\(\begin{align*} \int e^{\sqrt x}dx \end{align*}\) |
Connaissances:
- Changement de variable
Posons \(u = \sqrt x \Rightarrow x= u² \Rightarrow dx = 2u.du\)
\(\begin{align*} I & = \int e^{\sqrt x}dx \\
& = \int e^u.2u.du \\
& = \int 2ue^udu \end{align*}\)
Procédons à une IPP:
| \(D\) | \(I\) | ||
| \(+\) | \(2u\) | \(e^u\) | |
| \(\searrow\) | |||
| \(-\) | \(2\) | \(\rightarrow\) | \(e^u\) |
\(\begin{align*} I & = \int 2ue^udu \\
& = 2u.e^u - \int 2e^u.du \\
& = 2ue^u - 2e^u + C \\
& = 2e^u(u-1) + C \\
& = 2e^{\sqrt x}(\sqrt x-1) + C \end{align*}\)
\[\boxed {\begin{align*} I & = 2e^{\sqrt x}(\sqrt x-1) + C(\in \mathbb R) \end{align*}}\]