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x7→ √xR+
x7→ ln xR∗
+
x7→ xln x]0 ; 1]
f: [0 ; 1[ →Rf
f:R∗
+→R
∀x > 0, f(nx)−−−−→
n→∞ 0
f+∞
f: [a;b]→R
σ[a;b]f
f σ
t7→ sin 1
tt > 0t= 0
[0 ; 1]

R2
1
dt
t2R1
0
dt
1+t2R1/2
0
dt
√1−t2
R2π
0cos2tdtR2 1ln tdtR1 0
t
√1+t2dt
I=Zπ/4
0
ln(1 + tan x) dx
Rtet2dtRln t
tdtRdt tln t
Rcos tsin tdtRtan tdtRcos3tdt
Rt2
1+t3dtRt
√1+t2dtRt
1+t4dt
Rdt
it+1 Retcos tdtRtsin tetdt
Rtln tdtRtarctan tdtRtsin3tdt
R(t2−t+ 1)e−tdtR(t−1) sin tdtR(t+ 1) ch tdt
R1
0ln(1 + t2) dtRe
1tnln tdt
n∈NReπ
1sin(ln t) dt
R1
0arctan tdtR1/2
0arcsin tdtR1
0tarctan tdt
Z1
0
ln(1 + t2) dt
(a, b)∈R2µ∈R∗
+f∈ C2([a;b],R)
∀x∈[a;b],|f0(x)| ≥ µ f0
Zb
a
e2iπf(t)dt≤1
µπ

Rdt
√t+√t3Rln tdt
t+t(ln t)2Re2tdt
et+1
Zdt
t√t2−1
Re
1
dt
t+t(ln t)2Re
1
dt
t√ln t+1 R1
0
dt
et+1
R1
0√1−t2dtR1 0t2√1−t2dtR2 1
ln t
√tdt
Zπ/2
0
cos t
cos t+ sin tdt=Zπ/2
0
sin t
cos t+ sin tdt=π 4
Z1
0
dt
√1−t2+t
f∈ C ([0 ; 1],R)
Zπ
0
tf(sin t) dt=π
2Zπ
0
f(sin t) dt
In=Zπ
0
xsin2n(x)
sin2n(x) + cos2n(x)dx
a b ab > 0
I(a, b) = Zb
a
1−x2
(1 + x2)√1 + x4dx
I(−b, −a)I(1/a, 1/b)I(1/a, a)I(a, b)
a, b > 1I(a, b)v=x+ 1/x
v= 1/t
a, b
ab > 0
Rπ
0
sin t
3+cos2tdt
R2
1
dt
√t+2t
R2
1
ln(1+t)−ln t
t2dt
Z√3
0
arcsin2t
1 + t2dt
f:R→R
g:R→RC1

g(x) = Rx2
2xf(t) dtg(x) = Rx
0xf(t) dt g(x) =
Rx
0f(t+x) dt
ϕ:R→R
ϕ(t) = sh t
tt6= 0 ϕ(0) = 1
f:R→R
f(x) = Z2x
x
ϕ(t) dt
f f
f f0(x)
f
g:R→R
x∈R
f(x) = Zx
0
sin(x−t)g(t) dt
f
f0(x) = Zx
0
cos(t−x)g(t) dt
f y00 +y=g(x)
f:R→RC1F:R∗→R
∀x6= 0, F (x) = 1
2xZx
−x
f(t) dt
F
FR∗F0(x)
F F 0(0) = 0
fR R
∀(x, y)∈R2, f(x)−f(y) = Z2y+x
2x+y
f(t) dt
fC1f
x∈]0 ; 1[
ϕ(x) = Zx2
x
dt
ln t
ϕ
Z1
0
x−1
ln xdx
f(x) = Zx2
x
dt
ln t
f0++∞+∞f(x)/x
f(x) ln 2 x
fC∞R∗
+R+
f
f:x7→ Z2x
x
et
tdt
R∗

f
f:x∈R∗7→ Z2x
x
ch t
tdt
f f ]0 ; +∞[
f
fC1R+
f∈ C1(R,R)g:R∗→R
g(x) = 1
xZx
0
f(t) dt
g
C1R
x7→ Z2x
x
dt
√1 + t2+t4
x→0x→ ±∞
x∈[1 ; +∞[
F(x) = Zx
1
t
√t3−1dt
F[1 ; +∞[
C∞]1 ; +∞[
F0(x)
F1F1
F+∞
F[1 ; +∞[
F−1]0 ; +∞[
yy0=py3−1
F−10
Pn
k=1 n
n2+k2Pn
k=1 k
n2+k2Pn
k=1 1
√n2+2kn
Sn=
n
X
k=1
√k
(2n)!
nnn!1
n
Qn
k=1 1 + k
n1/n Qn
k=1 1 + k
n21/n
n
X
k=1
sin k
nsin k n2n
X
k=1
sin21
√k+n
n→+∞