annexes.maths

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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=xx3
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(α1)
2! x2+···+α(α1) ···(αn+ 1)
n!xn+ O xn+1
1
1x
= 1 + x+x2+x3+···+xn+ O xn+1
ln(1 x)=xx2
2x3
3x4
4− ··· − xn
n+ O xn+1
1
1 + x
= 1 x+x2x3+···+ (1)nxn+ O xn+1
ln(1 + x)=xx2
2+x3
3x4
4+···+ (1)n1xn
n+ O xn+1
1 + x= 1 + x
2x2
8+··· + (1)n11×3× ··· × (2n3)
2×4× ··· × 2nxn+ O xn+1
1
1 + x
= 1 x
2+3
8x2− ··· + (1)n1×3× ··· × (2n1)
2×4× ··· × 2nxn+ O xn+1
Arctan x=xx3
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× ···(2n1)
2×4× ··· × 2n
x2n+1
2n+ 1 + O x2n+3
Argsh x=x1
2
x3
3+···+ (1)n1×3× ···(2n1)
2×4× ··· × 2n
x2n+1
2n+ 1 + O x2n+3
th x=xx3
3+2
15x517
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!xnaC, x R
sh x=
P
n=0
1
(2n+ 1)! x2n+1 xR
ch x=
P
n=0
1
(2n)! x2nxR
sin x=
P
n=0
(1)n
(2n+ 1)! x2n+1 xR
cos x=
P
n=0
(1)n
(2n)! x2nxR
(1 + x)α= 1 +
P
n=1
α(α1) ···(αn+ 1)
n!xn(αR)x]1 ; 1 [
1
ax
=
P
n=0
1
an+1 xn(aC)x]−|a|;|a|[
1
(ax)2=
P
n=0
n+ 1
an+2 xn(aC)x]−|a|;|a|[
1
(ax)k=
P
n=0
Ck1
n+k1
an+kxn(aC)x]−|a|;|a|[
ln(1 x)=
P
n=1
1
nxnx[1 ; 1 [
ln(1 + x)=
P
n=1
(1)n1
nxnx]1 ; 1 ]
1 + x= 1 + x
2+
P
n=2
(1)n11×3× ··· × (2n3)
2×4× ··· × (2n)xnx]1 ; 1 [
1
1 + x
= 1 +
P
n=1
(1)n1×3× ··· × (2n1)
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× ··· × (2n1)
2×4× ··· × (2n)
x2n+1
2n+ 1 x]1 ; 1 [
Argsh x=x+
P
n=1
(1)n1×3× ··· × (2n1)
2×4× ··· × (2n)
x2n+1
2n+ 1 x]1 ; 1 [
3
Dérivées usuelles
Fonction Dérivée Dérivabilité
xnnZnxn1R
xααRαxα1R
+
eαx αCαeαx R
axaR
+axln aR
ln |x|1
xR
logax a R
+r{1}1
xln aR
cos xsin xR
sin xcos xR
tan x1 + tan2x=1
cos2xR r nπ
2+kπ kZo
cotan x1cotan 2x=1
sin2xR r πZ
ch xsh xR
sh xch xR
th x1th 2x=1
ch 2x
R
coth x1coth 2x=1
sh 2x
R
Arcsin x1
1x2]1 ; 1 [
Arccos x1
1x2]1 ; 1 [
Arctan x1
1 + x2R
Argsh x1
x2+ 1
R
Argch x1
x21] 1 ; +[
Argth x1
1x2]1 ; 1 [
4
Primitives usuelles
I Polynômes et fractions simples
Fonction Primitive Intervalles
(xx0)nx0R
nZ r {−1}
(xx0)n+1
n+ 1
nN:xR
nZ r (N∪ {−1}) :
x];x0[,]x0;+[
(xx0)αx0R
αC r {−1}
(xx0)α+1
α+ 1 ]x0;+[
(xz0)nz0C r R
nZ r {−1}
(xz0)n+1
n+ 1 R
1
xaaRln |xa|];a[,]a;+[
1
x(a+ ib)aR, b R
1
2ln (xa)2+b2
+ i Arctan xa
b
R
II Fonctions usuelles
Fonction Primitive Intervalles
ln x x(ln x1) ] 0 ; +[
eαx αC1
αeαx R
sin xcos xR
cos xsin xR
tan xln |cos x|iπ
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
2sin 2x
4R
cos2xx
2+sin 2x
4R
tan2xtan xxiπ
2+kπ ;π
2+kπ h
cotan 2xcotan xx]kπ ; (k+ 1)π[
sh 2xsh 2x
4x
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
4iπ
2+kπ ;π
2+kπ h
1
sh xln th x
2]; 0 [ ,] 0 ; +[
1
ch x2 Arctan exR
1
sin2x= 1 + cotan 2xcotan x]kπ ; (k+ 1)π[
1
cos2x= 1 + tan2xtan xiπ
2+kπ ;π
2+kπ h
1
sh 2x= coth 2x1coth x]; 0 [ ,] 0 ; +[
1
ch 2x= 1 th 2xth xR
1
sin4xcotan xcotan 3x
3]kπ ; (k+ 1)π[
1
cos4xtan x+tan3x
3iπ
2+kπ ;π
2+kπ h
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