Calcul de primitives

K R C
IRf:I→K
F:I→Kf I I
∀x∈I, F 0(x) = f(x).
Rf(x)dx=F(x), x ∈I F f I
R
BRf(x)dx
Rxdx=x2
2, x ∈RRxdx=x2 2+ 3, x ∈R
0 = 3 x7→ x2
2x7→ x2 2+ 3
x7→ xR
F f I c ∈KF+c
f I
F G f I
c∈KI G =F+c
F:I→Kf:I→K
f I
{F+c, c ∈K}.
f:I→Kg:I→KF G
f g I (α, β)∈KαF +βG
αf +βg I

Fonction Param`etre Primitive Intervalle d´ef .
xnn∈Nxn+1
n+1 R
xpp∈Z\ {−1},xp+1
p+1 R∗
+ou R∗ −
xaa∈R\ {−1},xa+1
a+1 R∗
+
1
xln(|x|)R∗
+ou R∗ −
eλx λ∈C∗eλx
λR
Fonction Primitive Intervalle d´ef .
cos xsin xR
sin x−cos xR
1
√1−x2dxarcsin(x) ] −1,1[
1
1+x2dxarctan(x)R
u:I→J I F :J→KJ
u0×F0◦u F ◦u I
f:R→KF f I
x7→ 1
aF(ax +b)x7→ f(ax +b)
•Rcos xsin xdx=1
2sin2(x)x∈R
u0u1
2u2
•Rx
1+x2dx=1 2ln (1 + x2)x∈R
u0
uuln(u)
•Rtan(x)dx=−ln (cos(x)) x∈−π
2,π 2
u0
u
•Rxex2dx=1
2ex2x∈R
u0eueu
•R1
a2+x2dx=1 a.arctan x ax∈R

ϕ
f:t7→ eαt cos(ωt) et g:t7→ eαt sin(ωt)α∈R, ω ∈R∗.
λ∈C∗t7→ 1
λeλt t7→ eλt
eαt cos(ωt) = eαtRe eiωt= Re e(α+iω)tet eαt sin(ωt) = eαtIm eiωt= Im e(α+iω)t.
φ:t7→ e(α+iω)tΦ : t7→ 1
α+iω e(α+iω)t
F:t7→ Re (Φ(t)) G:t7→ Im (Φ(t)) eαt cos(ωt)
eαt sin(ωt)
∀t∈R(Re (f(t)) = eαt
α2+ω2(αcos(ωt) + ωsin(ωt))
Im (f(t)) = eαt
α2+ω2(αsin(ωt)−ωcos(ωt)) .
a, b, c Ra6= 0
x7→ 1
ax2+bx +c.
ax2+bx +c=a x+b
2a2
+c
a−b2 4a2!=a x+b 2a2
−∆
4a2!,
∆ = b2−4ac
•∆>0α β α < β
x7→ 1
ax2+bx +c=1 a(x−α)(x−β).
A
x−α+B x−βA, B
1
a(x−α)(x−β)=1 a(β−α)1 x−β−1 x−α,
x∈]−∞, α[ ]α, β[ ]β, +∞[
Z1
a(x−α)(x−β)dx=1 a(β−α)(ln (|x−β|)−ln (|x−α|)) = 1 β−αln 
x−β
x−α.
•∆ = 0
γ x 7→ 1
a(x−γ)2
x7→ − 1
x−γ

•∆<0A=b
2aB=q|∆| 4a2
1
ax2+bx +c=1 a((x+A)2+B2)=1 aB ·1/B
1 + x+A
B2.
Z1
ax2+bx +c=1 aB arctan x+A Bx∈R.
IRf:I→KI a ∈I
F:x7→ Zx
a
f(t)dt
f I f a
I
•f:R→R
t7→ e−t2F:R→R x7→ Rx
0e−t2dtfR
x0= 0 ∈R
•g:R∗
+→R
t7→ ln(t)G:R→R
x7→ Rx
1ln(t)dtgR
x0= 1 ∈R∗
+
x > 0
G(x) = Rx
1g(t)dt=xln(x)−x+ 1
Rx
0e−t2dt F :x7→ Rx 0e−t2dt f

f: [a, b]→RF f [a, b]
Zb
a
f(t)dt=F(b)−F(a).
F[a, b]Fb
a=F(b)−F(a)
Zπ
2
0
cos tsin3tdt
u v
(uv)0=u0v+uv0, u0v= (uv)0−uv0.
(uv)0(uv)uv0
u0v
u0v u0v
u0v uv0u v
f:I→RC1I
I f0I
BC1
u v C1[a, b]
Zb
a
u0(t)v(t)dt=uvb
a−Zb
a
u(t)v0(t)dt.
Z1
0
te−tdt
f:x7→ xcos(x) et g:x7→ ln(x).