Question

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

Answer 1

$I={\int }_0^{\frac{\pi }{2}}\frac{\sqrt{sinx}}{\sqrt{sinx}+\sqrt{cosx}}dx$ ------(1)

$I={\int }_0^{\frac{\pi }{2}}\frac{\sqrt{cosx}}{\sqrt{sinx}+\sqrt{cosx}}dx$ ------ (2)

adding (1) & (2)

$2I={\int }_0^{\frac{\pi }{2}}\frac{\sqrt{sinx}+\sqrt{cosx}}{\sqrt{sinx}+\sqrt{cosx}}dx$

$2I={\int }_0^{\frac{\pi }{2}}1.dx$

$2I=[x{]}_0^{\frac{\pi }{2}}=\frac{\pi }{2}$

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