Solutions - Dimitri Zuchowski

Examen 1, solutions
NYB Calcul int´egral
Professeur : Dimitri Zuchowski
Question 1.(32%) ´ Evaluer les limites suivantes
a) lim
x→0
e3x−3x−1
1−cos x
0
0
= lim
x→0
3e3x−3
sin x
0
0
= lim
x→0
9>1 e3x
: 1
cos x= 9
b) lim
x→π
−2
π−x−tan x 2∞−∞
= lim
x→π
− 2 cos x
2−(π−x) sin x 2
(π−x) cos x
2!
0
0
= lim
x→π
− −
sin x
2+
sin x
2−1 2(π−x) cos x 2
−1
2(π−x) sin x 2−cos x 2!
0
0
= lim
x→π
−
1
4
: 0
(π−x) sin x
2+1 2
* 0
cos x
2
−1
4
: 0
(π−x) cos x
2+1 2* 1
sin x
2+1 2* 1
sin x
2
= 0
c) lim
x→1−
(ln x)1
1−x= 0∞= 0
d) lim
x→0(cos x+ 7x)1
x=A, ln A= lim
x→0ln (cos x+ 7x)1
x= lim
x→0
ln(cos x+ 7x)
x
0
0
= lim
x→0
−sin x+7
cos x+7x
1= 7, A =e7
Question 2.(63%) Calculer les int´egrales ind´efinie suivantes
a) Z4x3−7x2+3
8x−√11 dx =x4−7x3 3+3 ln |x| 8−√11x+C
b)
Zλ(3λ−9)33 dλ u = 3λ−9, du = 3dλ
=1
9Z(u+ 9)u33 du =1 9Zu34 + 9u33 du λ =u+ 9 3
=1
9(u35 35 +9u34 34 ) + C=(3λ−9)35 315 +(3λ−9)34 34 ) + C

c)
Z3θsin2(θ2+π)dθ u =θ2+π, du = 2θdθ
= 3 Z
θsin2(u)du
2 θ=3 2Z1−cos(2u)du 2v= 2u, dv = 2du
=3
4Z1−cos(v)dv 2=3 8(v−sin(v)) + C
=3(θ2+π)
4−3 sin(2(θ2+π)) 8+C
d)
Z2sec(x)
cos xcot xdx u = sec x, du = sec xtan xdx
Z2u
cos xcot xsec xtan xdu =Z2udu
2u
ln 2 +C=2sec x ln 2 +C
e) Zcsc2(6x3+ 7) −cot2(6x3+ 7) dx =Zdx =x+C
f) Z5x3+x2+ 1
x2+ 1 dx =Z5x+ 1 −5x x2+ 1 dx =5x2 2+x−5 ln |x2+ 1| 2+C
g)
Z(ln(cos(2x)))4
cot(2x)dx u = ln(cos(2x)), du =−2 sin(2x)dx cos(2x)
=−1
2Zu4cos(2x) cot(2x) sin(2x)du =−1 2Zu4du
=−u5
10 +C=−ln(cos(2x))5 10 +C
Question 3. y0=sin x
√y⇒Z√ydy =Zsin xdx ⇒2y3
2
3=−cos x+C. Et
quand x= 0, y = 4 on trouve que C=2(4)3
2
3+ cos(0) = 19 3d’o`u la solution
y=3
s−3 cos x
2+19 22
2