Exercices corrigés
| Exercice 1. Calculer l’intégrale {I=\displaystyle\int_{0}^{1}\dfrac{x\ln x}{(x^2+1)^2}\text{d}x}. |
| Exercice 2. Calculer {I_{\lambda,n}=\displaystyle\int_{0}^{1}\!\!x^\lambda\ln^nx\,\text{d}x} {(\lambda>0,n\in\mathbb{N})} |
| Exercice 3. Calculer {I_n=\displaystyle\int_{0}^{\pi/2}\sin^{2n}x\,\text{d}x} et {J_n=\displaystyle\int_{0}^{\pi} x\sin^{2n}x\,\text{d}x}. |
| Exercice 4. Soit {I_n^m=\displaystyle\int_{0}^{1} x^m(1-x)^n\,\text{d}x}, {(m,n)\in\mathbb{N^2})}
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| Exercice 5. Calculer {I(a)=\displaystyle\int_{1/a}^{a}\dfrac{\ln x}{1+x^2}\,\text{d}x} {(a>0)} |