Question

Evaluate $\underset{-\frac{\pi }{2}}{\overset{\frac{\pi }{2}}{\int }}\frac{\cos x}{1+e^x}dx$

Answer 1

use property $\underset{a}{\overset{b}{\int }}f\left(x\right)dx=\underset{a}{\overset{b}{\int }}f(a+b-x)dx$ using this $I=\underset{-\frac{\pi }{2}}{\overset{\frac{\pi }{2}}{\int }}\frac{\cos x}{1+e^x}$

$I=\underset{-\frac{\pi }{2}}{\overset{\frac{\pi }{2}}{\int }}\frac{\cos (-x)}{1+e^{-}x}dx$

add above two equation $2I=\underset{-\frac{\pi }{2}}{\overset{\frac{\pi }{2}}{\int }}\frac{\cos x(1+e^x)}{1+e^x}dx$

$I=\underset{-\frac{\pi }{2}}{\overset{\frac{\pi }{2}}{\int }}\frac{\cos x}{2}dx={{\left[\frac{\sin x}{2}\right]}^{\frac{\pi }{2}}}_{\frac{-\pi }{2}}=1$