\(\begin{align*} \int (ln(x))²dx \end{align*}\) |
Connaissances:
- Changement de variable
- suivi d'une IPP multiple
- Intégration par parties directement
Changement de variable:
Posonx \(u=lnx \Rightarrow \begin{cases}x=e^u \\ du=1/x.dx \Rightarrow dx = x.du=e^udu \end{cases}\)
\(\begin{align*} I & = \int (ln(x))²dx = \int u².e^u.du\end{align*}\)
| \(D\) | \(I\) | ||
| \(+\) | \(u²\) | \(e^u\) | |
| \(\searrow\) | |||
| \(-\) | \(2u\) | \(e^u\) | |
| \(\searrow\) | |||
| \(+\) | \(2\) | \(e^u\) | |
| \(\searrow\) | |||
| \(-\) | \(0\) | \(\rightarrow\) | \(e^u\) |
\(\begin{align*} I & = \int (ln(x))²dx = \int u².e^u.du \\
& = e^u(u²-2u+2)+C \\
& = e^{lnx}((lnx)²-2lnx+2)+C \\
& = x(ln²x-2lnx+2) +C \end{align*}\)
Intégration par parties directe:
| \(D\) | \(I\) | ||
| \(+\) | \(ln²x\) | \(1\) | |
| \(\searrow\) | |||
| \(-\) | \(2\frac{lnx}{x}\) | \(\rightarrow\) | \(x\) |
\(\begin{align*} I & = \int (ln(x))²dx \\
& = x.ln²x-\int 2\frac{lnx}{x} \times x.dx \\
& = x.ln²x-2\int lnx .dx\\
\end{align*}\)
Faisons une 2ème intégration par partie:
| \(D\) | \(I\) | ||
| \(+\) | \(lnx\) | \(1\) | |
| \(\searrow\) | |||
| \(-\) | \(\frac{1}{x}\) | \(\rightarrow\) | \(x\) |
\(\begin{align*} I & = \int (ln(x))²dx \\
& = x.ln²x-2(xlnx-\int 1 \times dx ) \\
& = x.ln²x-2(xlnx-x) +C \\
& = x. ln²x-2xlnx+2x) +C \\
& = x(ln²x-2lnx+2) +C
\end{align*}\)
\[\boxed {\begin{align*} I &= x(ln²x-2lnx+2) +C(\in \mathbb R)\end{align*}}\]