T.D. - Alain Camanes

2015/2016
Partie I : Primitives classiques
xα, α 6=−1xα+1
α+1 Rα∈N,R∗
+R∗ −α∈Z−\{−1},R? +α∈R\Z
1
xln |x|R∗
−R∗ +
ln |x|xln |x| − xR?
+R? −
ax, a > 0, a 6= 1 ax
ln aR
sin x−cos xR
cos xsin xR
tan x−ln |cos x|]−π
2+kπ, π 2+kπ[, k ∈Z
sinh xcosh xR
cosh xsinh xR
tanh xln cosh xR
tan2xtan x−x]−π
2+kπ, π 2+kπ[, k ∈Z
1
cosh2xtanh xR
1
sinh2x−coth xR∗
1
cos2xtan x]−π
2+kπ, π 2+kπ[, k ∈Z
1
sin2x−cotan x]kπ, π +kπ[, k ∈Z
1
sin xln |tan x 2|]kπ, π +kπ[, k ∈Z
1
cos xln |tan(x 2+π 4)|]−π 2+kπ, π 2+kπ[, k ∈Z
tanh2x x −tanh xR
1
x2+1 arctan xR
1
√1−x2arcsin x]−1,1[
Exercice 1. (Fonctions usuelles) a
1. f1(x) = x3+5x2−4
x2
2. f2(x) = cotan(x)
3. f3(x) = coth(x)
4. f4(x) = 1
a2+x2
5. f5(x) = 1
a2−x2
6. f6(x) = 1
√a2−x2
7. f7(x) = 1
9+x2
8. f8(x) = 1
√25−16x2
Partie II : Techniques élémentaires
Exercice 2. (u0·f0(u))
1. f1(x) = 8x2
(x3+2)3
2. f2(x) = x√1−2x2
3. f3(x) = (ex+ 1)3ex
4. f4(x) = sin(x)
3+sin2(x)
5. f5(x) = x2
√1−x6
6. f6(x) = x
x4+3
7. f7(x) = 1
x2+10x+30

Exercice 3. (Changements de variable)
1. f1(x) = 1
ex+1 2. f2(x) = √ex−13. f3(x) = 1
x3
2+1
Exercice 4. (Intégrations par parties)
1. f1(x) = arcsin(x)
2. f2(x) = x2ln(x)
3. f3(x) = x√1 + x
4. f4(x) = xtan2(x)
5. f5(x) = ln(x2+ 2)
6. f6(x) = √1 + xln(x)
7. f7(x) = xarctan2(x)
8. f8(x) = earccos(x)
Partie III : Fractions rationnelles
Primitives de la forme Zdx
(x−a)n.
∗n= 1 Zdx
x−a= ln |x−a|
∗n>2Zdx
(x−a)n=1 (1 −n)(x−a)n−1.
Primitives de la forme Zax +b
x2+px +qdx =a 2Z2x+p x2+px +qdx +b−ap 2Zdx
x+p
22+4q−p2 4
.
∗4q−p2>0
∗4q−p2<0
Exercice 5. (Fractions rationnelles)
1. f1(x) = x+2
x+1
2. f2(x) = x2+2x
(x+1)2
3. f3(x) = 1
x2−9
4. f4(x) = 1
x3+1
5. f5(x) = x2−3x−1
x3+x2−2x
6. f6(x) = 2x3
(x2+1)2
7. f7(x) = 2x−7
x2+9
8. f8(x) = x+1
x2−4x+8
Partie IV : Polynômes, Exponentielles et Trigonométrie
ZP(x)eαx cos(ωx)dx
ZP(x)eax dx a ∈C
∗a= 0
∗a6= 0 Q(x)eax Q
P aQ(x) + Q0(x) = P(x)
Exercice 6. n∈N
1. f1(x) = xcos x
2. f2(x) = x2e−3x
3. f3(x) = sinh xcos x4. f4(x) = xnex
Partie V : Fonctions rationnelles en cosinus et sinus
Fonctions polynomiales des fonctions trigonométriques.
Exercice 7. (Fonctions trigonométriques) a, b ab 6= 0 a26=b2
1. f1(x) = sin(x) sin(3x)2. f2(x) = sin3(x)3. f3(x) = cos(ax) cos(bx)

Règles de Bioche : ZF(cos x, sin x)dx.
(i)ω(x) = F(cos x, sin x)dx
∗ω(−x) = ω(x)t= cos x
∗ω(π−x) = ω(x)t= sin x
∗ω(π+x) = ω(x)t= tan x
(ii)t= tan x
2
Exercice 8. (Règles de Bioche)
1. f1(x) = 1
cos x+2
2. f2(x) = sin3x
2+cos x
3. f3(x) = tan x
1+sin2x
4. f4(x) = 4 sin x
(1+cos x)(3+cos 2x)
Partie VI : Fractions rationnelles et Exponentielle
t=excosh sinh
Exercice 9.
1. f1(x) = 1
1+sinh x+2 cosh x2. f2(x) = 1 sinh x
Partie VII : Fractions rationnelles et Racines
Primitives de la forme ZF x, n
rax +b
cx +d!dx.
t=n
qax+b
cx+d
Exercice 10.
1. f1(x) = q1+x
1−x2. f2(x) = √1+x−4
√x+1
√1+x+4 √x+1
Partie VIII : Primitives de la forme ZFx, pax2+bx +cdx
ax+b
2a2+c−b2 4a
∗a=α2c−b2
4a=β2x=β αsinh t−b a
1 + sinh2= cosh2
∗a=α2c−b2
4a=−β2x=β αcosh t−b a
sinh2= cosh2−1
∗a=−α2c−b2
4a=β2x=β αsin t−b a
sin2= 1 −cos2
argsh argch
Exercice 11.
1. f1(x) = √x2−1
2. f2(x) = x2√1−x2
3. f3(x) = √2x2−3x+ 5
4. f4(x) = x
√2x−x2

Partie IX : Divers
Exercice 12. (-)
1. I1=Z1
0p1−x2dx
2. I2=Z2
1
ln t
√tdt
3. I3=Ze
1
du
u+u(ln u)2
4. I4=Z1
0
dt
et+ 1
5. I5=Z2
1
ln(1 + x)−ln x
x2dx
6. I6=Z1
0
ln(1 + t2)dt
7. I7=Ze
1
xnln x dx
8. I8=Z1/2
0
arcsin t dt
9. I9=Zeπ
1
sin(ln t)dt