Definite Integration Question 452
Question: $ \int_0^{\pi /2}{\frac{\sqrt{\cos x}}{\sqrt{\sin x}+\sqrt{\cos x}}dx=} $
[MNR 1989; UPSEAT 2002]
Options:
A) 0
B) $ \frac{\pi }{2} $
C) $ \frac{\pi }{4} $
D) None of these
Show Answer
Answer:
Correct Answer: C
Solution:
Let $ I=\int_0^{\pi /2}{\frac{\sqrt{\cos x}}{\sqrt{\sin x}+\sqrt{\cos x}}dx} $ …..(i) and $ I=\int_0^{\pi /2}{\frac{\sqrt{\cos ( \frac{\pi }{2}-x )}}{\sqrt{\sin ( \frac{\pi }{2}-x )}+\sqrt{\cos ( \frac{\pi }{2}-x )}}dx} $
$ I=\int_0^{\pi /2}{\frac{\sqrt{\sin x}}{\sqrt{\cos x+}\sqrt{\sin x}}}dx $ …..(ii)
Adding (i) and (ii), we get $ 2I=\int_0^{\pi /2}{(1)dx=\frac{\pi }{2}\Rightarrow I=\frac{\pi }{4}} $ .
BETA
