Question

The value of ${\int }_0^{\pi /2}\frac{\sqrt{\cos x}}{\sqrt{\sin x}+\sqrt{\left(\cos x\right)}}dx$

• A. 0
• B. $\pi /2$
• C. $\pi /4$
• D. None of these.

Answer 1

The correct option is C $\pi /4$

Let $I={\int }_0^{\pi /2}\frac{\sqrt{\cos x}}{\sqrt{\sin x}+\sqrt{\left(\cos x\right)}}dx$ ...(1)
Using integration property
$I={\int }_0^{\pi /2}\frac{\sqrt{\cos (\frac{\pi }{2}-x)}}{\sqrt{\sin (\frac{\pi }{2}-x)}+\sqrt{\left(\cos (\frac{\pi }{2}-x)\right)}}dx$
$={\int }_0^{\pi /2}\frac{\sqrt{\sin x}}{\sqrt{\cos x}+\sqrt{\sin x}}dx$ ...(2)
Adding (1) and (2), we get
$2I={\int }_0^{\pi /2}dx=\frac{\pi }{2}\Rightarrow I=\frac{\pi }{4}$