\(\begin{align*} \int tanx.ln(cosx).dx \end{align*}\) |
Connaissances:
- Changement de variable
- trigonométrie
\(\begin{align*} I & = \int tanx.ln(cosx).dx \\
& \text{Posons : }u= ln(cosx) \Rightarrow du =-\frac{sinx}{cosx} .dx = -tanx.dx \\
I & = \int tanx.u.\frac{-du}{tanx} \\
& = - \int u.du = -u²/2+C \\
& = -\big(ln(cosx) \big)²/2 + C \\
\end{align*}\)
\[\boxed {\begin{align*} I =\frac{1}{2}\big(ln(cosx) \big)² + C(\in \mathbb R) \end{align*}}\]