Question

${\int }_0^{\pi }\frac{e^{cosx}}{e^{cosx}+e^{-cosx}}dx$

Answer 1

$I={\int }_0^{\pi }\frac{e^{\cos x}}{e^{\cos x}+e^{-\cos x}}.............\left(i\right)I={\int }_0^{\pi }\frac{e^{\cos (\pi -x)}}{e^{\cos (\pi -x)}+e^{-\cos (\pi -x)}}I={\int }_0^{\pi }\frac{e^{-\cos x}}{e^{-\cos x}+e^{\cos x}}...........\left(ii\right)$

Adding $\left(i\right)$ and $\left(ii\right)$

$2I={\int }_0^{\pi }1.dx2I=\pi -0I=\frac{\pi }{2}$