Question

Using properties of definite integrals, evaluate ${\int }_0^{\frac{\pi }{2}}\frac{\sqrt{\sin x}}{\sqrt{\sin x}+\sqrt{\cos x}}dx$

Answer 1

$I={\int }_0^{\pi /2}\frac{\sqrt{\sin x}}{\sqrt{\sin x}+\sqrt{\cos x}}dx....\left(1\right)$

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

$I={\int }_0^{\pi /2}\frac{\sqrt{\cos x}}{\sqrt{\cos x}+\sqrt{\sin x}}....\left(2\right)$

Adding equation $\left(1\right)$ and $\left(2\right)$,

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

$2I={\int }_0^{\pi /2}1dx=[x{]}_0^{\pi /2}$

$I=\frac{1}{2}[\frac{\pi }{2}-0]$

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