Exercice IG3 – Existence et calcul d'intégrales

Existence et calcul des intégrales suivantes :

1. \(\displaystyle \int_{0}^{+\infty} \frac{dx}{(x+1)(x+2)}\)
2. \(\displaystyle \int_{0}^{+\infty} \frac{x^4}{x^{10}+1} \, dx\)
3. \(\displaystyle \int_{0}^{1} \frac{x^2}{\sqrt{1-x^2}} \, dx\)
4. \(\displaystyle \int_{0}^{+\infty} \frac{e^{-\sqrt{x}}}{\sqrt{x}} \, dx\)
5. \(\displaystyle \int_{0}^{1} \frac{\ln x}{\sqrt{1-x}} \, dx\)
6. \(\displaystyle \int_{0}^{+\infty} \frac{dx}{x^2+x+1}\)
7. \(\displaystyle \int_{0}^{+\infty} \frac{dx}{(x^2+1)^2}\)
8. \(\displaystyle \int_{0}^{+\infty} \frac{dx}{5\ln x + 3\ln x + 4}\)
9. \(\displaystyle \int_{0}^{+\infty} \frac{x\ln x}{(1+x^2)^2} \, dx\)
10. \(\displaystyle \int_{0}^{+\infty} \frac{dx}{(1+x^2)^{3/2}}\qquad \) (\(t = \frac{1}{x}\))
11. \(\displaystyle \int_{0}^{\pi/2} \frac{dx}{1+(\tan x)^\alpha}\) (\(\alpha > 0\))
12. \(\displaystyle \int_{0}^{+\infty} \frac{dx}{(1+x^2)(1+x^n)}\), \(n\in\mathbb{N}\qquad \) (\(t = \frac{1}{x}\))
13. \(\displaystyle \int_{1}^{+\infty} \frac{dx}{x\sqrt{x^2-1}}\quad \) ( \(x = \operatorname{ch} t\))
14. \(\displaystyle \int_{0}^{+\infty} \frac{\ln x}{1+x+x^2} \, dx\qquad \) (\(t = \frac{1}{x}\))
15. \(\displaystyle \int_{0}^{+\infty} \ln\left(1 + \frac{3}{1+x^2}\right) \, dx\qquad\) ( intégration par parties)