Question Number 151345 by peter frank last updated on 20/Aug/21
$$\int_{\mathrm{0}} ^{\pi} \frac{\mathrm{e}^{\mathrm{cos}\:\mathrm{x}} }{\mathrm{e}^{\mathrm{cos}\:\mathrm{x}} +\mathrm{e}^{−\mathrm{cos}\:\mathrm{x}} }\mathrm{dx} \\ $$
Answered by Olaf_Thorendsen last updated on 20/Aug/21
$$\mathrm{I}\:=\:\int_{\mathrm{0}} ^{\pi} \frac{{e}^{\mathrm{cos}{x}} }{{e}^{\mathrm{cos}{x}} +{e}^{−\mathrm{cos}{x}} }\:{dx}\:\:\:\left(\mathrm{1}\right) \\ $$$$\mathrm{Let}\:{u}\:=\:\pi−{x}\:: \\ $$$$\mathrm{I}\:=\:\int_{\mathrm{0}} ^{\pi} \frac{{e}^{−\mathrm{cos}{u}} }{{e}^{−\mathrm{cos}{u}} +{e}^{\mathrm{cos}{u}} }\:{du}\:\:\:\left(\mathrm{2}\right) \\ $$$$\frac{\left(\mathrm{1}\right)+\left(\mathrm{2}\right)}{\mathrm{2}}\::\:\mathrm{I}\:=\:\frac{\pi}{\mathrm{2}} \\ $$