Exercice 9. Calculer les primitives suivantes par intégration par parties :

• 1. $\int x e^x \, dx$
• 2. $\int t \sin t \, dt$
• 3. $\int \ln x \, dx$
• 4. $\int \ln(2s + 3) \, ds$
• 5. $\int x \cos x \, dx$
• 6. $\int \ln(1 + u^2) \, du$
• 7. $\int \arctan x \, dx$
• 8. $\int \ln^2 s \, ds$
• 1. $\int \frac{\ln x}{x^2} \, dx$
• 10. $\int u \ln u \, du$
• 11. $\int e^x \sin x \, dx$

a)

$$\int x e^{x}\,dx$$ Posons $u=x,;dv=e^{x}dx$. Alors $du=dx,;v=e^{x}$.

$$\int x e^{x}\,dx = x e^{x} - \int e^{x}\,dx = e^{x}(x-1)+C.$$

b)

$$\int t\sin t\,dt$$ $u=t,;dv=\sin t,dt$.

$$\int t\sin t,dt = -t\cos t + \sin t + C.$$

c)

$$\int \ln x\,dx$$ $u=\ln x,;dv=dx$.

$$\int \ln x\,dx = x\ln x - x + C.$$

d)

$$\int \ln(2s+3)\,ds$$ $u=\ln(2s+3),;dv=ds$.

Après simplifications :

$$\int \ln(2s+3)\,ds = \left(s+\frac{3}{2}\right)\ln(2s+3) - s + C.$$

e)

$$\int x\cos x\,dx$$

$$\int x\cos x\,dx = x\sin x + \cos x + C.$$

f)

$$\int \ln(1+u^{2})\,du$$

$$\int \ln(1+u^{2})\,du = u\ln(1+u^{2}) - 2u + 2\arctan u + C.$$

g)

$$\int \arctan x\,dx$$

$$\int \arctan x\,dx = x\arctan x - \frac12\ln(1+x^{2}) + C.$$

h)

$$\int \ln^{2}s\,ds$$

$$\int \ln^{2}s\,ds = s\left(\ln^{2}s - 2\ln s + 2\right) + C.$$

i)

$$\int \frac{\ln x}{x^{2}},dx$$

$$\int \frac{\ln x}{x^{2}}\,dx = -\frac{\ln x + 1}{x} + C.$$

j)

$$\int u\ln u\,du$$

$$\int u\ln u\,du = \frac{u^{2}}{2}\ln u - \frac{u^{2}}{4} + C.$$

k)

$$\int e^{x}\sin x\,dx$$

On trouve :

$$\int e^{x}\sin x\,dx = \frac{e^{x}}{2}(\sin x - \cos x) + C.$$