Question

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

Answer 1

$I=\underset{0}{\overset{\frac{\pi }{2}}{\int }}\frac{\sqrt{\sin x}}{\sqrt{\sin x}+\sqrt{\cos x}}dx\cdots \left(1\right)$
Using $\underset{a}{\overset{b}{\int }}f\left(x\right)dx=\underset{a}{\overset{b}{\int }}f(a+b-x)dx$, we have
$I=\underset{0}{\overset{\frac{\pi }{2}}{\int }}\frac{\sqrt{\sin (\frac{\pi }{2}-x)}}{\sqrt{\sin (\frac{\pi }{2}-x)}+\sqrt{\cos (\frac{\pi }{2}-x)}}dx$
$I=\underset{0}{\overset{\frac{\pi }{2}}{\int }}\frac{\sqrt{\cos x}}{\sqrt{\sin x}+\sqrt{\cos x}}dx\cdots \left(2\right)$
On adding $\left(1\right)$ and $\left(2\right)$, we get
$2I=\underset{0}{\overset{\frac{\pi }{2}}{\int }}dx\Rightarrow I=\frac{\pi }{4}$