Question Number 62414 by mathmax by abdo last updated on 20/Jun/19
$${find}\:\int\:\:\:\:\:\frac{{e}^{{x}} }{\sqrt{{e}^{\mathrm{2}{x}} −\mathrm{1}}}{dx} \\ $$
Answered by $@ty@m last updated on 20/Jun/19
$${Put}\:{e}^{{x}} ={t} \\ $$$$\int\frac{{dt}}{\sqrt{{t}^{\mathrm{2}} −\mathrm{1}}}=\mathrm{ln}\:\mid{t}+\sqrt{{t}^{\mathrm{2}} −\mathrm{1}}\mid \\ $$$$=\mathrm{ln}\:\mid{e}^{{x}} +\sqrt{{e}^{\mathrm{2}{x}} −\mathrm{1}}\mid+\boldsymbol{\mathrm{C}} \\ $$