Question

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

• A. $\frac{\pi }{4}$
• B. $\frac{\pi }{2}$
• C. $0$
• D. $1$

Answer 1

The correct option is A $\frac{\pi }{4}$

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

$={\int }_0^{a}f\left(x\right)dx={\int }_0^{a}f(a-x)dx$

${\int }_0^{\frac{\pi }{2}}\frac{\sqrt{\sin x}}{\sqrt{\sin x}+\sqrt{\cos x}}dx={\int }_0^{\frac{\pi }{2}}\frac{\sqrt{\cos x}}{\sqrt{\sin x}+\sqrt{\cos x}}dx$
$2I={\int }_0^{\frac{\pi }{2}}1\cdot dx$

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

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