Question

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

Answer 1

Let $I={\int }_0^{\frac{\pi }{2}}\frac{\sqrt{\sin x}}{\sqrt{\sin x}+\sqrt{\cos x}}dx$ .........(1)
$\Rightarrow I={\int }_0^{\frac{\pi }{2}}\frac{\sqrt{\sin (\frac{\pi }{2}-x)}}{\sqrt{\sin (\frac{\pi }{2}-x)}+\sqrt{\cos (\frac{\pi }{2}-x)}}dx,(\because {\int }_0^{a}f(x)dx={\int }_0^{a}f(a-x\left)dx\right)$
$\Rightarrow I={\int }_0^{\frac{\pi }{2}}\frac{\sqrt{\cos x}}{\sqrt{\cos x}+\sqrt{\sin x}}dx$ ......... (2)

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

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

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

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

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