Correction Exercice — Primitives (TSE)

Énoncé

Déterminez les primitives des fonctions suivantes :

1. \( f(x) = -x^3 + 6x^2 + 10x - 4 \)

Technique : Intégration directe.

\[ F(x) = \int (-x^3 + 6x^2 + 10x - 4) \, dx = -\frac{x^4}{4} + 2x^3 + 5x^2 - 4x + C \]

2. \( f(x) = (x^2 - 6x + 7)^4 (2x - 6) \)

Technique : Substitution \( u = x^2 - 6x + 7 \), \( du = (2x - 6) \, dx \).

\[ F(x) = \frac{(x^2 - 6x + 7)^5}{5} + C \]

3. \( f(x) = (2x - 1)^3 \)

Technique : Substitution \( u = 2x - 1 \), \( du = 2 \, dx \).

\[ F(x) = \frac{(2x - 1)^4}{8} + C \]

4. \( f(x) = (x - 3)(x^2 - 6x + 7)^4 \)

Technique : Substitution \( u = x^2 - 6x + 7 \), \( du = (2x - 6) \, dx \).

\[ F(x) = \frac{(x^2 - 6x + 7)^5}{10} + C \]

5. \( f(x) = (-x + 3)(x^2 - 6x + 7)^4 \)

Technique : Substitution \( u = x^2 - 6x + 7 \), \( du = (2x - 6) \, dx \).

\[ F(x) = -\frac{(x^2 - 6x + 7)^5}{10} + C \]

6. \( f(x) = 5x(1 - x^2)^6 \)

Technique : Substitution \( u = 1 - x^2 \), \( du = -2x \, dx \).

\[ F(x) = -\frac{5(1 - x^2)^7}{14} + C \]

7. \( f(x) = (-6x^2 + 2)(x^3 - x + 7)^4 \)

Technique : Substitution \( u = x^3 - x + 7 \), \( du = (3x^2 - 1) \, dx \).

\[ F(x) = -\frac{2(x^3 - x + 7)^5}{5} + C \]

8. \( f(x) = \frac{6x + 3}{(x^2 + x + 1)^4} \)

Technique : Substitution \( u = x^2 + x + 1 \), \( du = (2x + 1) \, dx \).

\[ F(x) = -\frac{1}{(x^2 + x + 1)^3} + C \]

9. \( f(x) = \frac{x(2x^2 + 1)}{\sqrt{x^4 + x^2 + 9}} \)

Technique : Substitution \( u = x^4 + x^2 + 9 \), \( du = (4x^3 + 2x) \, dx \).

\[ F(x) = \sqrt{x^4 + x^2 + 9} + C \]

10. \( f(x) = \frac{-x}{\sqrt{x^2 - 9}} \)

Technique : Substitution \( u = x^2 - 9 \), \( du = 2x \, dx \).

\[ F(x) = -\sqrt{x^2 - 9} + C \]

11. \( f(x) = \sqrt{-5x + 2} \)

Technique : Substitution \( u = -5x + 2 \), \( du = -5 \, dx \).

\[ F(x) = -\frac{2(-5x + 2)^{3/2}}{15} + C \]

12. \( f(x) = \frac{4 + \sqrt{x}}{\sqrt{x}} \)

Technique : Simplification algébrique.

\[ F(x) = 8\sqrt{x} + x + C \]

13. \( f(x) = \frac{2x^5 - 3x^3 + 2}{x^2} \)

Technique : Simplification algébrique.

\[ F(x) = \frac{x^4}{2} - \frac{3x^2}{2} - \frac{2}{x} + C \]

14. \( f(x) = \frac{2}{x^3 + 3x^2 + 3x + 1} \)

Technique : Substitution \( u = x + 1 \), \( du = dx \).

\[ F(x) = -\frac{1}{(x + 1)^2} + C \]

15. \( f(x) = -2x + 1 - \frac{3}{1 - 3x + 3x^2 - x^3} \)

Technique : Décomposition en éléments simples.

\[ F(x) = -x^2 + x + \frac{3}{2(1 - x)^2} + C \]

16. \( f(x) = \frac{x^3 + x^2 + 4x + 4}{x^3(x + 2)^2} \)

Technique : Décomposition en éléments simples.

\[ F(x) = \ln|x| + \frac{1}{x} + \frac{1}{2x^2} + \ln|x + 2| + \frac{1}{x + 2} + C \]

17. \( f(x) = 2x + 3 + \frac{7x^2}{(5 - x^3)^3} \)

Technique : Intégration directe et substitution \( u = 5 - x^3 \), \( du = -3x^2 \, dx \).

\[ F(x) = x^2 + 3x + \frac{7}{6(5 - x^3)^2} + C \]

18. \( f(x) = \frac{2x^3 + 5x^2 + 4x + 4}{(x + 1)^2} \)

Technique : Division polynomiale et décomposition en éléments simples.

\[ F(x) = x^2 + x + 2\ln|x + 1| - \frac{1}{x + 1} + C \]

Étude de fonction — Série TSE
Intégrale - serie TSE exo20

ð Tableau de Rappel des Formules de Primitives

Ce tableau regroupe les formules essentielles utilisées dans les exercices.

Type de fonction	Primitive
\( x^n \) (avec \( n \neq -1 \))	\[ \int x^n dx = \frac{x^{n+1}}{n+1} + C \]
\( \frac{1}{x} \)	\[ \int \frac{1}{x} dx = \ln\|x\| + C \]
\( e^x \)	\[ \int e^x dx = e^x + C \]
\( a^x \) (avec \( a>0 \))	\[ \int a^x dx = \frac{a^x}{\ln(a)} + C \]
\( u'(x)\,[u(x)]^n \)	\[ \int u'(x)u^n dx = \frac{u^{n+1}}{n+1} + C \]
\( \frac{u'(x)}{u(x)} \)	\[ \int \frac{u'}{u} dx = \ln\|u\| + C \]
\( \sqrt{x} \)	\[ \int \sqrt{x} dx = \frac{2}{3}x^{3/2} + C \]
\( \frac{1}{\sqrt{x}} \)	\[ \int \frac{1}{\sqrt{x}} dx = 2\sqrt{x} + C \]

Correction Exercice — Primitives (TSE)

Énoncé

1. \( f(x) = -x^3 + 6x^2 + 10x - 4 \)

2. \( f(x) = (x^2 - 6x + 7)^4 (2x - 6) \)

3. \( f(x) = (2x - 1)^3 \)

4. \( f(x) = (x - 3)(x^2 - 6x + 7)^4 \)

5. \( f(x) = (-x + 3)(x^2 - 6x + 7)^4 \)

6. \( f(x) = 5x(1 - x^2)^6 \)

7. \( f(x) = (-6x^2 + 2)(x^3 - x + 7)^4 \)

8. \( f(x) = \frac{6x + 3}{(x^2 + x + 1)^4} \)

9. \( f(x) = \frac{x(2x^2 + 1)}{\sqrt{x^4 + x^2 + 9}} \)

10. \( f(x) = \frac{-x}{\sqrt{x^2 - 9}} \)

11. \( f(x) = \sqrt{-5x + 2} \)

12. \( f(x) = \frac{4 + \sqrt{x}}{\sqrt{x}} \)

13. \( f(x) = \frac{2x^5 - 3x^3 + 2}{x^2} \)

14. \( f(x) = \frac{2}{x^3 + 3x^2 + 3x + 1} \)

15. \( f(x) = -2x + 1 - \frac{3}{1 - 3x + 3x^2 - x^3} \)

16. \( f(x) = \frac{x^3 + x^2 + 4x + 4}{x^3(x + 2)^2} \)

17. \( f(x) = 2x + 3 + \frac{7x^2}{(5 - x^3)^3} \)

18. \( f(x) = \frac{2x^3 + 5x^2 + 4x + 4}{(x + 1)^2} \)

ð Tableau de Rappel des Formules de Primitives

ð Tableau de Rappel des Formules de Primitives