\(\begin{align*} \int x^3e^{x^2}.dx \end{align*}\) |
Connaissances:
- Intégration par parties
\(\begin{align*} I & = \int x^3e^{x²}dx \\
& = \int x² \times x.e^{x²}dx \\ \end{align*}\)
| \(D\) | \(I\) | ||
| \(+\) | \(x²\) | \(x.e^{x²}\) | |
| \(\searrow\) | |||
| \(-\) | \(2x\) | \(\rightarrow\) | \(\frac{1}{2}e^{x²}\) |
\(\begin{align*} I & = \frac{1}{2}x²e^{x²} - \int 2x.\frac{1}{2}e^{x²}.dx \\
& = \frac{1}{2}x²e^{x²}-\frac{1}{2} \int 2x.e^{x²}.dx \\
& = \frac{1}{2}x²e^{x²} -\frac{1}{2}e^{x²} +C \\
& = \frac{1}{2}e^{x²}(x²-1) + C
\end{align*}\)
\[\boxed {\begin{align*} I =\frac{1}{2}e^{x²}(x²-1) + C(\in \mathbb R) \end{align*}}\]