Formulaire : Développements Limités, Dérivées et Primitives

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 =
∞
#
n=0
an
n!xna∈C,x∈R
sh x=
∞
#
n=0
1
(2n+1)! x2n+1 x∈R
ch x=
∞
#
n=0
1
(2n)! x2nx∈R
sin x=
∞
#
n=0
(−1)n
(2n+1)! x2n+1 x∈R
cos x=
∞
#
n=0
(−1)n
(2n)! x2nx∈R
(1 + x)α=1+
∞
#
n=1
α(α−1) ···(α−n+1)
n!xn(α∈R)x∈]−1;1[
1
a−x
=
∞
#
n=0
1
an+1 xn(a∈C∗)x∈]−|a|;|a|[
1
(a−x)2=
∞
#
n=0
n+1
an+2 xn(a∈C∗)x∈]−|a|;|a|[
1
(a−x)k=
∞
#
n=0
Ck−1
n+k−1
an+kxn(a∈C∗)x∈]−|a|;|a|[
ln(1 −x)=−
∞
#
n=1
1
nxnx∈[−1;1[
ln(1 + x)=
∞
#
n=1
(−1)n−1
nxnx∈]−1;1]
√1+x=1+
x
2+
∞
#
n=2
(−1)n−11×3×···×(2n−3)
2×4×···×(2n)xnx∈]−1;1[
1
√1+x
=1+
∞
#
n=1
(−1)n1×3×···×(2n−1)
2×4×···×(2n)xnx∈]−1;1[
Arctan x=
∞
#
n=0
(−1)n
2n+1 x2n+1 x∈[−1;1]
Argth x=
∞
#
n=0
1
2n+1 x2n+1 x∈]−1;1[
Arcsin x=x+
∞
#
n=1
1×3×···×(2n−1)
2×4×···×(2n)
x2n+1
2n+1 x∈]−1;1[
Argsh x=x+
∞
#
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∗
+
eαxα∈CαeαxR
axa∈R∗
+axln aR
ln |x|1
xR∗
logaxa∈R∗
+!{1}1
xln aR∗
cos x−sin xR
sin xcos xR
tan x1+tan
2x=1
cos2xR!$π 2+kπ%%%k∈Z&
cotan x−1−cotan 2x=−1
sin2xR!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
IPolynômesetfractionssimples
Fonction Pr imitive Interva lles
(x−x0)nx0∈R
n∈Z!{−1}
(x−x0)n+1
n+1
n∈N:x∈R
n∈Z!(N∪{−1}):
x∈]−∞;x0[,]x0;+∞[
(x−x0)αx0∈R
α∈C!{−1}
(x−x0)α+1
α+1 ]x0;+∞[
(x−z0)nz0∈C!R
n∈Z!{−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 Pr imitive Interva lles
ln x x(ln x−1) ]0;+∞[
eαxα∈C∗1
αeαxR
sin x−cos xR
cos xsin xR
tan x−ln |cos x|)−π
2+kπ;π 2+kπ*
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 Interva lles
sin2xx
2−sin 2x 4R
cos2xx
2+sin 2x 4R
tan2xtan x−x)−π
2+kπ;π 2+kπ*
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,%%%)−π 2+kπ;π 2+kπ*
1
sh xln %%%th x 2%%%]−∞;0[ ,]0;+∞[
1
ch x2ArctanexR
1
sin2x=1+cotan
2x−cotan x]kπ;(k+1)π[
1
cos2x=1+tan
2xtan x)−π
2+kπ;π 2+kπ*
1
sh 2x=coth2x−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 3)−π 2+kπ;π 2+kπ*