Question

The value of the integral $ \int^{ \pi / 2 }_ 0 \frac {\sqrt { \cot \, x }}{ \sqrt { \cot \, x } + \sqrt { \tan \, x }} \, dx \, is $

• A. $ \frac{\pi}{ 4 } $
• B. $ \frac{\pi}{ 2 } $
• C. $ \pi $
• D. None of these

Answer 1

The Correct Option is A

$ Let \, \, I = \int^{ \pi / 2 }_ 0 \frac {\sqrt { cot \, x }}{ \sqrt { cot \, x } + \sqrt { tan \, x }} \, dx \, is ..................(1)$
$ \Rightarrow \, \, I = \int^{ \pi / 2 }_ 0 \frac {\sqrt { tan \, x }}{ \sqrt { cot \, x } + \sqrt { tan \, x }} \, dx \, is ..................(2)$
On adding Eqs. (i) and (ii), we get
$ 2I = \int^{ \pi / 2 }_ 0 \, 1 \, dx $
$ \therefore I = \frac{\pi}{4 } $