Question

Evaluate :

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

Answer 1

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

$I=\underset{0}{\overset{\pi /2}{\int }}\frac{\sqrt{\cos x}}{\sqrt{\sin x}+\sqrt{\cos x}}dx$ ......(2) $\left(\underset{0}{\overset{a}{\int }}f\right(x)dx=\underset{0}{\overset{a}{\int }}f(a-x\left)dx\right)$

Adding 1 and 2, we get,

$2I=\underset{0}{\overset{\pi /2}{\int }}dx$

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

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