Question

${\int }_0^{\pi /2}\frac{\cos x}{\cos x+\sin x}dx$.

Answer 1

We have,

$I={\int }_0^{\pi /2}\frac{\cos x}{\cos x+\sin x}dx$

$={\int }_0^{\pi /2}\frac{\cos (\frac{\pi }{2}-x)}{\cos x+\sin x}dx$

$={\int }_0^{\pi /2}\frac{\sin x}{\cos x+\sin x}dx$

$\Rightarrow 2I={\int }_0^{\pi }2\frac{\cos x+\sin x}{\cos x+\sin x}dx=[\frac{\pi }{2}-0]=\frac{\pi }{2}$

$\Rightarrow I=\frac{\pi }{4}.$

Hence, this is the answer.