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Question Number 152420 by tabata last updated on 28/Aug/21

$$\int_{\mathrm{0}} ^{\:\frac{\pi}{\mathrm{2}}} \:\frac{{e}^{{sinx}} }{{e}^{{cosx}} +{e}^{{sinx}} }{dx}\:\:\:? \\$$

Answered by mindispower last updated on 28/Aug/21

$${f}\left({a}+{b}−{x}\right)+{f}\left({x}\right)={c} \\$$ $$\int_{{a}} ^{{b}} {f}\left({x}\right){dx}=\frac{\mathrm{1}}{\mathrm{2}}{c}\left({b}−{a}\right) \\$$

Answered by Lordose last updated on 28/Aug/21

$$\Omega\:=\:\int_{\mathrm{0}} ^{\frac{\boldsymbol{\pi}}{\mathrm{2}}} \frac{\mathrm{e}^{\mathrm{sin}\left(\mathrm{x}\right)} }{\mathrm{e}^{\mathrm{cos}\left(\mathrm{x}\right)} +\mathrm{e}^{\mathrm{sin}\left(\mathrm{x}\right)} }\mathrm{dx}\:\:\:\:\left(\mathrm{1}\right) \\$$ $$\Omega\:\overset{\boldsymbol{\mathrm{king}}'\boldsymbol{\mathrm{s}}\:\boldsymbol{\mathrm{rule}}} {=}\int_{\mathrm{0}} ^{\frac{\boldsymbol{\pi}}{\mathrm{2}}} \frac{\mathrm{e}^{\mathrm{cos}\left(\mathrm{x}\right)} }{\mathrm{e}^{\mathrm{cos}\left(\mathrm{x}\right)} +\mathrm{e}^{\mathrm{sin}\left(\mathrm{x}\right)} }\mathrm{dx}\:\:\:\left(\mathrm{2}\right) \\$$ $$\frac{\left(\mathrm{1}\right)\:+\:\left(\mathrm{2}\right)}{\mathrm{2}}\::: \\$$ $$\Omega\:=\:\frac{\mathrm{1}}{\mathrm{2}}\int_{\mathrm{0}} ^{\frac{\boldsymbol{\pi}}{\mathrm{2}}} \mathrm{1dx}\:=\:\frac{\boldsymbol{\pi}}{\mathrm{4}} \\$$