Intégration par parties :$${{\int f(x)g'(x)dx}}={{fg-\int f'(x)g(x)dx}}$$
Démonstration : voir Dérivée - Dérivation
$$\begin{align}(fg)'&=f'g+fg'\\ \implies fg'&=(fg)'-f'g\\ \implies \int fg'&=\int(fg)'-\int f'g\\ &=fg-\int f'g\end{align}$$
Exemples :
1. \(\int xe^xdx=\int f(x)g'(x)dx\)
\(=xe^x-\int1.e^xdx\)
\(=xe^x-e^x+k=e^x(x-1)+\underset{k\in\Bbb R}{k}\)
2. \(\int\ln x\,dx=\int1.\ln x\,dx=x\ln x-\int\frac1x.1dx\)
\(=x\ln x-x+k\)
3. \(\int\arcsin xdx=\int1.\arcsin x\,dx=x\arcsin x-\int {x\over\sqrt{1-x^2} }dx\)
\(=x\arcsin x+\sqrt{1-x^2}+k\)