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R+∞
0
te−√t
1+t2dt
R1
0
ln t
√(1−t)3dtR+∞
0
dt
et−1
R+∞
0e−(ln t)2dtR+∞
0e−tarctan tdt
R+∞
0t+ 2 −√t2+ 4t+ 1 dt
]1 ; +∞[
f(x) = √ln x
(x−1)√x
Z3
2
f(x) dx≤ln 3
2
Z+∞
0
ln(th t) dt
t7→ sin t t 7→ sin t
t[0 ; +∞[
f: [1 ; +∞[→R
∀x, a ≥1,0≤f(x)≤a
x2+1 a2
f[1 ; +∞[
f: [0 ; +∞[→R
g: [0 ; +∞[→R
g(x) = f(x) sin x
f g
f: [0 ; +∞[→R
f(x+ 1)
f(x)−−−−−→
x→+∞`∈[0 ; 1[
R+∞
0f(t) dt
f: [0 ; 1] →R
f(x) = x2cos 1/x2x∈]0 ; 1] f(0) = 0
f[0 ; 1] f0
]0 ; 1]
gR∗
+
g(x) = 1
xZx
0
f(t) dt
fR+
g
g0(x)f(x)g(x)x > 0
0< a < b
Zb
a
g2(t) dt= 2 Zb
a
f(t)g(t) dt+ag2(a)−bg2(b)
sZb
a
g2(t) dt≤sZ+∞
0
f2(t) dt+sag2(a) + Z+∞
0
f2(t) dt
Z+∞
0
g2(t) dt

α∈R
Z+∞
0
t−sin t
tαdt
a−1x= tan t
Zπ/2
0
dt
1 + asin2(t)=π
2√1 + a
α > 0
XZπ
0
dt
1+(nπ)αsin2(t)
XZ(n+1)π
nπ
dt
1 + tαsin2(t)
Z+∞
0
dt
1 + tαsin2(t)
f: [0 ; +∞[→RC1
f2f02f+∞
f: [0 ; +∞[→R
f
Zx+1
x
f(t) dt−−−−−→
x→+∞0
f:R+→R R+
f+∞
xf(x)x→+∞f
R+f+∞
f: [0 ; +∞[→R
(xn)
xn→+∞xnf(xn)→0
f∈ C0(R+,R+)
f∈ C2([0 ; +∞[,R)f f 00
f0(x)→0x→+∞
f.f0
g:R+→R
∀ε > 0,∃M∈R,Z+∞
0g(t)dt−ZM 0g(t)dt≤ε
g
fC2[0 ; +∞[f00 [0 ; +∞[
R+∞
0f(t) dt
lim
x→+∞f0(x) = 0 lim x→+∞f(x) = 0
Xf(n)Xf0(n)

R+∞
0
dt
(t+1)(t+2)
R+∞
0
dt
(et+1)(e−t+1) R+∞
0ln1 + 1
t2dtR+∞
0e−√tdt
R+∞
0
ln t
(1+t)2dt
R+∞
0
dt
√et+1
R+∞
1
dt
sh tR+∞
0
tln t
(t2+1)2dt
R+∞
1
dt
t2√1+t2R1
0
ln t
√tdt
R+∞
0
e−√t
√tdt
Rπ/2
0sin xln(sin x) dx
R1
0
ln t
√1−tdt
R+∞
0
dx
(x+1) 3
√x
R+∞
0
√1+x−1
x(1+x)dx
R+∞
0
(1+x)1/3−1
x(1+x)2/3dx
R2π
0
dx
2+cos x
R2π
0
sin2(x)
3 cos2(x)+1 dx
R1
0
xdx
√x−x2
J=Z+∞
0
tdt
1 + t4
I=Z+∞
0
dt
1 + t4=Z+∞
0
t2dt
1 + t4
1 + t4I
Z+∞
−∞
dt
(1 + t2)(1 + it)
I=Z+∞
0
sin3t
t2dt
x > 0
I(x) = Z+∞
x
sin3t
t2dt
sin 3a= 3 sin a−4 sin3a
I(x) = 3
4Z3x
x
sin t
t2dt
I
(a, b)∈R2a < b f ∈ C0(R,R)`−∞
R+∞
0f
Z+∞
−∞
(f(a+x)−f(b+x)) dx
Z+∞
0
arctan(2x)−arctan x
xdx
Z+∞
−∞
dx
1 + x4+x8

∀x∈R,ex≥1 + x
∀t∈R,1−t2≤e−t2≤1
1 + t2
n∈N∗
I=Z+∞
0
e−t2dt, In=Z1
0
(1 −t2)ndt Jn=Z+∞
0
dt
(1 + t2)n
In≤I
√n≤Jn
Wn=Zπ/2
0
cosnxdx
In=W2n+1 Jn+1 =W2n
WnWn+2
un= (n+ 1)WnWn+1
WnI
I=Z+∞
0
t[1/t] dt
f: ]0 ; 1] →Rn∈N∗
Sn=1
n
n
X
k=1
fk
n
f]0 ; 1] (Sn)
Z]0;1]
f(t) dt= lim
n→+∞Sn
a > 0
I(a) = Z+∞
0
sin(t)e−at dt
a > 0u=a/t
I(a) = Z+∞
0
ln t
a2+t2dt
Z+∞
0
ln t
t2+a2dt
a > 0
a b
Z+∞
0√t+a√t+1+b√t+ 2dt
a > 0
I(a) = Z+∞
0
(t− btc)e−at dt
fRlimx→+∞f(x) = `

a > 0
Z+∞
0
f(x+a)−f(x) dx
Z+∞
−∞
arctan(x+a)−arctan(x) dx
Zπ
0
sin2t
(1 −2xcos t+x2)(1 −2ycos t+y2)dt
x, y ∈]−1 ; 1[
Z+∞
−∞
1
1+(t+ib)2dt
α∈R
Z+∞
0
dx
x2+αx + 1
a, b > 0
I(a, b) = Z+∞
−∞
dt
(t2+a2)(t2+b2)
P Q R[X]QRdeg P≤deg Q−2
RRP/Q
P/Q
f: [0 ; +∞[→R
Z+∞
1
f(t)
tdt
0< a < b
x > 0
Z+∞
x
f(at)−f(bt)
tdt=Zbx
ax
f(t)
tdt
Z+∞
0
f(at)−f(bt)
tdt
t= e−x
I=Z1
0
1 + t2
1 + t4dt
Z+∞
0
1 + x2
1 + x4dx
x= et
Z+∞
0
dx
1 + x4