Exercice 43

\( \begin{align*}  \int sinh^3x.dxdx \end{align*}\)

Connaissances:

  • trigonométrie hyperbolique

\( \begin{align*}  I & = \int sinh^3x.dx \\
& = \int sinh²x.sinhx.dx \\
& = \int (cosh²x-1)sinhx.dx \\
& = \int cosh²x.sinhx.dx - \int sinhx.dx \\
& = \frac{1}{3}cosh^3x-coshx +C \end{align*}\)

 

\[\boxed{ \begin{align*} I = \frac{1}{3}cosh^3x-coshx +C(\in \mathbb R) \end{align*}}\]