Intégration par parties.

[F(t)]b
a=f(b)−f(a)
F f [a,b]
Zb
a
f(t) dt= [F(t)]b
a
f[a,b]
x∈[a,b]
x7→ Zx
a
f(t) dt
f[a,b]a
(uv)0=u0v+uv0
u0v
u0v= (uv)0−uv0
Zb
a
u0(t)v(t) dt=Zb
a
(uv)0(t)−u(t)v0(t) dt
Zb
a
u0(t)v(t) dt=Zb
a
(uv)0(t) dt−Zb
a
u(t)v0(t) dt
(uv) (uv)0
Zb
a
u0(t)v(t) dt= [u(t)v(t)]b
a−Zb
a
u(t)v0(t) dt

u v [a,b]
Zb
a
u0(t)v(t) dt= [u(t)v(t)]b
a−Zb
a
u(t)v0(t) dt
I=Ze
1
(t+ 1) ln(t) dt
I
u v0
u v0
u0(t) = t+ 1 v(t) = ln(t)
u(t) = 1
2t2+t v0(t) = 1 t
u v [1,e]
I=1
2t2+t×ln(t)e
1−Ze
11
2t2+t×1 tdt
=1
2e2+ e −Ze
1
1
2t+ 1 dt
=1
2e2+ e −1 4t2+te
1
=1
2e2+ e −1 4e2−e + 1 4+ 1
I=1
4e2+5 4

g(t) ln(t)
g(t)et
g(t) sin(t)g(t) cos(t)
g(t) sinh(t)g(t) cosh(t)
I=Z1
0
xetdt
I=Z4
1
(3t+ 1)√tdt
I=Zln 2
0
(t+ 2)etdt
I=Z2
1
ln tdt
I=Z1
0
(2t−1)etdt
I=Ze
1
t2(1 −2 ln t) dt
I=Zπ
6
0
(2t−1) sin tdt
I=Ze
1
tln tdt
I=tet1
0−Z1
0
etdt
= e −et1
0
= e −(e −1)
= 1

fR
∀x∈R, f(x) = (x2+ 2x+ 2) exp(−x)
a b c F :R→R
x7→ (ax2+bx +c)e−x
F f R
I=Z1
0
(t2+ 2t+ 2) exp(−t) dt
I
a b c F f
F f x ∈R
F0(x) = f(x)
(2ax +b)e−x+ (ax2+bx +c)−e−x=f(x)
2ax +b−ax2−bx −ce−x=f(x)
−ax2+ (2a−b)x−c+be−x=f(x)
e−x6= 0
−ax2+ (2a−b)x−c+b=x2+ 2x+ 2
a=−1 2a−b= 2 b−c= 2
a=−1b=−4c=−6
a=−1b=−2c=−6
x7→ (−x2−4x−6)e−xRF0(x) =
(x2+ 2x+ 2) exp(−x)
a b c

I
I=F(1) −F(0)
= 5e−1−2
I= 5e−1−2
(Wn)n∈N
∀n∈N, Wn=Zπ
2
0
sinn(t) dt
(Wn)n∈N
(Wn)
(Wn)n∈N
n≥2
Wnu0(t) = sin(t)v(t) = sinn−1(t)
nWn= (n−1)Wn−2.
p∈N
W2p=(2p−1)(2p−3) . . . 1
[(2p)(2p−2) . . . 2]2·π
2W2p+1 =[(2p)(2p−2) . . . 2]2 (2p+ 1)(2p−1) . . . 3
p
(2p+ 1)W2pW2p+1 =π
2
(Wn)n∈N0