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Question Number 62414 by mathmax by abdo last updated on 20/Jun/19

find ∫     (e^x /(√(e^(2x) −1)))dx

$${find}\:\int\:\:\:\:\:\frac{{e}^{{x}} }{\sqrt{{e}^{\mathrm{2}{x}} −\mathrm{1}}}{dx} \\ $$

Answered by $@ty@m last updated on 20/Jun/19

Put e^x =t  ∫(dt/(√(t^2 −1)))=ln ∣t+(√(t^2 −1))∣  =ln ∣e^x +(√(e^(2x) −1))∣+C

$${Put}\:{e}^{{x}} ={t} \\ $$$$\int\frac{{dt}}{\sqrt{{t}^{\mathrm{2}} −\mathrm{1}}}=\mathrm{ln}\:\mid{t}+\sqrt{{t}^{\mathrm{2}} −\mathrm{1}}\mid \\ $$$$=\mathrm{ln}\:\mid{e}^{{x}} +\sqrt{{e}^{\mathrm{2}{x}} −\mathrm{1}}\mid+\boldsymbol{\mathrm{C}} \\ $$

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