Question

Evaluate the integral

${\int }_{-3\pi /4}^{5\pi /4}\left(\frac{sinx+cosx}{e^{x-\pi /4}+1}\right)dx$

• A. $0$
• B. $1$
• C. $2$
• D. None of these

Answer 1

The correct option is A $0$

Let $I={\int }_{-3\pi /4}^{5\pi /4}\left(\frac{sinx+cosx}{e^{x-\pi /4}+1}\right)dx$
$={\int }_{-3\pi /4}^{5\pi /4}\frac{\sqrt{2}cos(x-\frac{\pi }{4})}{e^{x-\frac{\pi }{4}}+1}dx$
Substitute $x-\frac{\pi }{4}=t$ or $dx=dt$
$\therefore I={\int }_{-\pi }^{\pi }\frac{\sqrt{2}\cos t}{e^t+1}dt$........(1)
Replacing t by $\pi +(-\pi )-t$ or $-t$, we get
$I={\int }_{-\pi }^{\pi }\frac{\sqrt{2}\cos (-t)}{e^{-t}+1}dt={\int }_{-\pi }^{\pi }\frac{e^t\sqrt{2}\cos \left(t\right)}{e^t+1}dt$......(2)
Adding equations (1) and (2), we get
$2I=\sqrt{2}{\int }_{-\pi }^{\pi }\cos tdtorI=0$