Exercice 42

\( \begin{align*}  \int sinh²(x).dx \end{align*}\)

Connaissances:

  • trigonométrie hyperbolique
    • \(sinh²x = -\frac{1}{2}+\frac{coshx}{2}\)

\( \begin{align*} sinh²x & = \big( \frac{e^x-e^{-x}}{2} \big)^2 \\
& = \frac{e^{2x}-2 + e^{2x}}{4}  \\
& = -\frac{1}{2}+ \frac{e^{2x}+ e^{2x}}{4} \\
& = -\frac{1}{2} + \frac{cosh(2x)}{2} \\ \\
\Rightarrow I & = \int -\frac{1}{2} + \frac{cosh(2x)}{2}.dx \\
& = -\frac{x}{2} + \frac{sinh(2x)}{4} + C \end{align*}\)
 \[ \boxed{\begin{align*}  I =-\frac{x}{2} + \frac{sinh(2x)}{4} + C(\in \mathbb R) \end{align*}}\]