Z dx x ln x−√ 10 x+√ 10 6

MPSI 2 Semaine 8

Exercice 1 Intégration directe 1.

Z

p2pxdx=2x 3

p2px

2.

Z

x(x+a)(x+b)dx= x⁴

4 +(a+b)x³

3 +abx² 2

3.

Z

a²³ −x²³³

dx=a²x−9

5a^4/3x^5/3+9

7a^2/3x^7/3−x³ 3

4.

Z

tan²(x)dx= tan(x)−x

5.

Z dx

x²−10 = 1 2√

10ln

x−√ 10 x+√

10

6.

Z

3^xe^xdx= 3^xe^x 1 + ln(3).

Exercice 2 Changement de variable

1.

Z √

x+ ln(x)

x dx= 2√

x+ln²(x) 2

2.

Z dx

(a+b)−(a−b)x² = 1 2√

a²−b²ln

√a+b+x√ a−b

√a+b−x√ a−b

pour0< b < a 3.

Z x²+ 5x+ 7

x+ 3 dx= x²

2 + 2x+ ln|x+ 3|

4.

Z 3x+ 1

√5x²+ 1dx=3 5

p5x²+ 1 + 1

√5ln(x√ 5 +p

5x²+ 1)

5.

Z xdx

√a⁴−x⁴ = 1 2arcsin

x² a²

6.

Z r

arcsin(x) 1−x² dx= 2

3 q

arcsin³(x)

7.

Z dx q

(1 +x²) ln(x+√ 1 +x²)

= 2 q

ln(x+p 1 +x²)

8.

Z

p1 + 3 cos²(x) sin(2x)dx=−2 9

p(1 + 3 cos²(x))³

9.

Z x³dx x⁸+ 5 = 1

4√

5arctan x⁴

√5

10.

Z e^arctan(x)+xln(1 +x²) + 1

1 +x² dx=e^arctan(x)+ln²(1 +x²)

4 + arctan(x)

11.

Z sin(x)−cos(x)

sin(x) + cos(x)dx=−ln|sin(x) + cos(x)|

12.

Z arcsin(x) +x

√1−x² dx= arcsin²(x)

2 −p

1−x²

Feuille d’exercices 7 Page 1/3

MPSI 2 Semaine 8

Exercice 3 Intégration par parties 1.

Z ln(x)

√x dx= 2√

xln(x)−4√ x

2.

Z

(x²+ 5x+ 6) cos(2x)dx=2x²+ 10x+ 11

4 sin(2x) +2x+ 5

4 cos(2x)

3.

Z

x²ln(x)dx=x³

3 ln(x)−x³ 9

4.

Z

sin(ln(x))dx=x

2(sin(ln(x))−cos(ln(x))) 5.

Z xln

1−x 1 +x

dx=x²−1 2 ln

1−x 1 +x

−x 6.

Z ln²(x)

x² dx=−ln²(x) + 2 ln(x) + 2 x

7.

Z ln(ln(x))

x dx= ln(x) (ln(ln(x))−1) 8.

Z

x²arctan(3x)dx= x³

3 arctan(3x)−x² 18+ 1

162ln(9x²+ 1).

Exercice 4 Intégration des fractions rationnelles 1.

Z x+ 1

(x²+ 4x+ 5)²dx=− x+ 3

2(x²+ 4x+ 5)−1

2arctan(x+ 2)

2.

Z dx

x³−2x²+x = ln|x| −ln|x−1| − 1 x−1 3.

Z dx

(x²−4x+ 3)(x²+ 4x+ 5) = 1

52ln|x−3| − 1

20ln|x−1|+ 1

65ln(x²+ 4x+ 5) + 7

130arctan(x+ 2)

4.

Z x³−1

4x³−xdx= x 4 + 1

16ln

x¹⁶ (2x−7)⁷(2x+ 1)⁹

5.

Z 2x−3

(x²−3x+ 2)³dx=− 1 2(x²−3x+ 2)² 6.

Z x⁷+x³

x¹²−2x⁴+ 1dx=1

2ln|x⁴−1| −1

4ln|x⁸+x⁴−1| − 1 2√

5ln

2x⁴+ 1−√ 5 2x⁴+ 1 +√

5

7.

Z x²dx

(x−1)¹⁰ =− 1

9(x−1)⁹− 1

4(x−1)⁸− 1 7(x−1)⁷ 8.

Z dx

x⁸+x⁶− 1 5x⁵ + 1

3x³ −1

x−arctan(x)

Feuille d’exercices 7 Page 2/3

MPSI 2 Semaine 8

Exercice 5 Intégration des fonctions trigonométriques

1.

Z

cos³(x)dx= sin(x)−sin³(x) 3

2.

Z cos⁵(x)

sin³(x)dx=sin²(x)

2 − 1

2 sin²(x)−ln(sin²(x)) 3.

Z dx

ptan(x) = 1 2√

2ln tan²(x)−√

2 tan(x) + 1 tan²(x) +√

2 tan(x) + 1

!

− 1

√2arctan

√2 tan(x) tan²(x)−1

!

4.

Z dx

3 + 5 cos(x)= 1 4ln

tan ^x₂

−2 tan ^x₂

+ 2

5.

Z 1 + tan(x)

1−tan(x)dx=−ln|sin(x) + cos(x)|

6.

Z dx

cos(x) + 2 sin(x) + 3= arctan

1 + tanx 2

7.

Z 1−sin(x) + cos(x)

1 + sin(x)−cos(x)dx=−x+ 2 ln

tan ^x₂ 1 + tan ^x₂

8.

Z dx

sinh(x) cosh²(x) = ln

tanhx 2

+ 1

cosh(x)

9. Calculer pour n = 3 et n = 4 les intégrales In =

Z dx

cosⁿ(x). On a I4 = tan(x) + ^tan₃^3(x) et I3= sin(x)

2 cos⁽x)+1 2ln

tan(x) + 1 sin(x)

. 10.

Z

tan²x 2 +π

4

dx= 2 tanx 2 +π

4 −x.

11.

Z

xarctan(2x+ 3)dx= (x²−2) arctan(2x+ 3)

2 +3 ln(2x²+ 6x+ 5)

8 −x

4.

Feuille d’exercices 7 Page 3/3