Exercice 22

\(\begin{align*} \int \frac{1}{x² \sqrt{x²+1}}dx \end{align*}\)

\(\begin{align*}
I & = \int \frac{1}{x² \sqrt{x²+1}}dx \\
& = \int \frac{1}{x² \sqrt{x²(1+x^{-2})}}dx \\
& = \int \frac{1}{x^3 \sqrt{(1+x^{-2})}}dx \\
& =\int \frac{x^{-3}}{ \sqrt{(1+x^{-2})}}dx \\
\end{align*}\)

On remarque que du \(x^{-3}\) ressemble a une dérivée de \(x^{-2} \). Procédons à un changement de variable \(u=1+x^{-2}\)

\(\begin{align*} \begin{cases}  u=1+x^{-2} \\ du=-2.x^{-3}dx \end{cases} \Rightarrow dx=\frac{du}{-2.x^{-3}}\end{align*}\)

\(\begin{align*}
I & =\int \frac{x^{-3}}{ \sqrt{(1+x^{-2})}}dx \\
& = \int \frac{x^{-3}}{ \sqrt{u}} \times \frac{du}{-2.x^{-3}} \\
& = -\int \frac{1}{2 \sqrt{u}}.du \\
& = - \sqrt{u}+C
\end{align*}\)

\[\boxed{I=- \sqrt{1+x^{-2}}+C (\in \mathbb R)}\]