Exercice 47

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

Connaissances:

  • trigonométrie hyperbolique

\( \begin{align*} I & = \int sech(x).dx \\
& = \int \frac{1}{coshx}dx \\
& = \int \frac{1 \times coshx}{coshx \times coshx}dx \\
& = \int \frac{coshx}{cosh²x}dx \\
& = \int \frac{coshx}{1+sinh²x}dx \\
& u=sinhx \Rightarrow du = coshx \\
I & = \int \frac{du}{1+u²} \\
 & = tan^{-1}u + C \\
& = tan^{-1}(sinhx) + C \end{align*}\)

 

\[\boxed{\begin{align*} I = tan^{-1}(sinhx) + C(\in \mathbb R) \end{align*} }\]