Question

Evaluate:

$\underset{0}{\overset{\frac{\pi }{2}}{\int }}\frac{\sin x}{\sin x+\cos x}dx$.

Answer 1

Let $I=$ $\underset{0}{\overset{\frac{\pi }{2}}{\int }}\frac{\sin x}{\sin x+\cos x}dx$......(1)

or, $I=$ $\underset{0}{\overset{\frac{\pi }{2}}{\int }}\frac{\cos x}{\cos x+\sin x}dx$......(2) [ Using property of definite integral]

Now adding (1) and (2) we get,

$2I={\int }_0^{\frac{\pi }{2}}dx$

or, $2I=\frac{\pi }{2}$

or, $I=\frac{\pi }{4}$.