Développements limités usuels en 0

1
Développements limités usuels en 0
ex= 1 + x
1! +x2 2! +···+xn n!+ O xn+1
sh x=x+x3
3! +···+x2n+1 (2n+ 1)! + O x2n+3
ch x= 1 + x2
2! +x4 4! +···+x2n (2n)! + O x2n+2
sin x=x−x3
3! +···+ (−1)nx2n+1 (2n+ 1)! + O x2n+3
cos x= 1 −x2
2! +x4 4! +···+ (−1)nx2n (2n)! + O x2n+2
(1 + x)α= 1 + αx +α(α−1)
2! x2+···+α(α−1) ···(α−n+ 1) n!xn+ O xn+1
1
1−x
= 1 + x+x2+x3+···+xn+ O xn+1
ln(1 −x)=−x−x2
2−x3 3−x4 4− ··· − xn n+ O xn+1
1
1 + x
= 1 −x+x2−x3+···+ (−1)nxn+ O xn+1
ln(1 + x)=x−x2
2+x3 3−x4 4+···+ (−1)n−1xn n+ O xn+1
√1 + x= 1 + x
2−x2 8+··· + (−1)n−11×3× ··· × (2n−3) 2×4× ··· × 2nxn+ O xn+1
1
√1 + x
= 1 −x
2+3 8x2− ··· + (−1)n1×3× ··· × (2n−1) 2×4× ··· × 2nxn+ O xn+1 
Arctan x=x−x3
3+···+ (−1)nx2n+1 2n+ 1 + O x2n+3 
Argth x=x+x3
3+···+x2n+1 2n+ 1 + O x2n+3 
Arcsin x=x+1
2
x3
3+···+1×3× ···(2n−1) 2×4× ··· × 2n
x2n+1
2n+ 1 + O x2n+3 
Argsh x=x−1
2
x3
3+···+ (−1)n1×3× ···(2n−1) 2×4× ··· × 2n
x2n+1
2n+ 1 + O x2n+3 
th x=x−x3
3+2 15x5−17 315x7+ O x9
tan x=x+1
3x3+2 15x5+17 315x7+ O x9

2
Développements en série entière usuels
eax =
∞
P
n=0
an
n!xna∈C, x ∈R
sh x=
∞
P
n=0
1
(2n+ 1)! x2n+1 x∈R
ch x=
∞
P
n=0
1
(2n)! x2nx∈R
sin x=
∞
P
n=0
(−1)n
(2n+ 1)! x2n+1 x∈R
cos x=
∞
P
n=0
(−1)n
(2n)! x2nx∈R
(1 + x)α= 1 +
∞
P
n=1
α(α−1) ···(α−n+ 1)
n!xn(α∈R)x∈]−1 ; 1 [
1
a−x
=
∞
P
n=0
1
an+1 xn(a∈C∗)x∈]−|a|;|a|[
1
(a−x)2=
∞
P
n=0
n+ 1
an+2 xn(a∈C∗)x∈]−|a|;|a|[
1
(a−x)k=
∞
P
n=0
Ck−1
n+k−1
an+kxn(a∈C∗)x∈]−|a|;|a|[
ln(1 −x)=−
∞
P
n=1
1
nxnx∈[−1 ; 1 [
ln(1 + x)=
∞
P
n=1
(−1)n−1
nxnx∈]−1 ; 1 ]
√1 + x= 1 + x
2+
∞
P
n=2
(−1)n−11×3× ··· × (2n−3)
2×4× ··· × (2n)xnx∈]−1 ; 1 [
1
√1 + x
= 1 +
∞
P
n=1
(−1)n1×3× ··· × (2n−1)
2×4× ··· × (2n)xnx∈]−1 ; 1 [
Arctan x=
∞
P
n=0
(−1)n
2n+ 1 x2n+1 x∈[−1 ; 1 ]
Argth x=
∞
P
n=0
1
2n+ 1 x2n+1 x∈]−1 ; 1 [
Arcsin x=x+
∞
P
n=1
1×3× ··· × (2n−1)
2×4× ··· × (2n)
x2n+1
2n+ 1 x∈]−1 ; 1 [
Argsh x=x+
∞
P
n=1
(−1)n1×3× ··· × (2n−1)
2×4× ··· × (2n)
x2n+1
2n+ 1 x∈]−1 ; 1 [

3
Dérivées usuelles
Fonction Dérivée Dérivabilité
xnn∈Znxn−1R∗
xαα∈Rαxα−1R∗
+
eαx α∈Cαeαx R
axa∈R∗
+axln aR
ln |x|1
xR∗
logax a ∈R∗
+r{1}1
xln aR∗
cos x−sin xR
sin xcos xR
tan x1 + tan2x=1
cos2xR r nπ 2+kπ k∈Zo
cotan x−1−cotan 2x=−1
sin2xR r πZ
ch xsh xR
sh xch xR
th x1−th 2x=1
ch 2x
R
coth x1−coth 2x=−1
sh 2x
R∗
Arcsin x1
√1−x2]−1 ; 1 [
Arccos x−1
√1−x2]−1 ; 1 [
Arctan x1
1 + x2R
Argsh x1
√x2+ 1
R
Argch x1
√x2−1] 1 ; +∞[
Argth x1
1−x2]−1 ; 1 [

4
Primitives usuelles
I Polynômes et fractions simples
Fonction Primitive Intervalles
(x−x0)nx0∈R
n∈Z r {−1}
(x−x0)n+1
n+ 1
n∈N:x∈R
n∈Z r (N∪ {−1}) :
x∈]−∞;x0[,]x0;+∞[
(x−x0)αx0∈R
α∈C r {−1}
(x−x0)α+1
α+ 1 ]x0;+∞[
(x−z0)nz0∈C r R
n∈Z r {−1}
(x−z0)n+1
n+ 1 R
1
x−aa∈Rln |x−a|]−∞;a[,]a;+∞[
1
x−(a+ ib)a∈R, b ∈R∗
1
2ln (x−a)2+b2
+ i Arctan x−a
b
R
II Fonctions usuelles
Fonction Primitive Intervalles
ln x x(ln x−1) ] 0 ; +∞[
eαx α∈C∗1
αeαx R
sin x−cos xR
cos xsin xR
tan x−ln |cos x|i−π
2+kπ ;π 2+kπ h
cotan xln |sin x|]kπ ; (k+ 1)π[
sh xch xR
ch xsh xR
th xln(ch x)R
coth xln |sh x|]−∞; 0 [ ,] 0 ; +∞[

Primitives usuelles 5
III Puissances et inverses de fonctions usuelles
Fonction Primitive Intervalles
sin2xx
2−sin 2x 4R
cos2xx
2+sin 2x 4R
tan2xtan x−xi−π
2+kπ ;π 2+kπ h
cotan 2x−cotan x−x]kπ ; (k+ 1)π[
sh 2xsh 2x
4−x 2R
ch 2xsh 2x
4+x 2R
th 2x x −th xR
coth 2x x −coth x]−∞; 0 [ ,] 0 ; +∞[
1
sin xln tan x 2]kπ ; (k+ 1)π[
1
cos xln tan x 2+π 4i−π 2+kπ ;π 2+kπ h
1
sh xln th x 2]−∞; 0 [ ,] 0 ; +∞[
1
ch x2 Arctan exR
1
sin2x= 1 + cotan 2x−cotan x]kπ ; (k+ 1)π[
1
cos2x= 1 + tan2xtan xi−π 2+kπ ;π 2+kπ h
1
sh 2x= coth 2x−1−coth x]−∞; 0 [ ,] 0 ; +∞[
1
ch 2x= 1 −th 2xth xR
1
sin4x−cotan x−cotan 3x
3]kπ ; (k+ 1)π[
1
cos4xtan x+tan3x 3i−π 2+kπ ;π 2+kπ h