0
ET II 297(6)
2
2
1.
−i∞
Γ(s − k − λ) Γ λ + μ − s + 12 Γ λ − μ − s + 12 z s ds
Re(k + λ) < 0,
1
z
− k − μ Γ 12 − k + μ z λ e 2 W k,μ (z)
Re λ > |Re μ| − 12 , |arg z| < 32 π
ET II 302(29)
= 2πi Γ
2
γ+i∞
2.
γ−i∞
Γ(α + s) Γ(−s) Γ(1 − c − s)xs ds = 2πi Γ(α) Γ(α − c + 1)Ψ(α, c; x)
− Re α < γ < min (0, 1 − Re c) , − 32 π < arg x < 32 π
EH I 256(5)
6.422
The gamma function, the exponential, and powers
γ+i∞
3.
Γ(−s) Γ(β + s)ts ds = 2πi Γ(β)(1 + t)−β
653
[0 > γ > Re(1 − β),
|arg t| < π]
γ−i∞
EH I 256, BU 75
∞i
4.
t−p
2
Γ
−∞i
i∞
5.
Γ(s) Γ
−i∞
6.
2ν +
1
4
√
2
t−p−2
− s Γ 12 ν −
1
2
z t dt = 2πie 4 z Γ(−p) D p (z)
|arg z| < 34 π, p is not a positive integer
1
4
−s
z2
2
WH
s
ds
1
1
1
3 2
= 2πi · 2 4 − 2 ν z − 2 e 4 z Γ 12 ν + 14 Γ 12 ν − 14 D ν (z)
|arg z| < 34 π, ν = 12 , − 12 , − 32 , . . . EH II 120
c+i∞
3
c−i∞
1
Γ(−t)
1 −s 1
−1
Γ 2 ν + 12 s Γ 1 + 12 ν − 12 s
ds = 4πi J ν (x)
2x
[x > 0, − Re ν < c < 1]
−c+i∞
7.
−c−i∞
ν+2s
1
Γ(−ν − s) Γ(−s) − 12 iz
ds = −2π 2 e 2 iνπ H (1)
ν (z)
+
π
|arg(−iz)| < ,
2
EH II 21(34)
,
0 < Re ν < c
EH II 83(34)
−c+i∞
8.
−c−i∞
Γ(−ν − s) Γ(−s)
1 ν+2s
1
ds = 2π 2 e− 2 iνπ H (2)
ν (z)
2 iz
+
π
|arg(iz)| < ,
2
,
0 < Re ν < c
EH II 83(35)
9.
10.
1 ν+2s
i∞
x
ds = 2πi J ν (x)
Γ(−s) 2
[x > 0, Re ν > 0]
EH II 83(36)
Γ(ν + s + 1)
−i∞
i∞
5
Γ(−s) Γ(−2ν − s) Γ ν + s + 12 (−2iz)s ds = −π 2 e−i(z−νπ) sec(νπ)(2z)−ν H (1)
ν (z)
−i∞
|arg(−iz)| < 32 π, 2ν = ±1, ±3 . . .
EH II 83(37)
11.
5
Γ(−s) Γ(−2ν − s) Γ ν + s + 12 (2iz)s ds = π 2 ei(z−νπ) sec(νπ)(2z)−ν H (2)
ν (z)
−i∞
|arg(iz)| < 32 π, 2ν = ±1,
i∞
±3 . . .
EH II 84(38)
i∞
12.
Γ(s) Γ
−i∞
1
2
3
3
1
− s − ν Γ 12 − s + ν (2z)s ds = 2 2 π 2 iz 2 ez sec(νπ) K ν (z)
|arg z| < 32 π, 2ν = ±1,
±3, . . .
EH II 84(39)
− 12 +i∞
13.
− 12 −i∞
Γ(−s) 2s
x ds = 4π
s Γ(1 + s)
∞
2x
J 0 (t)
dt
t
[x > 0]
MO 41
654
The Gamma Function and Functions Generated by It
14.
6.423
i∞
Γ(α + s) Γ(β + s) Γ(−s)
Γ(α) Γ(β)
(−z)s ds = 2πi
F (α, β; γ; z)
Γ(γ + s)
Γ(γ)
−i∞
[For arg(−z) < π, the path of integration must separate the poles of the integrand at the points
s = 0, 1, 2, 3, . . . from the poles s = −α − n and s = −β − n (for n = 0, 1, 2, . . . ).]
15.
16.
δ+i∞
Γ(α + s) Γ(−s)
2πi Γ(α)
(−z)s ds =
1 F 1 (α; γ; z)
Γ(γ
+
s)
Γ(γ)
δ−i∞
+ π
,
π
− < arg(−z) < , 0 > δ > − Re α, γ = 0, 1, 2, . . .
2
2
&
%
2
i∞
+
,
Γ 12 − s
1
1
1
z s ds = 2πiz 2 2π −1 K 0 4z 4 − Y 0 4z 4
Γ(s)
−i∞
EH I 62(15), EH I 256(4)
[z > 0]
i∞
Γ λ + μ − s + 12 Γ λ − μ − s + 12 s
z
z ds = 2πiz λ e− 2 W k,μ (z)
Γ(λ
−
k
−
s
+
1)
−i∞
+
ET II 303(33)
17.
Re λ > |Re μ| − 12 ,
18.
Γ(k − λ + s) Γ λ + μ − s +
Γ μ − λ + s + 12 +
−i∞
i∞
1
2
z s ds = 2πi
Re(k − λ) > 0,
m
-
i∞
19.
−i∞
j=1
q
j=m+1
Γ (bj − s)
n
-
Γ (1 − aj + s)
j=1
Γ (1 − bj + s)
p
j=n+1
Γ (aj − s)
|arg z| <
ET II 302(30)
1
Γ k + μ + 2 λ −z
z e 2 M k,μ (z)
Γ(2μ + 1)
Re(λ + μ) > − 12 ,
π,
2
|arg z| <
π,
2
ET II 302(31)
!
! a1 , . . . , ap
!
z
z s ds = 2πi G pq
mn
! b1 , . . . , bq
p + q < 2(m + n);
Re ak < 1,
|arg z| < m + n − 12 p − 12 q π;
k = 1, . . . , n; Re bj > 0, j = 1, . . . , m
ET II 303(34)
6.423
∞
1.
e−αx
0
2.
3.
∞
dx
= ν e−α
Γ(1 + x)
dx
= eβα ν e−α , β
Γ(x + β + 1)
0
∞
xm
dx = μ e−α , m Γ(m + 1)
e−αx
Γ(x + 1)
0
MI 39, EH III 222(16)
e−αx
MI 39, EH III 222(16)
[Re m > −1]
MI 39, EH III 222(17)
6.433
Gamma functions and trigonometric functions
4.
6.424
∞
655
xm
dx = enα μ e−α , m, n Γ(m + 1)
MI 39, EH III 222(17)
Γ(x
+
n
+
1)
0
α+β−2
θ
1
∞
2 cos
1
R(x) exp[(2πn + θ)xi] dx
2
=
exp θ(β − α)i
R(t) exp(2πnti) dt
Γ(α + x) Γ(β − x)
Γ(α + β − 1)
2
−∞
0
[Re(α + β) > 1, −π < θ < π, n is an integer, R(x + 1) = R(x)] ET II 299(16)
e−αx
6.43 Combinations of the gamma function and trigonometric functions
6.431
−∞
1.
−∞
sin rx dx
=
Γ(p + x) Γ(q − x)
2 cos
r p+q−2
r(q − p)
sin
2
2
Γ(p + q − 1)
=0
[|r| > π]
[r is real;
r p+q−2
r(q − p)
2 cos
cos
cos rx dx
2
2
=
Γ(p + q − 1)
−∞ Γ(p + x) Γ(q − x)
2.
[|r| < π]
Re(p + q) > 1]
=
[m is an even integer]
α+β−2
2
Γ(α + β − 1)
[m is an odd integer]
[Re(α + β) > 1]
6.433
1.
2.
∞
sin πx dx
=
Γ(α
+
x)
Γ(β
−
x) Γ(γ + x) Γ(δ − x)
−∞
MO 10a, ET II 299(13, 14)
∞
dx
sin(mπx)
=0
sin(πx)
Γ(α
+
x)
Γ(β − x)
−∞
MO 10a, ET II 298(9, 10)
[|r| > π]
[r is real;
6.432
Re(p + q) > 1]
∞
=0
[|r| < π]
∞
sin
+π
ET II 298(11, 12)
,
(β − α)
2
α+β
γ+δ
2Γ
Γ
Γ(α + δ − 1)
2
2
[α + δ = β + γ, Re(α + β + γ + δ) > 2]
cos πx dx
=
Γ(α
+
x)
Γ(β
− x) Γ(γ + x) Γ(δ − x)
−∞
cos
+π
ET II 300(22)
,
(β − α)
2
α+β
γ+δ
2Γ
Γ
Γ(α + δ − 1)
2
2
[α + δ = β + γ, Re(α + β + γ + δ) > 2]
ET II 301(23)
656
The Gamma Function and Functions Generated by It
6.441
6.44 The logarithm of the gamma function∗
6.441
p+1
1.
ln Γ(x) dx =
p
1
2.
1
ln 2π + p ln p − p
2
1
ln Γ(1 − x) dx =
ln Γ(x) dx =
0
0
1
3.
ln Γ(x + q) dx =
0
z
FI II 784
1
ln 2π
2
1
ln 2π + q ln q − q
2
FI II 783
[q ≥ 0]
NH 89(17), ET II 304(40)
z(z + 1)
z
ln 2π −
+ z ln Γ(z + 1) − ln G(z + 1),
2
2
∞
z
z
z(z + 1) Cz 2 where G(z + 1) = (2π) 2 exp −
1+
−
2
2
k
4.
ln Γ(x + 1) dx =
0
k
k=1
n
5.
ln Γ(α + x) dx =
0
n−1
k=0
1
6.442
[a > 0;
WH
n = 1, 2, . . .]
ET II 304(41)
exp(2πnxi) ln Γ(a + x) dx = (2πni)−1 [ln a − exp(−2πnai) Ei(2πnai)]
0
6.443
1
1
(a + k) ln(a + k) − na + n ln(2π) − n(n − 1)
2
2
[a ≥ 0;
z2
exp −z +
2k
n = ±1, ±2, . . .]
ET II 304(38)
1
1
[ln(2πn) + C]
NH 203(5), ET II 304(42)
2πn
0
1
1
1
1
π
1
+ 2 1 + + ··· +
ln Γ(x) sin(2n + 1)πx dx =
ln
+
(2n + 1)π
2
3
2n − 1
2n + 1
0
1.
ln Γ(x) sin 2πnx dx =
2.
ET II 305(43)
1
3.
ln Γ(x) cos 2πnx dx =
0
4.
1
8
0
1
5.
1
4n
2
ln Γ(x) cos(2n + 1)πx dx = 2
π
NH 203(6), ET II 305(44)
%
∞
ln k
1
(C + ln 2π) + 2
2
2
(2n + 1)
4k − (2n + 1)2
&
NH 203(6)
k=2
sin(2πnx) ln Γ(a + x) dx = −(2πn)−1 [ln a + cos(2πna) ci(2πna) − sin(2πna) si(2πna)]
0
[a > 0;
6.
1
n = 1, 2, . . .]
ET II 304(36)
cos(2πnx) ln Γ(a + x) dx = −(2πn)−1 [sin(2πna) ci(2πna) + cos(2πna) si(2πna)]
0
[a > 0;
n = 1, 2, . . .]
ET II 304(37)
∗ Here, we are violating our usual order of presentation of the formulas in order to make it easier to examine the
integrals involving the gamma function.
6.457
The incomplete gamma function
657
6.45 The incomplete gamma function
6.451
1.
2.
6.452
∞
1
Γ(β)(1 + α)−β
α
0
∞
1
1
e−αx Γ(β, x) dx = Γ(β) 1 −
α
(α + 1)β
0
∞
1.
e−αx γ(β, x) dx =
e−μx γ ν,
0
x2
8a2
dx =
[β > 0]
MI 39
[β > 0]
MI 39
2
1 −ν−1
2
Γ(2ν)e(aμ) D −2ν (2aμ)
μ
|arg a| <
π
,
4
1
Re ν > − ,
2
Re μ > 0
ET I 179(36)
∞
2.
e−μx γ
0
∞
1 x2
,
4 8a2
√
2
2 a
dx = √ e(aμ) K 14 a2 μ2
μ
3
4
1
1
a
√
dx = 2a 2 ν μ 2 ν−1 K ν (2 μa)
x
0 ∞
√ 1
1
e−βx γ ν, α x dx = 2− 2 ν αν β − 2 ν−1 Γ(ν) exp
6.453
6.454
e−μx Γ ν,
0
+
π
|arg a| < ,
4
,
Re μ > 0
+
,
π
|arg a| < , Re μ > 0
2
α2
α
D −ν √
8β
2β
[Re β > 0, Re ν > 0]
ET I 179(35)
ET I 179(32)
ET II 309(19), MI 39a
6.455
∞
1.
xμ−1 e−βx Γ(ν, αx) dx =
0
∞
2.
xμ−1 e−βx γ(ν, αx) dx =
0
6.456
∞
1.
∞
2.
1.
2.
αν Γ(μ + ν)
α
2 F 1 1, μ + ν; ν + 1;
μ+ν
ν(α + β)
α+β
[Re(α + β) > 0, Re β > 0, Re(μ + ν) > 0]
ET II 308(15)
1
4x
dx =
e−αx (4x)ν− 2 Γ ν,
1
4x
dx =
1
1
0
6.457
ET II 309(16)
e−αx (4x)ν− 2 γ ν,
0
αν Γ(μ + ν)
β
2 F 1 1, μ + ν; μ + 1;
μ+ν
μ(α + β)
α+β
[Re(α + β) > 0, Re μ > 0, Re(μ + ν) > 0]
√
√ γ (2ν, α)
π
1
αν+ 2
√
√
π Γ (2ν, α)
1
αν+ 2
√
√ γ (2ν + 1, α)
e
dx = π
1
αν+ 2
0
√
∞
√ Γ (2ν + 1, α)
(4x)ν
1
e−αx √ Γ ν + 1,
dx = π
1
4x
x
αν+ 2
0
∞
ν
−αx (4x)
1
√ γ ν + 1,
4x
x
MI 39a
MI 39a
MI 39
MI 39
658
The Gamma Function and Functions Generated by It
∞
6.458
1−2ν
x
2
exp αx
0
6.458
3
1
b2
b
2
−ν ν−1
2
sin(bx) Γ ν, αx dx = π 2 α
Γ 2 − ν exp
D 2ν−2
1
8α
(2α) 2
3π
, 0 < Re ν < 1
|arg α| <
2
ET II 309(18)
6.46–6.47 The function ψ(x)
x
6.461
ψ(t) dt = ln Γ(x)
1
6.462
1
ψ(α + x) dx = ln α
0 ∞
x−α [C + ψ(1 + x)] = −π cosec(πα) ζ(α)
0
1
e2πnxi ψ(α + x) dx = e−2πnαi Ei(2πnαi)
6.463
6.464
[α > 0]
ET II 305(1)
[1 < Re α < 2]
ET II 305(6)
[α > 0;
n = ±i, ±2, . . .]
ET II 305(2)
0
6.465
1.
1
8
0
%
&
∞
ln k
2
C + ln 2π + 2
ψ(x) sin πx dx = −
π
4k 2 − 1
k=2
(see 6.443 4)
1
2.
0
1
ψ(x) sin(2πnx) dx = − π
2
∞
6.466
[n = 1, 2, . . .]
NH 204
ET II 305(3)
−1
[ψ(α + ix) − ψ(α − ix)] sin xy dx = iπe−αy 1 − e−y
0
[α > 0,
6.467 1.
y > 0]
ET I 96(1)
1
sin(2πnx) ψ(α + x) dx = sin(2πnα) ci(2πnα) + cos(2πnα) si(2πnα)
0
[α ≥ 0;
n = 1, 2, . . .]
ET II 305(4)
1
cos(2πnx) ψ(α + x) dx = sin(2πnα) si(2πnα) − cos(2πnα) ci(2πnα)
2.
0
[α > 0;
1
6.468
0
6.469 1.
1
ψ(x) sin2 πx dx = − [C + ln(2π)]
2
1
ψ(x) sin πx cos πx dx = −
0
2.8
1
ET II 305(5)
NH 204
π
4
n
1 − n2
1 n−1
= ln
2 n+1
ψ(x) sin πx sin(nπx) dx =
0
n = 1, 2, . . .]
NH 204
[n is even]
[n > 1 is odd]
NH 204(8)a
6.511
6.471
Bessel functions
∞
1.
x−α [ln x − ψ(1 + x)] dx = π cosec(πα) ζ(α)
659
[0 < Re α < 1]
ET II 306(7)
0
∞
2.
x−α [ln(1 + x) − ψ(1 + x)] dx = π cosec(πα) ζ(α) − (α − 1)−1
0
[0 < Re α < 1]
∞
[ψ(x + 1) − ln x] cos(2πxy) dx =
3.
0
6.472
∞
1.
ET II 306(8)
1
[ψ(y + 1) − ln y]
2
ET II 306(12)
x−α (1 + x)−1 − ψ (1 + x) dx = −πα cosec(πα) ζ(1 + α) − α−1
0
[|Re α| < 1]
∞
2.
ET II 306(9)
x−α x−1 − ψ (1 + x) dx = −πα cosec(πα) ζ(1 + α)
0
[−2 < Re α < 0]
∞
6.473
x−α ψ (n) (1 + x) dx = (−1)n−1
0
π Γ(α + n)
ζ(α + n)
Γ(α) sin πα
[n = 1, 2, . . . ;
ET II 306(10)
0 < Re α < 1]
ET II 306(11)
6.5–6.7 Bessel Functions
6.51 Bessel functions
6.511
∞
J ν (bx) dx =
1.
0
∞
2.
0
1
b
νπ
1
Y ν (bx) dx = − tan
b
2
[Re ν > −1,
b > 0]
[|Re ν| < 1,
b > 0]
ET II 22(3)
WA 432(7), ET II 96(1)
a
3.
J ν (x) dx = 2
0
a
J 12 (t) dt = 2 S
0
J − 12 (t) dt = 2 C
0
6.
ET II 333(1)
WA 599(4)
√ a
a
J 0 (x) dx = a J 0 (a) +
0
[Re ν > −1]
√ a
a
5.
J ν+2k+1 (a)
k=0
4.
∞
WA 599(3)
πa
[J 1 (a) H0 (a) − J 0 (a) H1 (a)]
2
[a > 0]
ET II 7(2)
660
Bessel Functions
6.512
a
J 1 (x) dx = 1 − J 0 (a)
7.
[a > 0]
ET II 18(1)
0
∞
8.
J 0 (x) dx = 1 − a J 0 (a) +
a
πa
[J 0 (a) H1 (a) − J 1 (a) H0 (a)]
2
[a > 0]
ET II 7(3)
[a > 0]
ET II 18(2)
∞
9.
J 1 (x) dx = J 0 (a)
a
b
10.
Y ν (x) dx = 2
a
11.
a
I ν (x) dx = 2
ET II 339(46)
∞
[Re ν > −1]
(−1)n I ν+2n+1 (a)
ET II 364(1)
n=0
∞
K 0 (ax) =
π
2a
[a > 0]
K 20 (ax) =
π2
4a
[a > 0]
0
13.∗
[Y ν+2n+1 (b) − Y ν+2n+1 (a)]
n=0
0
12.∗
∞
∞
0
6.512
∞
1.11
0
μ+ν+1
b2
μ+ν+1 ν−μ+1
2
,
; ν + 1; 2
J μ (ax) J ν (bx) dx = bν a−ν−1
F
μ−ν+1
2
2
a
Γ(ν + 1) Γ
2
[a > 0, b > 0, Re(μ + ν) > −1, b < a.
Γ
For a > b, the positions of μ and ν should be reversed.]
ET II 48(6)
∞
2.7
ν−n−1
Γ(ν)
β
F
αν−n n! Γ(ν − n)
1
= (−1)n
2α
=0
J ν+n (αt) J ν−n−1 (βt) dt =
0
ν, −n; ν − n;
[0 < β < α]
[0 < β = α]
[0 < α < β]
0
β
αν
1
=
2β
[β < α]
[β = α]
=0
[β > α]
[Re ν > 0]
∞
MO 50
ν−1
J ν (αx) J ν−1 (βx) dx =
4.
β
α2
[Re(ν) > 0]
∞
3.8
2
J ν+2n+1 (ax) J ν (bx) dx = bν a−ν−1 P (ν,0)
1−
n
0
=0
2
2b
a2
WA 444(8), KU (40)a
[Re ν > −1 − n,
0 −1 − n,
0 < a < b]
ET II 47(5)
6.513
Bessel functions
∞
5.
J ν+n (ax) Y ν−n (ax) dx = (−1)n+1
0
∞
J 1 (bx) Y 0 (ax) dx = −
0
a
J ν (x) J ν+1 (x) dx =
0
8.
∞
0
∞
K 0 (ax) J 1 (bx) =
0
10.
∗
b2
b−1
ln 1 − 2
π
a
∞
k J n (ka) J n (kb) dk =
Re ν > − 12 ,
a > 0,
n = 0, 1, 2, . . .
[J ν+n+1 (a)]
2
[0 −1]
ET II 338(37)
n=0
9
9.∗
1
2a
ET II 347(57)
6.
7.
661
∞
[n = 0, 1, . . .]
b2
1
ln 1 + 2
2b
a
K 0 (ax) I 1 (bx) = −
0
1
δ(b − a)
a
b2
1
ln 1 − 2
2b
a
JAC 110
[a > 0,
b > 0]
[a > 0,
b > 0]
6.513
∞
1.
0
1 + ν + 2μ
2
[J μ (ax)] J ν (bx) dx = a2μ b−2μ−1
1 + ν − 2μ
2
[Γ(μ + 1)] Γ
2
⎡ ⎛
Γ
2
1−
⎢ ⎜ 1 − ν + 2μ 1 + ν + 2μ
⎜
,
; μ + 1;
×⎢
⎣F ⎝
2
2
[Re ν + Re 2μ > −1,
∞
2.
0
b−1
Γ
[J μ (ax)] K ν (bx) dx =
2
2
2μ + ν + 1
2
Γ
⎞⎤2
4a2
1 − 2 ⎟⎥
b ⎟⎥
⎠⎦
2
0 < 2a < b]
ET II 52(33)
&2
%
2μ − ν + 1
4a2
−μ
1+ 2
P 1 ν− 1
2
2
2
b
[2 Re μ > |Re ν| − 1, Re b > 2|Im a|]
ET II 138(18)
∞
3.
0
4.
0
∞
ν + 2μ + 1
4a2
4a2
2
−μ
−μ
I μ (ax) K μ (ax) J ν (bx) dx =
1 + 2 Q 1 ν− 1
1+ 2
P 1 ν− 1
2
2
2
2
ν − 2μ + 1
b
b
bΓ
2
[Re a > 0, b > 0, Re ν > −1, Re(ν + 2μ) > −1] ET II 65(20)
eμπi Γ
νπ
π
sec
J μ (ax) J −μ (ax) K ν (bx) dx =
P μ1 ν− 1
2
2
2b
2
2
4a
4a2
1+ 2
1 + 2 P −μ
1
1
2 ν− 2
b
b
[|Re ν| < 1, Re b > 2|Im a|]
ET II 138(21)
662
Bessel Functions
e2μπi Γ
∞
5.
2
[K μ (ax)] J ν (bx) dx =
0
bΓ
z
J μ (x) J ν (z − x) dx = 2
6.
0
∞
6.514
1 + ν + 2μ %
&2
4a2
2
−μ
1+ 2
Q 1 ν− 1
2
2
1 + ν − 2μ
b
2 Re a > 0, b > 0, Re 12 ν ± μ > − 12
(−1)k J μ+ν+2k+1 (z)
[Re μ > −1,
ET II 66(28)
Re ν > −1]
WA 414(2)
k=0
z
7.
J μ (x) J −μ (z − x) dx = sin z
[−1 < Re μ < 1]
WA 415(4)
J μ (x) J 1−μ (z − x) dx = J 0 (z) − cos(z)
[−1 < Re μ < 2]
WA 415(4)
0
8.
z
0
9.
∗
∞
1
b
2
=
arcsin
πb
J 20 (ax) J 1 (bx) =
0
6.514
Jν
√
√
a
2
−1
2ab
Y 2ν 2 ab + K 2ν
Y ν (bx) dx = b
x
π
a > 0,
∞
2.
0
Jν
0
∞
Yν
0
Yν
0
6.
Re ν > − 12
ET II 57(9)
b > 0,
− 12 < Re ν <
3
2
+ 1 √ ,
+
√ ,
1
1
1
a
K ν (bx) dx = b−1 e 2 i(ν+1)π K 2ν 2e 4 iπ ab + b−1 e− 2 i(ν+1)π K 2ν 2e− 4 πi ab
x
a > 0, Re b > 0, |Re ν| < 52
,
√
√
a
2b−1 +
π
J ν (bx) dx = −
K 2ν 2 ab − Y 2ν 2 ab
x
π
2
a > 0, b > 0,
|Re ν| <
1
2
√
a
Y ν (bx) dx = −b−1 J 2ν 2 ab
x
a > 0,
b > 0,
|Re ν| <
1
2
ET II 110(14)
∞
Yν
0
b > 0,
ET II 62(37)a
∞
5.
a > 0,
ET II 141(31)
4.
ET II 110(12)
∞
3.
[2a > b > 0]
√
a
J ν (bx) dx = b−1 J 2ν 2 ab
x
0
b
2a
Jν
∞
1.
[b > 2a > 0]
√
√
1
1
1
1
a
K ν (bx) dx = −b−1 e 2 νπi K 2ν 2e 4 πi ab − b−1 e− 2 νπi K 2ν 2e− 4 πi ab
x
a > 0, Re b > 0, |Re ν| < 52
ET II 143(37)
6.516
Bessel functions
∞
7.
Kν
0
√
a
K ν (bx) dx = πb−1 K 2ν 2 ab
x
Jμ
√
√
a
a
Yμ
K 0 (bx) dx = −2b−1 J 2μ 2 ab K 2μ 2 ab
x
x
∞
1.
0
∞
2.
0
ker2ν
√
√
3νπ
2 ab + cos
kei2ν 2 ab
2
Re a > 0, b > 0, |Re ν| < 12
Kν
∞
0
3νπ
2
ET II 113(28)
8.
6.515
a
−1
sin
Y ν (bx) dx = −2b
x
663
[Re a > 0,
3.
0
ET II 146(54)
[a > 0, Re b > 0]
+
,
√
√
1
1
a 2
K 0 (bx) dx = 2πb−1 K 2μ 2e 4 πi ab K 2μ 2e− 4 πi ab
Kμ
x
[Re a > 0,
∞
Re b > 0]
H (1)
μ
2
a
x
H (2)
μ
2
a
x
ET II 143(42)
Re b > 0]
ET II 147(59)
√
√
J 0 (bx) dx = 16π −2 b−1 cos μπ K 2μ 2eπi/4 a b K 2μ 2e−πi/4 a b
,
+
π
|arg a| < , b > 0, |Re μ| < 14
4
ET II 17(36)
6.516
∞
1.
√ J 2ν a x J ν (bx) dx = b−1 J ν
0
√ J 2ν a x Y ν (bx) dx = −b−1 Hν
0
∞
3.
0
∞
4.10
0
5.
a > 0,
b > 0,
Re ν > − 12
ET II 58(16)
∞
2.
a2
4b
∞
√ π
J 2ν a x K ν (bx) dx = b−1 I ν
2
a2
4b
2
a
4b
− Lν
b > 0,
Re ν > − 12
Re b > 0,
Re ν > − 12
√ a
a
1
1
J −ν
Y 2ν a x J ν (bx) dx = J ν
cot(2πν) −
cosec(2πν)
b
4b
2b
4b
a4
3 3
23ν−3 a2−2ν bν−2 Γ ν − 12 1 F 2 1; , − ν;
−
3/2
2 2
64b2
π
[a > 0, b > 0]
2
=
b−1
sec(νπ) J −ν
2
ET II 144(45)
2
√ Y 2ν a x Y ν (bx) dx
0
ET II 111(18)
2
a
4b
a > 0,
a2
4b
MC
a2
a2
+ cosec(νπ) H−ν
− 2 cot(2νπ) Hν
4b
4b
a > 0, b > 0, |Re ν| < 12
ET II 111(19)
664
Bessel Functions
∞
6.
0
⎡
−1
√ πb ⎣
cosec(2νπ) L−ν
Y 2ν a x K ν (bx) dx =
2
− tan(νπ) I ν
∞
7.
0
∞
0
∞
9.
∞
10.
0
6.517
√ K 2ν a x Y ν (bx) dx
1 −1
= − πb
sec(νπ) J −ν
4
√ K 2ν a x K ν (bx) dx =
z
J0
0
∞
⎤
a2
4b
2
a ⎦
sec(νπ)
Kν
π
4b
ET II 144(46)
Re b > 0, |Re ν| < 12
2
2
a
a
− Y −ν
4b 4b
Re a > 0, b > 0, Re ν > − 12
a2
4b
a2
4b
+ Lν
a2
π
+
L−ν
− Lν
2 sin(νπ)
4b
Re b > 0, |Re ν| < 12
a2
4b
Re b > 0, Re ν > − 12
a2
4b
2
π
Jν (z) + Nν2 (z)
8 cos νπ
J 2ν (2z sin x) dx =
π 2
J (z)
2 ν
π/2
0
ET II 147(63)
ET II 147(60)
MO 48
π 2
J (z)
2 ν
0
2.
Kν
a2
a2
− cosec(νπ) H−ν
+ 2 cosec(2νπ) Hν
4b
4b
Re a > 0, b > 0, |Re ν| < 12
ET II 114(34)
J 2ν (2z cos x) dx =
π/2
1.
Iν
a2
4b
2
0
−
− cot(2νπ) Lν
z 2 − x2 dx = sin z
K 2ν (2z sinh x) dx =
πb−1
4 cos(νπ)
√ πb−1
I 2ν a x K ν (bx) dx =
2
6.518
6.519
a
4b
√ 1 −1
K 2ν a x J ν (bx) dx = πb sec(νπ) H−ν
4
0
2
a2
4b
ET II 70(22)
8.
6.517
− 12 < Re ν <
Re z > 0,
Re ν > − 12
Re ν > − 12
1
2
MO 45
WH
WA 42(1)a
6.52 Bessel functions combined with x and x2
6.521
1.
1
β J ν−1 (β) J ν (α) − α J ν−1 (α) J ν (β)
α2 − β 2
α J ν (β) J ν (α) − β J ν (α) J ν (β)
=
β 2 − α2
x J ν (αx) J ν (βx) dx =
0
[α = β,
ν > −1]
[α = β,
ν > −1]
WH
Bessel functions combined with x and x2
6.522
2.
∞
10
x K ν (ax) J ν (bx) dx =
0
bν
aν (b2 + a2 )
665
[Re a > 0,
b > 0,
Re ν > −1]
ET II 63(2)
∞
π(ab)−ν a2ν − b2ν
x K ν (ax) K ν (bx) dx =
2 sin(νπ) (a2 − b2 )
0
3.
4.
a
−1
x J ν (λx) K ν (μx) dx = μ2 + λ2
0
5.∗
7.
∞
x K 20 (ax) =
1
2a2
[a > 0]
∞
b
a (a2 + b2 )
∞
x K 0 (ax) I 0 (bx) =
0
9.∗
∞
x K 1 (ax) I 1 (bx) =
0
10.
∗
∞
∞
∞
[a > b > 0]
b
a (a2 − b2 )
[a > b > 0]
[a > 0]
x2 K 1 (ax) =
2
a3
[a > 0]
x2 K 0 (ax) J 1 (bx) =
0
13.∗
∞
x2 K 1 (ax) J 0 (bx) =
0
14.
∗
∞
x2 K 0 (ax) I 1 (bx) =
0
15.∗
∞
x2 K 1 (ax) I 0 (bx) =
0
6.522
Notation: 1 =
1.8
0
∞
2b
2
[a > 0,
2
[a > b > 0]
2
[a > b > 0]
2
[a > b > 0]
(a2 + b2 )
2a
(a2
+ b2 )
2b
(a2
− b2 )
2a
(a2
− b2 )
ET II 367(26)
b > 0]
1
− b2
a2
π
2a3
0
12.∗
[a > 0,
x2 K 0 (ax) =
0
11.∗
+ λa J ν+1 (λa) K ν (μa) − μa J ν (λa) K ν+1 (μa)
[a > 0]
0
λ
μ
π
2a2
x K 1 (ax) J 1 (bx) =
8.∗
ET II 145(48)
ν
x K 1 (ax) =
0
∗
Re(a + b) > 0]
[Re ν > −1]
∞
0
6.∗
[|Re ν| < 1,
b > 0]
,
,
1 +
1 +
(b + c)2 + a2 − (b − c)2 + a2 , 2 =
(b + c)2 + a2 + (b − c)2 + a2
2
2
2
x [J μ (ax)] K ν (bx) dx = Γ μ + 12 ν + 1 Γ μ − 12 ν + 1 b−2
+
− 1
1 , −μ +
1 ,
2 −2 2
2 −2 2
1
+
4a
P
1
+
4a
× 1 + 4a2 b−2 2 P −μ
b
b
1
1
− ν
ν
2
[Re b > 2|Im a|,
2
2 Re μ > |Re ν| − 2]
ET II 138(19)
666
Bessel Functions
∞
2.
0
∞
3.11
0
6.522
2e2μπi Γ 1 + 12 ν + μ
x [K μ (ax)] J ν (bx) dx = 1 b 4a2 + b2 2 Γ 12 ν − μ
(1 + 4a2 b−2 ) Q −μ
(1 + 4a2 b−2 )
× Q −μ
1
1
ν
ν−1
2
2
b > 0, Re a > 0, Re 12 ν ± μ > −1
2
ν
ν
−ν
x K 0 (ax) J ν (bx) J ν (cx) dx = r1−1 r2−1 (r2 − r1 ) (r2 − r1 ) = ν 2 1 2 ,
2 (2 − 1 )
+
2
2
2
2
r1 = a + (b − c) , r2 = a + (b + c) , c > 0, Re ν > −1,
ET II 66(27)a
Re a > |Im b|
,
ET II 63(6)
∞
4.10
− 1
x I 0 (ax) K 0 (bx) J 0 (cx) dx = a4 + b4 + c4 − 2a2 b2 + 2a2 c2 + 2b2 c2 2
0
[Re b > Re a,
5.10
c > 0]
ET II 16(27)
alternatively, with a and c interchanged
∞
1
x I 0 (cx) K 0 (bx) J 0 (ax) dx = 2
[Re b > Re c, a > 0]
−
21
0
2
∞
− 1
x J 0 (ax) K 0 (bx) J 0 (cx) dx = a4 + b4 + c4 − 2a2 c2 + 2a2 b2 + 2b2 c2 2
0
alternatively, with a and b interchanged
∞
1
x J 0 (bx) K 0 (ax) J 0 (cx) dx = 2
2 − 21
0
∞
x J 0 (ax) Y 0 (ax) J 0 (bx) dx = 0
6.
c > 0]
[Re a > |Im b|,
c > 0]
− 12
[0 < 2a 2|Im a|,
9.
0
ET II 138(20)
x K μ− 12 (ax) K μ+ 12 (ax) J ν (bx) dx
+
1 , −μ− 12 +
1 ,
2e2μπi Γ 12 ν + μ + 1
−μ+ 12 2 −2 2
2 −2 2
1
+
4a
1
+
4a
=−
Q
b
b
Q
1
1
1
1
1
ν−
2 ν− 2
b Γ 12 ν − μ b2 + 4a2 2 2 2
b > 0, Re a > 0, Re ν > −1, |Re μ| < 1 + 12 Re ν
ET II 67(29)a
0
2 Re μ > |Re ν| − 3]
∞
8.
8
1
−2 − 2
3+ν
3−ν
1 + 4a2 b
Γ μ+
b
2
2
+
, 1 1 +
,
1
1
2 ν− 2
2 ν− 2
×P −μ
1 + 4a2 b−2 P −μ−1
1 + 4a2 b−2
x J μ (ax) J μ+1 (ax) K ν (bx) dx = Γ μ +
ET II 15(25)
[0 |Im a|,
∞
− 1
x I 12 ν (ax) K 12 ν (ax) J ν (bx) dx = b−1 b2 + 4a2 2
[b > 0,
Re a > 0,
Re ν > −1]
ET II 65(16)
Bessel functions combined with x and x2
6.522
∞
10.
x J 12 ν (ax) Y
0
1
2ν
(ax) J ν (bx) dx
=0
= −2π −1 b−1 b2 − 4a2
− 12
[a > 0,
Re ν > −1,
0 0,
Re ν > −1,
2a −1,
0 0,
Re ν > −1,
2a < b]
x I 12 (ν−μ) (ax) K 12 (ν+μ) (ax) J ν (bx) dx = 2−μ a−μ b−1 b2 + 4a2
[b > 0,
13.
[a > 0,
ET II 52(32)
∞
12.
667
Re a > 0,
Re ν > −1,
− 12
+
1 ,μ
b + b2 + 4a2 2
Re(ν − μ) > −2]
μ
ν
ET II 66(23)
ν−μ
μ−ν
(cos ψ)
(sin ϕ) (sin ψ) (cos ϕ)
x J μ (xa sin ϕ) K ν−μ (ax cos ϕ cos ψ) J ν (xa sin ψ) dx =
2
2
2
a 1 − sin ϕ ,sin ψ
+
π
π
ET II 64(10)
a > 0, 0 < ϕ < , 0 < ψ < , Re μ > −1, Re ν > −1
2
2
∞
x J μ (xa sin ϕ cos ψ) J ν−μ (ax) J ν (xa cos ϕ sin ψ) dx
0
μ
15.10
16.10
17.11
−ν
−μ
−1
= −2π −1 a−2 sin(μπ) (sin ϕ) (sin ψ) (cos ϕ) (cos ψ) [cos(ϕ + ψ) cos(ϕ − ψ)]
+
,
π
a > 0, 0 < ϕ < , 0 < ψ < 12 π, Re ν > −1
ET II 54(39)
2
∞
23ν (abc)ν Γ ν + 12
ν+1
x
J ν (bx) K ν (ax) J ν (cx) dx = √
2ν+1
π (22 − 21 )
0
[Re a > |Im b|, c > 0]
∞
1
3ν
ν
2 (abc) Γ ν + 2
xν+1 I ν (cx) K ν (bx) J ν (ax) dx = √
2ν+1
π (22 − 21 )
0
[Re b > |Im a| + |Im c|]
∞
tν−μ−ρ+1 J μ (ct) J ν (bt) K ρ (at) dt
0
μ−ν+ρ−1
1 1+2ν−2ρ 2
1 − x2 22 − x2
x
21+ν−μ−ρ
dx
= μ ν ρ
μ−ν
c b a Γ (μ − ν + ρ) 0
(b2 − x2 )
,
,
1 +
1 +
1 =
(b + c)2 + a2 − (b − c)2 + a2 , 2 =
(b + c)2 + a2 + (b − c)2 + a2
2
2
[Re a > |Im b|, c > 0]
ν
668
Bessel Functions
18.
∞
11
6.523
tμ−ν+ρ+1 J μ (ct) J ν (bt) K ρ (at) dt
ν−μ−ρ−1
1 1+2μ+2ρ 2
1 − x2 22 − x2
x
21+μ−ν+ρ aρ
dx
= μ ν
ν−μ
c b Γ (ν − μ − ρ) 0
(c2 − x2 )
,
,
+
+
1 1 1 =
(b + c)2 + a2 − (b − c)2 + a2 , 2 =
(b + c)2 + a2 + (b − c)2 + a2
2
2
[Re a > |Im b|, c > 0]
0
∞
6.523
0
+
−1 2
−1 , b
ln
x 2π −1 K 0 (ax) − Y 0 (ax) K 0 (bx) dx = 2π −1 a2 + b2
+ b − a2
a
[Re b > |Im a|, Re(a + b) > 0]
ET II 145(50)
6.524
∞
1.
0
x J 2ν (ax) J ν (bx) Y ν (bx) dx = 0
= −(2πab)−1
0 < a < b,
Re ν > − 12
0 − 12
2
x [J 0 (ax) K 0 (bx)] dx =
0
b −a
b 2 + a2
2
1
π
−
arcsin
8ab 4ab
2
[a > 0,
1.10
ET II 352(14)
∞
2.
6.525
b > 0]
ET II 373(9)
,
,
1 +
1 +
(b + c)2 + a2 − (b − c)2 + a2 , 2 =
(b + c)2 + a2 + (b − c)2 + a2
2
2
∞
,− 32
+ 2
2
a + b2 + c2 − 4a2 c2
x2 J 1 (ax) K 0 (bx) J 0 (cx) dx = 2a a2 + b2 − c2
Notation: 1 =
0
[c > 0,
Re b ≥ |Im a|,
Re a > 0]
ET II 15(26)
2.10
alternatively, with a and b interchanged
∞
2b a2 + b2 − c2
2
x J 1 (bx) K 0 (ax) J 0 (cx) dx =
[Re a > |Im b|, Re b > 0,
3
(22 − 21 )
0
∞
,− 32
+ 2
2
a + b2 + c2 − 4a2 b2
x2 I 0 (ax) K 1 (bx) J 0 (cx) dx = 2b b2 + c2 − a2
c > 0]
0
∞
3.10
0
6.526
1.
0
∞
x2 I 0 (cx) K 0 (bx) J 0 (ax) dx =
2b a2 + b2 − c2
(22 − 21 )
x J 12 ν ax2 J ν (bx) dx = (2a)−1 J 12 ν
3
[Re b > |Re a|,
c > 0]
[Re a > |Im b|,
c > 0]
ET II 16(28)
b2
4a
[a > 0,
b > 0,
Re ν > −1]
ET II 56(1)
Bessel functions combined with x and x2
6.527
∞
2.
0
x J 12 ν ax2 Y ν (bx) dx
−1
= (4a)
∞
0
xY
0
1
2ν
xY
0
π
K ν (bx) dx =
8a cos
νπ
2
H− 12 ν
ax2 J ν (bx) dx = −(2a)−1 H 12 ν
1
2ν
b2
4a
∞
6.
0
∞
7.
0
b2
4a
∞
8.
0
2
π
x K 12 ν ax J ν (bx) dx =
I1
4a 2 ν
b2
4a
− L 12 ν
b2
4a
[Re a > 0,
2.
3.
∞
Re ν > −1]
b > 0,
ET II 68(9)
⎡
π ⎣
cosec(νπ) L− 12 ν
x K 12 ν ax2 Y ν (bx) dx =
4a
b2
4a
b2
4a
x K 12 ν ax2 K ν (bx) dx
π
=
8a
1.
Re ν > −1]
Re b > 0,
b2
b2
b2
νπ
J − 12 ν
− sin
− H 12 ν
4a
2
4a
4a
[a > 0, Re b > 0, |Re ν| < 1] ET II 141(28)
− cot(νπ) L 12 ν
νπ
1
K 12 ν
− sec
π
2
[Re a > 0,
6.527
Re ν > −1]
Re b > 0,
[a > 0,
π
νπ
H− 12 ν
cos
4a sin(νπ)
2
νπ
I 12 ν
− tan
2
b2
4a
− Y − 12 ν
[a > 0,
ax2 K ν (bx) dx
=
1
2ν
b2
b2
νπ
νπ
J 12 ν
H− 12 ν
− tan
+ sec
2
4a
2
4a
[a > 0, b > 0, Re ν > −1] ET II 109(9)
ET II 61(35)
∞
5.
b2
4a
ET II 140(27)
∞
4.
2
x J 12 ν ax
3.
Y
669
νπ
K 12 ν
sec
2
b2
4a
1
x2 J 2ν (2ax) J ν− 12 x2 dx = a J ν+ 12 a2
2
0
∞
1
x2 J 2ν (2ax) J ν+ 12 x2 dx = a J ν− 12 a2
2
0
∞
1
x2 J 2ν (2ax) Y ν+ 12 x2 dx = − a Hν− 12 a2
2
0
b > 0,
b2
4a
⎤
b2 ⎦
4a
|Re ν| < 1]
ET II 112(25)
b2
b2
1
+ π cosec(νπ) L
− L2ν
4a
4a
[Re a > 0, |Re ν| < 1]
ET II 146(52)
− 12 ν
a > 0,
Re ν > − 12
[a > 0,
Re ν > −2]
ET II 355(35)
[a > 0,
Re ν > −2]
ET II 355(36)
ET II 355(33)
670
Bessel Functions
∞
6.528
0
6.529
∞
1.
0
2.
x K 14 ν
x2
4
I 14 ν
x2
4
J ν (bx) dx = K 14 ν
x2
4
√ √ 2a
1
x J ν 2 ax K ν 2 ax J ν (bx) dx = b−2 e− b
2
6.528
b2
4
[b > 0,
I 14 ν
ν > −1]
[Re a > 0,
b > 0,
MO 183a
Re ν > −1]
ET II 70(23)
a
x J λ (2a) I λ (2x) J μ
2 a2 − x2 I μ 2 a2 − x2 dx
0
=
a2λ+2μ+2
2 Γ(λ + 1) Γ(μ + 1) Γ(λ + μ + 2)
λ+μ+1
λ+μ+3
; λ + 1, μ + 1, λ + μ + 1,
; −a4
× 1F 4
2
2
[Re λ > −1, Re μ > −1] ET II 376(31)
6.53–6.54 Combinations of Bessel functions and rational functions
6.531
∞
1.10
0
Y ν (bx)
dx
x+a
= −π J ν (ab) cot(πν) cosec(πν) − π J −ν (ab) cosec2 (πν) +
+
∞
2.
0
a2 b 2
3−ν 3+ν
ab
,
; −
1 F 2 1;
−1
2
2
4
ν2
tan
πν
a2 b 2
1
2−ν 2+ν
cot
,
; −
1 F 2 1;
ν
2
2
2
4
πν
2
[Re ν < 1, arg a = π, b > 0]
Y ν (bx)
2
dx = π cot(νπ) [Y ν (ab) + Eν (ab)] + Jν (ab) + 2 [cot(νπ)] [Jν (ab) − J ν (ab)]
x−a
[b > 0, a > 0, |Re ν| < 1]
MC
ET II 98(9)
∞
3.
0
,
+
1
1
π2
K ν (bx)
2
dx =
[cosec(νπ)] I ν (ab) + I −ν (ab) − e− 2 iνπ Jν (iab) − e 2 iνπ J−ν (iab)
x+a
2
[Re b > 0, |arg a| < π, |Re ν| < 1]
ET II 128(5)
6.532
1.11
0
∞
, 1+
,
νπ
J ν (x)
i+
π
S 0,ν (ia) − e−iνπ/2 K ν (a) =
i s 0,ν (ia) + sec
I ν (a)
dx =
2
2
x +a
a
a
2
2
[Re a > 0,
Re ν > −1]
6.536
Bessel functions and rational functions
∞
2.
0
⎡
νπ
Y ν (x)
π
1 ⎣
1
dx =
νπ − 2a tan 2 I ν (ab) − a K ν (ab)
x2 + a2
cos
2
⎤
νπ
b sin
2 2
3−ν 3+ν a b ⎦
2
,
;
+
1 F 2 1;
1 − ν2
2
2
4
[b > 0,
∞
3.
0
∞
4.
0
∞
5.
0
∞
6.
0
6.533 1.
ET II 99(13)
x J 0 (ax)
dx = K 0 (ak)
x2 + k 2
[a > 0,
Re k > 0]
WA 466(5)
Y 0 (ax)
K 0 (ak)
dx = −
x2 + k 2
k
[a > 0,
Re k > 0]
WA 466(6)
J 0 (ax)
π
[I 0 (ak) − L0 (ak)]
dx =
2
2
x +k
2k
[a > 0,
Re k > 0]
WA 467(7)
z
J p (x) J q (z − x)
z
2.
0
J p+q (z)
dx
=
x
p
J p (x) J q (z − x)
dx =
x
z−x
∞
11
[J 0 (ax) − 1] J 1 (bx)
0
4.
|Re ν| < 1]
Y ν (bx)
νπ
π νπ J ν (ab) + tan
tan
[Jν (ab) − J ν (ab)] − Eν (ab) − Y ν (ab)
dx =
2
2
x −a
2a
2
2
[b > 0, a > 0, |Re ν| < 1]
0
3b
Re a > 0,
ET II 101(21)
3.
671
1 1
+
p q
J p+q (z)
z
a,
dx
b+
1
+
2
ln
=
−
x2
4
b
a2
=−
4b
⎧
b
1 1
b2
⎪
⎪
∞
, ; 2, 2
⎨
2F 1
dx
2a
2 2
a
=
[J 0 (ax) − 1] J 1 (bx)
2
b
2
⎪
x
0
⎪ E
−1
⎩
π
a2
∞
dx
=0
[1 − J 0 (ax)] J 0 (bx)
x
0
a
= ln
b
[Re p > 0,
Re q > −1]
WA 415(3)
[Re p > 0,
Re q > 0]
WA 415(5)
[0 < b < a]
[0 < a < b]
ET II 21(28)a
− 1 [0 < b < a]
[0 < a < b]
[0 < a < b]
[0 0]
ET II 340(5)
[Re a > 0,
b > 0,
Re ν > −1]
|arg a| < 14 π
ET II 342(26)
ET II 8(9), MO 46a
672
Bessel Functions
∞
6.537
0
6.538
∞
1.
0
+
x2 J 0 (bx)
1
dx = − 2 kei(ab)
4
4
x +a
a
b > 0,
|arg a| <
6.537
π,
4
MO 46a
% √ √ &
dx
a+b
2i ab
2i ab
J 1 (ax) J 1 (bx) 2 =
E
−K
x
π
|b − a|
|b − a|
[a > 0,
∞
2.8
x−1 J ν+2n+1 (x) J ν+2m+1 (x) dx = 0
0
b > 0]
ET II 21(30)
[m = n with m, n integers, ν > −1]
= (4n + 2ν + 2)−1
[m = n,
ν > −1]
EH II 64
6.539
1.
2.
π Y ν (b) Y ν (a)
−
2 = 2
J ν (b)
J ν (a)
a x [J ν (x)]
b
J ν (b)
dx
π J ν (a)
−
=
2
2 Y ν (a) Y ν (b)
a x [Y ν (x)]
b
dx
[J ν (x) = 0 for x ∈ [a, b]]
ET II 338(41)
[Y ν (x) = 0 for x ∈ [a, b]]
ET II 339(49)
3.
a
6.541
b
J ν (a) Y ν (b)
π
dx
= ln
x J ν (x) Y ν (x)
2
J ν (b) Y ν (a)
∞
x J ν (ax) J ν (bx)
1.
0
2.8
0
ET II 339(50)
dx
= I ν (bc) K ν (ac)
x2 + c2
= I ν (ac) K ν (bc)
[0 0,
Re ν > −1]
[0 < a < b,
Re c > 0,
Re ν > −1]
ET II 49(10)
∞
dx
x1−2n J ν (ax) J ν (bx) 2
x
+ c2
%
n
1
1
= − 2
I ν (bc) K ν (ac) −
c
2
=
−
1
c2
n
%
I ν (bc) K ν (ac) −
1
2ν
ν
b
a
2 2 p n−1−p 2 2 k &
n−1
a c /4
b c /4
π
sin(πν) p=0 p! Γ(1 − ν + p)
k! Γ(1 − ν + k)
k=0
b
a
ν n−1
p=0
2 2 p n−1−p 2 2 k &
b c /4
a c /4
p!(1 − ν)p
k!(1 + ν)k
[0 n − 1,
0 0,
6.544
Bessel functions and rational functions
3.
∞
8
0
xα−1
1
ρ J μ (cx) J ν (cx) dx =
2
2
2
(x + z )
×Γ
c
2
673
2ρ−α
(μ + ν + α)/2 − ρ, 1 + 2ρ − α
(μ⎛− ν − α)/2 + ρ + 1, (μ + ν − α)/2 + ρ + 1, (ν − μ − α)/2 + ρ + 1
α
μ+ν+α
μ−ν−α
1−α
+ ρ, 1 − + ρ, ρ; ρ + 1 −
,ρ + 1 +
,
2
2
2
2
⎞
μ+ν−α
ν − μ − α 2 2 ⎠ z α−2ρ cz μ+ν
ρ+1+
,ρ + 1 +
;c z
+
,
2
2
2
2
⎛
ρ − (α + μ + ν)/2, (α + μ + ν) /2
⎝1 + μ + ν , 1 + μ + ν
Γ
3F 4
ρ, μ + 1, ν + 1
2
2
⎞
α+μ+ν
α+μ+ν
;1 − ρ +
, μ + 1, ν + 1, μ + ν + 1; c2 z 2 ⎠
2
2
× 3F 4 ⎝
a1 , . . . , ap
Γ (a1 ) . . . Γ (ap )
, c > 0, Re z > 0, Re(α + μ + ν) > 0; Re(α − 2ρ) > 1
Γ
=
b1 , . . . , bq
Γ (b1 ) . . . Γ (bq )
∞
ν
J ν (ax) Y ν (bx) − J ν (bx) Y ν (ax)
π b
6.542
[0 0,
6.544
∞
Jν
1.
0
∞
2.
Jν
0
∞
3.
Jν
0
∞
4.
Yν
0
5.
∞
Yν
0
a
x
Yν
x
b
a
x
Jν
x
b
dx
1 2
K 2ν
=−
x2
a π
dx
1
= J 2ν
x2
a
− Y 2ν
dx
1 1
= e 2 iνπ K 2ν
2
x
a
a
x
Jν
x
b
dx
2
=
K 2ν
x2
aπ
√ 2 a
√
b
a > 0, b > 0,
Re μ > |Re ν| − 2]
|Re ν| <
1
2
EI II 357(47)
√
2 a
√
b
a
x
Kν
x
b
a
x
Kν
x
b
√
2 a
√
b
a ≥ b > 0,
√
2 a 1 iπ
√ e4
b
√
2 a
√
b
dx
4 1 i(ν+1)π
=
K 2ν
e2
x2
a
+
a > 0,
b > 0,
Re ν > − 12
ET II 57(10)
√
2 a − 1 iπ
1 − 1 iνπ
K 2ν √ e 4
+ e 2
a
b
Re b > 0, a > 0, |Re ν| < 12
π
Y 2ν
2
√ 2 a
√
b
a > 0, b > 0,
ET II 142(32)
|Re ν| <
1
2
ET II 62(38)
√
√
1
2 a 1 iπ
2 a 1
√ e4
+ e− 2 i(ν+1)π K 2ν √ e− 4 iπ
b
b
Re b > 0, a > 0, |Re ν| < 12
ET II 143(38)
674
Bessel Functions
∞
6.
Kν
0
∞
7.
Kν
0
∞
8.
Kν
0
a
x
Jν
x
b
√
1
dx
i 1 νπi
πi 2 a
2
4
√
=
K 2ν e
e
x2
a
b
a
x
Yν
x
b
dx
2
=
sin
x2
a
3
πν kei2ν
2
dx
π
= K 2ν
x2
a
√
2 a
√
b
a
x
Kν
x
b
6.551
√ a
√
b
|Re ν| < 52
− 14 πi 2
− 12 νπi
K 2ν e
−e
Re a > 0, b > 0,
ET II 70(19)
√
√ 3
2 a
2 a
√
πν ker2ν √
− cos
2
b b
Re a > 0, b > 0, |Re ν| < 52
ET II 113(29)
[Re a > 0,
Re b > 0]
ET II 146(55)
6.55 Combinations of Bessel functions and algebraic functions
6.55110
1
1/2
1.
x
0
∞
2.
1
√ −3/2 Γ 34 + 12 ν
J ν (xy) dx = 2y
Γ 14 + 12 ν
+y −1/2 ν − 12 J ν (y) S −1/2,ν−1 (y) − J ν−1 (y) S 1/2,ν (y)
y > 0, Re ν > − 32
x1/2 J ν (xy) dx = y −1/2 J ν−1 (y) S 1/2,ν (y) + 12 − ν J ν (y) S −1/2,ν−1 (y)
[y > 0]
6.552
∞
1.
J ν (xy)
0
dx
1/2
a2 )
(x2 +
= I ν/2
1
2 ay
K ν/2
1
2 ay
ET II 22(2)
[Re a > 0,
y > 0,
Re ν > −1]
ET II 23(11), WA 477(3), MO 44
∞
2.
Y ν (xy)
0
ET II 21(1)
dx
(x2
+
1/2
a2 )
=−
1
sec 12 νπ K ν/2 12 ay K ν/2 12 ay + π sin 12 νπ I ν/2 12 ay
π
[y > 0, Re a > 0, |Re ν| < 1]
ET II 100(18)
∞
3.
K ν (xy)
0
dx
(x2
+
π
sec 12 νπ
8
2
1/2
a2 )
=
J ν/2
1
2 ay
2
+ Y ν/2 12 ay
[Re a > 0,
2 Re y > 0,
|Re ν| < 1]
ET II 128(6)
1
4.
J ν (xy)
0
5.
dx
(1 −
1
Y 0 (xy)
0
6.
dx
(1 −
∞
J ν (xy)
1
1/2
x2 )
1/2
x2 )
dx
1/2
(x2 − 1)
=
2
π
J ν/2 12 y
2
[y > 0,
=
π
J 0 12 y Y 0 12 y
2
[y > 0]
ET II 102(26)a
[y > 0]
ET II 24(23)a
=−
π
J ν/2 12 y Y ν/2 12 y
2
Re ν > −1]
ET II 24(22)a
6.561
Bessel functions and powers
∞
7.
Y ν (xy)
1
dx
1/2
(x2 − 1)
=
675
2 2 π J ν/2 12 y
− Y ν/2 12 y
4
[y > 0]
∞
6.553
x−1/2 I ν (x) K ν (x) K μ (2x) dx =
Γ
1
0
4
ET II 102(27)
+ 12 μ Γ 14 − 12 μ Γ 14 + ν + 12 μ Γ 14 + ν − 12 μ
3
4 Γ 4 + ν + 12 μ Γ 34 + ν − 12 μ
|Re μ| < 12 , 2 Re ν > |Re μ| − 12
ET II 372(2)
6.554
∞
1.
x J 0 (xy)
0
x J 0 (xy)
0
dx
(1 −
∞
3.
x J 0 (xy)
1
x J 0 (xy)
0
5.
∞
11
[y > 0]
ET II 7(6)a
= a−1 e−ay
[y > 0,
Re a > 0]
1 ν √
2a π dx =
J ν (ak) K ν (ak)
2ν
(2k) Γ ν + 12
a > 0,
|arg k| >
3/2
a2 )
ν+1/2
(x4 + 4k 4 )
0
ET II 7(4)
= y −1 cos y
− 1)
xν+1 J ν (ax)
Re a > 0]
ET II 7(5)a
1/2
+
[y > 0,
= y −1 sin y
dx
(x2
= y −1 e−ay
[y > 0]
1/2
x2 )
dx
(x2
∞
4.
1/2
(a2 + x2 )
1
2.
dx
ET II 7(7)a
π
4,
Re ν > − 12
WA 473(1)
∞
6.555
0
∞
6.556
0
x1/2 J 2ν−1 ax1/2 Y ν (xy) dx = −
a
Hν−1
2y 2
a2
4y
a > 0,
+ 1/2 ,
dx
a
a
π
√
Y ν/2
J ν a x2 + 1
= − J ν/2
2
2
2
x2 + 1
y > 0,
Re ν > − 12
ET II 111(17)
[Re ν > −1,
a > 0]
MO 46
6.56–6.58 Combinations of Bessel functions and powers
6.561
1
1.
0
1
2.
0
3.
0
1
1
xν J ν (ax) dx = 2ν−1 a−ν π 2 Γ ν + 12 [J ν (a) Hν−1 (a) − Hν (a) J ν−1 (a)]
Re ν > − 12
1
xν Y ν (ax) dx = 2ν−1 a−ν π 2 Γ ν + 12 [Y ν (a) Hν−1 (a) − Hν (a) Y ν−1 (a)]
Re ν > − 12
1
xν I ν (ax) dx = 2ν−1 a−ν π 2 Γ ν + 12 [I ν (a) Lν−1 (a) − Lν (a) I ν−1 (a)]
Re ν > − 12
ET II 333(2)a
ET II 338(43)a
ET II 364(2)a
676
Bessel Functions
1
4.
0
1
5.
6.561
1
xν K ν (ax) dx = 2ν−1 a−ν π 2 Γ ν + 12 [K ν (a) Lν−1 (a) + Lν (a) K ν−1 (a)]
Re ν > − 12
xν+1 J ν (ax) dx = a−1 J ν+1 (a)
ET II 367(21)a
[Re ν > −1]
ET II 333(3)a
0
1
6.
xν+1 Y ν (ax) dx = a−1 Y ν+1 (a) + 2ν+1 a−ν−2 π −1 Γ(ν + 1)
0
1
7.
xν+1 I ν (ax) dx = a−1 I ν+1 (a)
[Re ν > −1]
ET II 339(44)a
[Re ν > −1]
ET II 365(3)a
[Re ν > −1]
ET II 367(22)a
0
1
8.
xν+1 K ν (ax) dx = 2ν a−ν−2 Γ(ν + 1) − a−1 K ν+1 (a)
0
1
x1−ν J ν (ax) dx =
9.
0
1
x1−ν Y ν (ax) dx =
10.
0
1
11.
a
ν−2
2ν−1
Γ(ν)
− a−1 J ν−1 (a)
aν−2 cot(νπ)
− a−1 Y ν−1 (a)
2ν−1 Γ(ν)
x1−ν I ν (ax) dx = a−1 I ν−1 (a) −
0
1
12.
ET II 333(4)a
[Re ν < 1]
ET II 339(45)a
aν−2
Γ(ν)
ET II 365(4)a
2ν−1
x1−ν K ν (ax) dx = 2−ν aν−2 Γ(1 − ν) − a−1 K ν−1 (a)
0
1
xμ J ν (ax) dx =
13.7
0
2μ Γ
[Re ν < 1]
ν+μ+1 aμ+1 Γ
2
+ a−μ {(μ + ν − 1) J ν (a) S μ−1,ν−1 (a) − J ν−1 (a) S μ,ν (a)}
ν−μ+1
2
[a > 0,
1 1
1
∞
μ
μ −μ−1 Γ 2 + 2 ν + 2 μ
x J ν (ax) dx = 2 a
Γ 12 + 12 ν − 12 μ
0
14.
∞
15.
0
16.
0
ET II 367(23)a
Re(μ + ν) > −1]
− Re ν − 1 < Re μ < 12 ,
1
−μ−1 Γ 12 + 12 ν + 12 μ
μ
μ
x Y ν (ax) dx = 2 cot 2 (ν + 1 − μ)π a
Γ 12 + 12 ν − 12 μ
|Re ν| − 1 < μ < 12 ,
ET II 22(8)a
a>0
EH II 49(19)
a>0
ET II 97(3)a
∞
xμ K ν (ax) dx = 2μ−1 a−μ−1 Γ
1+μ+ν
2
Γ
1+μ−ν
2
[Re (μ + 1 ± ν) > 0,
Re a > 0]
EH II 51(27)
6.564
Bessel functions and powers
∞
17.
0
∞
18.
0
Γ 12 q + 12
J ν (ax)
dx = ν−q q−ν+1 xν−q
2
a
Γ ν − 12 q + 12
Γ
Y ν (x)
dx =
xν−μ
1
2
677
−1 < Re q < Re ν −
1
2
WA 428(1), KU 144(5)
+ 12 μ Γ 12 + 12 μ − ν sin 12 μ − ν π
2ν−μ π
|Re ν| < Re(1 + μ − ν) < 32
WA 430(5)
1
x2m+n+1/2 K n+1/2 (αx) dx =
19.
0
6.562
∞
xμ Y ν (bx)
0
∞
2.
0
STR
k=0
1.
n
π (n + k)! γ(2m + n − k + 1, α)
2
k!(n − k)!
α2m+n+3/2 2k
dx
= (2a)μ π −1 sin 12 π(μ − ν) Γ 12 (μ + ν + 1) Γ 12 (1 + μ − ν) S −μ,ν (ab)
x+a
−2 cos 12 π (μ − ν) Γ 1 + 12 μ + 12 ν Γ 1 + 12 μ − 12 ν S −μ−1,ν (ab)
ET II 98(8)
b > 0, |arg a| < π, Re (μ ± ν) > −1, Re μ < 32
πk ν
xν J ν (ax)
dx =
[H−ν (ak) − Y −ν (ak)]
x+k
2 cos νπ
− 12 < Re ν < 32 ,
a > 0,
|arg k| < π
WA 479(7)
∞
3.
dx
x+a
μ − ν a2 b 2
μ+ν
μ−2
,1 −
;
Γ 12 (μ + ν) Γ 12 (μ − ν) b−μ 1 F 2 1; 1 −
=2
2
2
4
3
− μ + ν a2 b 2
3
−
μ
−
ν
,
;
−2μ−3 Γ 12 (μ − ν − 1) Γ 12 (μ + ν − 1) ab1−μ 1 F 2 1;
2
2
4
xμ K ν (bx)
0
−πaμ cosec[π(μ − ν)] {K ν (ab) + π cos(μπ) cosec[π(ν + μ)] I ν (ab)}
[Re b > 0,
∞
6.563
x−1 J ν (bx)
0
6.564
1.
∞
ν+1
x
0
|arg a| < π,
Re μ > |Re ν| − 1]
ET II 127(4)
dx
πa−μ−1
=
(x + a)1+μ
sin[(
⎧ + ν − μ)π] Γ(μ + 1)
ν+2m
∞
⎨
(−1)m 12 ab
Γ( + ν + 2m)
×
⎩
m! Γ(ν + m + 1) Γ ( + ν − μ + 2m)
m=0
⎫
μ+1−+m
⎬
1
∞ 1
Γ(μ + m + 1) sin 2 ( + ν − μ − m)π
2 ab −
1
m!
Γ
(μ
+
ν
−
+ m + 3) Γ 12 (μ − ν − + m + 3) ⎭
2
m=0
ET II 23(10), WA 479
b > 0, |arg a| < π, Re( + ν) > 0, Re( − μ) < 52
dx
J ν (bx) √
=
2
x + a2
2 ν+ 1
a 2 K ν+ 12 (ab)
πb
Re a > 0,
b > 0,
−1 < Re ν <
1
2
ET II 23(15)
678
Bessel Functions
∞
2.
1−ν
x
0
dx
J ν (bx) √
=
2
x + a2
6.565
,
π 1 −ν +
a2
I ν− 12 (ab) − Lν− 12 (ab)
2b
Re a > 0,
b > 0,
Re ν > − 12
ET II 23(16)
6.565
∞
1.
0
∞
2.
−ν− 12
ab
ab
Γ(ν + 1)
Iν
x−ν x2 + a2
J ν (bx) dx = 2ν a−2ν bν
Kν
Γ(2ν + 1) 2
2
Re a > 0, b > 0,
−ν− 12
xν+1 x2 + a2
0
∞
3.
−ν− 32
xν+1 x2 + a2
0
WA 477(4), ET II 23(17)
Re a > 0,
√
bν π
J ν (bx) dx = ν+1 ab 2
ae Γ ν + 32
[Re a > 0,
b > 0,
Re ν > − 12
ET II 24(18)
b > 0,
Re ν > −1]
ET II 24(19)
∞
4.
J ν (bx)xν+1
(x2 +
0
√ ν−1
πb
J ν (bx) dx = ν ab 2 e Γ ν + 12
Re ν > − 12
∞
5.
μ+1
a2 )
dx =
aν−μ bμ
K ν−μ (ab)
Γ(μ + 1)
−1 < Re ν < Re 2μ + 32 ,
2μ
a > 0,
b>0
MO 43
μ
xν+1 x2 + a2 Y ν (bx) dx = 2ν−1 π −1 a2μ+2 (1 + μ)−1 Γ(ν)b−ν
0
× 1 F 2 1; 1 − ν, 2 + μ;
a2 b 2
4
− 2μ aμ+ν+1 [sin(νπ)]
−1
× Γ(μ + 1)b−1−μ [I μ+ν+1 (ab) − 2 cos(μπ) K μ+ν+1 (ab)]
[b > 0,
ET II 100(19)
μ
2μ a1+μ−ν b−1−μ π
I −1−μ+ν (ab) cot[π(μ − ν)] cosec(πμ)
x1−ν x2 + a2 Y ν (bx) dx =
Γ(−μ)
2μ a1+μ−ν b−1−μ π
−
I 1+μ−ν (ab) cosec[π(μ − ν)] cosec(πν)
Γ(−μ)
2−1−ν a2+2μ bν
a2 b 2
+
cos(πν) Γ(−μ) 1 F 2 1; 2 + μ, 1 + ν;
4
(1 + μ)π
2
MC
Re ν < 1, Re(ν − 2μ) > −3, arg a = π, b > 0
∞
μ
x1+ν x2 + a2 K ν (bx) dx = 2ν Γ(ν + 1)aν+μ+1 b−1−μ S μ−ν,μ+ν+1 (ab)
0
7.
−1 < Re ν < −2 Re μ]
∞
6.10
Re a > 0,
0
[Re a > 0,
Re b > 0,
Re ν > −1]
ET II 128(8)
6.567
Bessel functions and powers
8.
∞
11
μ+1
(x2 + k 2 )
0
6.566
x−1 J ν (ax)
∞
1.
xμ Y ν (bx)
0
∞
2.
x2
xν+1 J ν (ax)
0
0
WA 477(1)
dx
= 2μ−2 π −1 b1−μ
+ a2
, +π
(μ − ν + 1) Γ 12 μ + 12 ν − 12 Γ 12 μ − 12 ν − 12
× cos
2
μ + 1 − ν a2 b 2
μ+1+ν
,2 −
;
× 1 F 2 1; 2 −
2
2
4
,
+
,
+
π
π
1 μ−1
cosec (μ + ν + 1) cot (μ − ν + 1) I ν (ab)
− πa
2
2
,
+π 2
−aμ−1 cosec (μ − ν + 1) K ν (ab)
2
b > 0, Re a > 0, |Re ν| − 1 < Re μ < 52
ET II 100(17)
x2
dx
= bν K ν (ab)
+ b2
a > 0,
Re b > 0,
−1 < Re ν <
3
2
xν K ν (ax)
2 ν−1
x2
dx
π b
[H−ν (ab) − Y −ν (ab)]
=
2
+b
4 cos νπ
a > 0,
Re b > 0,
Re ν > − 12
WA 468(9)
∞
4.
x−ν K ν (ax)
0
aν k +ν−2μ−2 Γ 12 + 12 ν Γ μ + 1 − 12 − 12 ν
dx =
2ν+1 Γ(μ + 1) Γ(ν + 1)
+ν +ν
a2 k 2
;
− μ, ν + 1;
× 1F 2
2 2
4
a2μ+2− Γ 12 ν + 12 − μ − 1
+
1
1
22μ+3− Γ μ + 2 + ν − 2
2
ν + a2 k 2
ν−
× 1 F 2 μ + 1; μ + 2 +
,μ + 2 −
;
2
2
4
a > 0, − Re ν < Re < 2 Re μ + 72 , Re k > 0
EH II 96(58)
∞
3.
679
2
dx
π
[Hν (ab) − Y ν (ab)]
= ν+1
x2 + b2
4b
cos νπ
a > 0,
Re b > 0,
Re ν <
1
2
WA 468(10)
∞
5.
0
x−ν J ν (ax)
x2
dx
π
= ν+1 [I ν (ab) − Lν (ab)]
2
+b
2b
a > 0,
Re b > 0,
Re ν > − 52
WA 468(11)
6.567
1.
0
1
μ
xν+1 1 − x2 J ν (bx) dx = 2μ Γ(μ + 1)b−(μ+1) J ν+μ+1 (b)
[b > 0,
Re ν > −1,
Re μ > −1]
ET II 26(33)a
680
Bessel Functions
1
2.
μ
xν+1 1 − x2 Y ν (bx) dx
0
= b−(μ+1) 2μ Γ(μ + 1) Y μ+ν+1 (b) + 2ν+1 π −1 Γ(ν + 1) S μ−ν,μ+ν+1 (b)
[b > 0,
3.
4.
6.567
Re μ > −1,
Re ν > −1]
μ
21−ν S ν+μ,μ−ν+1 (b)
[b > 0, Re μ > −1]
x1−ν 1 − x2 J ν (bx) dx =
bμ+1 Γ(ν)
0
1
μ
x1−ν 1 − x2 Y ν (bx) dx = b−(μ+1) 21−ν π −1 cos(νπ) Γ (1 − ν)
ET II 103(35)a
1
ET II 25(31)a
× S μ+ν,μ−ν+1 (b) − 2 cosec(νπ) Γ(μ + 1) J μ−ν+1 (b)
0
μ
[b > 0,
1
5.
0
Re μ > −1,
Re ν < 1]
ET II 104(37)a
μ
b2
x1−ν 1 − x2 K ν (bx) dx = 2−ν−2 bν (μ + 1)−1 Γ(−ν) 1 F 2 1; ν + 1, μ + 2;
4
+π2μ−1 b−(μ+1) cosec (νπ) Γ(μ + 1) I μ−ν+1 (b)
6.
7.
8.
9.
10.
11.
12.
13.
1
[Re μ > −1,
Re ν < 1]
π
Hν− 12 (b)
[b > 0]
2b
0
1
+
,
π
dx
cosec(νπ) cos(νπ) J ν+ 12 (b) − H−ν− 12 (b)
x1+ν Y ν (bx) √
=
2b
1 − x2
0
[b > 0, Re ν > −1]
1
+
,
π
dx
cot(νπ) Hν− 12 (b) − Y ν− 12 (b) − J ν− 12 (b)
x1−ν Y ν (bx) √
=
2b
1 − x2
0
[b > 0, Re ν < 1]
2
1
ν− 12
√
b
xν 1 − x2
J ν (bx) dx = 2ν−1 πb−ν Γ ν + 12 J ν
2
0
b > 0, Re ν > − 12
1
ν− 12
√
b
b
1
xν 1 − x2
Y ν (bx) dx = 2ν−1 πb−ν Γ ν +
Jν
Yν
2
2
2
0
b > 0, Re ν > − 12
1
ν− 12
√
b
b
1
xν 1 − x2
K ν (bx) dx = 2ν−1 πb−ν Γ ν +
Iν
Kν
2
2
2
0
Re ν > − 12
2
1
1
√ −ν
b
1
ν
2 ν− 2
−ν−1
x 1−x
I ν (bx) dx = 2
πb Γ ν +
Iν
2
2
0
1
ν−1
1
1
b
−ν− 2
− ν sin b
xν+1 1 − x2
J ν (bx) dx = 2−ν √ Γ
2
π
0
b > 0, |Re ν| < 12
x1−ν J ν (bx) √
dx
=
1 − x2
ET II 129(12)a
ET II 24(24)a
ET II 102(28)a
ET II 102(30)a
ET II 24(25)a
ET II 102(31)a
ET II 129(10)a
ET II 365(5)a
ET II 25(27)a
6.571
Bessel functions and powers
∞
14.
ν
x
x −1
2
ν− 12
Y ν (bx) dx = 2
1
ν− 12
2ν−1
1
xν x2 − 1
K ν (bx) dx = √ b−ν Γ ν +
2
π
∞
−ν− 12
√
x−ν x2 − 1
J ν (bx) dx = −2−ν−1 πbν Γ
1
16.
1
∞
17.8
1
6.568
∞
1.
ν− 12
2ν
x−ν+1 x2 − 1
J ν (bx) dx = √ b−ν−1 Γ
π
xν Y ν (bx)
0
dx
π
= aν−1 J ν (ab)
x2 − a2
2
xμ Y ν (bx)
0
|Re ν| <
1
+ ν cos b
2
b > 0,
|Re ν| <
1
2
b > 0,
− 12 < Re ν <
a > 0,
b
2
Yν
b
2
1
ET II 25(26)a
2
ET II 25(28)
5
2
+π
,
(μ − ν + 1)
dx
π
= aμ−1 J ν (ab) + 2μ π −1 aμ−1 cos
x2 − a2
2
2
μ−ν+1
μ+ν+1
×Γ
Γ
S −μ,ν (ab)
2 2
a > 0, b > 0, |Re ν| − 1 < Re μ < 52
ET II (101)(25)
1
xλ (1 − x)μ−1 J ν (ax) dx
6.569
0
=
6.571
1
− ν Jν
2
b > 0,
ET II 101(22)
∞
2.
b
b
b
b
Jν
J −ν
−Yν
Y −ν
2
2
2
2
1
|Re ν| < 2 , b > 0
ET II 103(32)a
2
b
Kν
2
Re b > 0, Re ν > − 12
ET II 129(11)a
√ −ν
1
πb Γ ν +
2
∞
15.
ν−2
681
∞
1.
+
0
2.
0
∞
+
x2 + a2
12
Γ(μ) Γ(1 + λ + ν)2−ν aν
Γ(ν + 1) Γ(1 + λ + μ + ν)
λ+1+ν λ+2+ν
λ+1+μ+ν λ+2+μ+ν
a2
,
; ν + 1,
,
;−
× 2F 3
2
2
2
2
4
[Re μ > 0, Re(λ + ν) > −1] ET II 193(56)a
,μ
ab
ab
dx
± x J ν (bx) √
= aμ I 12 (ν∓μ)
K 12 (ν±μ)
2
2
x2 + a 2
Re a > 0, b > 0, Re ν > −1, Re μ < 32
ET II 26(38)
,μ
1
dx
x2 + a2 2 − x Y ν (bx) √
2
x + a2
ab
ab
ab
ab
= aμ cot(νπ) I 12 (μ+ν)
K 12 (μ−ν)
− cosec(νπ) I 12 (μ−ν)
K 12 (μ+ν)
2 2
2
2
Re a > 0, b > 0, Re μ > − 32 , |Re ν| < 1 ET II 104(40)
682
Bessel Functions
∞
3.
+
0
6.572
∞
−μ
x
1.
,μ
dx
+ x K ν (bx) √
2
x + a2
ab
π2 μ
a cosec(νπ) J 12 (ν−μ)
=
4
2
x2 + a2
+
2
12
x +a
2
12
,μ
+a
0
6.572
Y − 12 (ν+μ)
ab
ab
ab
− Y 12 (ν−μ)
J − 12 (ν+μ)
2
2
2
[Re a > 0, Re b > 0]
ET II 130(15)
Γ 1+ν−μ
dx
2
W 12 μ, 12 ν (ab) M − 12 μ, 12 ν (ab)
J ν (bx) √
=
ab Γ(ν + 1)
x2 + a2
[Re a > 0, b > 0, Re(ν − μ) > −1]
ET II 26(40)
∞
2.
0
+
,μ
1
dx
x−μ x2 + a2 2 + a K ν (bx) √
2
x + a2
1+ν−μ 1−ν−μ Γ
2
2
W 12 μ, 12 ν (iab) W 12 μ, 12 ν (−iab)
=
2ab
Re b > 0, Re μ + |Re ν| < 1] ET II 130(18), BU 87(6a)
Γ
[Re a > 0,
∞
3.
x−μ
+
x2 + a2
12
−a
,μ
0
6.573
∞
1.
xν−M +1 J ν (bx)
0
k
-
dx
+ a2
Γ 1+ν+μ
ν−μ
1
2
tan
π M 12 μ, 12 ν (ab)
= − W − 12 μ, 12 ν (ab)
ab
Γ(ν + 1)
2
ν−μ
π W 12 μ, 12 ν (ab)
+ sec
2
Re a > 0, b > 0, |Re ν| < 12 + 12 Re μ
ET II 105(42)
Y ν (bx) √
x2
J μi (ai x) dx = 0
i=1 %
ai > 0,
k
M=
k
μi
&
i=1
ai 0,
k
i=1
ai < b < ∞,
k
-
aμi i
,
Γ (1 + μi )
i=1
&
0 < Re ν < Re M + 12 k +
3
2
M=
k
μi
i=1
WA 460(16)a, ET II 54(43)
6.576
Bessel functions and powers
6.574
∞
1.8
0
683
ν+μ−λ+1
2
J ν (αt) J μ (βt)t−λ dt =
−ν + μ + λ + 1
2λ β ν−λ+1 Γ
Γ(ν + 1)
2
ν+μ−λ+1 ν−μ−λ+1
α2
,
; ν + 1; 2
×F
2
2
β
[Re(ν + μ − λ + 1) > 0, Re λ > −1, 0 < α < β]
αν Γ
WA 439(2)a, MO 49
If we reverse the positions of ν and μ and at the same time reverse the positions of α and β, the
function on the right-hand side of this equation will change. Thus, the right-hand side represents
α
α
that is not analytic at = 1.
a function of
β
β
For α = β, we have the following equation:
∞
2.
0
ν+μ−λ+1
2
ν+μ+λ+1
ν−μ+λ+1
Γ
2
2
[Re(ν + μ + 1) > Re λ > 0, α > 0]
αλ−1 Γ(λ) Γ
J ν (αt) J μ (αt)t−λ dt =
2λ Γ
−ν + μ + λ + 1
2
Γ
MO 49, WA 441(2)a
3.∗
If μ − ν + λ + 1 (or ν − μ + λ + 1) is a negative integer, the right-hand side of equation 6.574 1 (or
6.574 3) vanishes. The cases in which the hypergeometric function F in 6.574 3 (or 6.574 1)
can be reduced to an elementary function are then especially important.
μ+ν−λ+1
∞
βν Γ
2
J ν (αt) J μ (βt)t−λ dt =
ν
−
μ
+
λ+1
0
2λ αμ−λ+1 Γ
Γ(ν + 1)
2
ν + μ − λ + 1 −ν + μ − λ + 1
β2
,
; μ + 1; 2
×F
2
2
α
[Re(ν + μ − λ + 1) > 0, Re λ > −1, 0 < β < α] MO 50, WA 440(3)a
If μ − ν + λ + 1 (or ν − μ + λ + 1) is a negative integer, the right-hand side of equation 6.754 1 (or
6.574 3) vanishes. The cases in which the hypergeometric function F in 6.754 3 (or 6.574 1)
can be reduced to an elementary function are then especially important.
6.575 1.11
∞
J ν+1 (αt) J μ (βt)tμ−ν dt = 0
0
=
2.
0
∞
2 ν−μ
[α < β]
α −β
β
Γ(ν − μ + 1)
2
2ν−μ αν+1
√
μ
[α ≥ β]
[Re(ν + 1) > Re μ > −1]
MO 51
J ν (x) J μ (x)
π Γ(ν + μ)
dx = ν+μ xν+μ
2
Γ ν + μ + 12 Γ ν + 12 Γ μ + 12
[Re(ν + μ) > 0]
KU 147(17), WA 434(1)
684
6.576
Bessel Functions
∞
1.
xμ−ν+1 J μ (x) K ν (x) dx =
0
1
2
6.576
Γ(μ − ν + 1)
[Re μ > −1,
Re(μ − ν) > −1]
ET II 370(47)
∞
2.11
aν b ν Γ ν +
x−λ J ν (ax) J ν (bx) dx =
0
∞
3.
0
∞
4.
∞
5.
ν−λ+μ+1
ν−λ−μ+1
Γ
2
2
x−λ K μ (ax) J ν (bx) dx =
λ+1 ν−λ+1
2
a
Γ(1 + ν)
ν−λ+μ+1 ν−λ−μ+1
b2
,
; ν + 1; − 2
×F
2
2
a
[Re (a ± ib) > 0, Re(ν − λ + 1) > |Re μ|] EH II 52(31), ET II 63(4), WA 449(1)
1−λ+μ+ν
1−λ−μ+ν
2−2−λ a−ν+λ−1 bν
Γ
Γ
Γ(1 − λ)
2
2
1−λ+μ−ν
1−λ−μ−ν
×Γ
Γ
2
2
1−λ+μ+ν 1−λ−μ+ν
b2
,
; 1 − λ; 1 − 2
×F
2
2
a
[Re a + b > 0, Re λ < 1 − |Re μ| − |Re ν|] ET II 145(49), EH II 93(36)
x−λ K μ (ax) K ν (bx) dx =
−λ
x
− 12 λ + 12 μ + 12 ν Γ 12 − 12 λ − 12 μ + 12 ν
2λ+1 Γ(ν + 1)a−λ+ν+1
2
1 1
− λ + 12 μ + 12 ν, 12 − 12 λ − 12 μ + 12 ν; ν + 1; ab 2
2 2
[Re (ν + 1 − λ ± μ) > 0, a > b]
EH II 93(35)
π(ν − μ − λ) ∞ −λ
2
sin
x K μ (ax) I ν (bx) dx
π
2
0
Re λ > −1, Re (ν − λ + 1 ± μ) > 0]
(see 6.576 5)
EH II 93(37)
bν Γ
K μ (ax) I ν (bx) dx =
0
×F
∞
6.
[a > b,
7.
0
∞
1
2
x−λ Y μ (ax) J ν (bx) dx =
0
8
ET II 47(4)
bν Γ
0
1+λ
2
1
4ab
1−λ
, ν + ; 2ν + 1;
ν+
2
2
(a + b)2
[a > 0, b > 0, 2 Re ν + 1 > Re λ > −1]
2λ (a + b)2ν−λ+1 Γ(ν + 1) Γ
×F
1−λ
2
xμ+ν+1 J μ (ax) K ν (bx) dx = 2μ+ν aμ bν
Γ(μ + ν + 1)
μ+ν+1
(a2 + b2 )
[Re μ > |Re ν| − 1,
Re b > |Im a|]
ET 137(16), EH II 93(36)
6.578
Bessel functions and powers
6.577 1.8
∞
[a > 0,
2.
∞
dx
= (−1)n cν−μ+2n I μ (ac) K ν (bc)
x2 + c2
Re c > 0, 2 + Re μ − 2n > Re ν > −1 − n, n ≥ 0 an integer]
ET II 49(13)
dx
= (−1)n cμ−ν+2n I ν (bc) K μ (ac)
+ c2
Re ν − 2n + 2 > Re μ > −n − 1, n ≥ 0 an integer]
ET II 49(15)
xν−μ+1+2n J μ (ax) J ν (bx)
0
8
b > a,
xμ−ν+1+2n J μ (ax) J ν (bx)
0
[b > 0,
6.578
∞
−1
x
1.
a > b,
x2
2−1 aλ bμ c−λ−μ− Γ
J λ (ax) J μ (bx) J ν (cx) dx =
×F4
Re(λ + μ + ν + ) > 0,
∞
2.
=
∞
3.
0
∞
4.
0
λ+μ−ν+
2
λ+μ−ν+ λ+μ+ν+
a2 b 2
,
; λ + 1, μ + 1; 2 , 2
2
2
c c
∞
5.
Re <

Table Of Integrals, Series And Products (7Ed , Elsevier, 2007 Gradshteyn I , Ryzhik 1220S)

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Table Of Integrals, Series And Products (7Ed , Elsevier, 2007 Gradshteyn I , Ryzhik 1220S)

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