A) \[\pi /3\]
B) \[\pi /6\]
C) \[\pi /12\]
D) \[\pi /2\]
Correct Answer: C
Solution :
\[I=\int_{\pi /6}^{\pi /3}{\frac{dx}{1+\sqrt{\cot x}}=}\int_{\pi /6}^{\pi /3}{\frac{\sqrt{\sin x}}{\sqrt{\sin x}+\sqrt{\cos x}}\,}dx\] ?..(i) \[I=\int_{\pi /6}^{\pi /3}{\frac{\sqrt{\cos x}}{\sqrt{\cos x}+\sqrt{\sin x}}\,}dx\] .....(ii) Adding (i) and (ii), \[2I=\int_{\,\pi /6}^{\,\pi /3}{dx}\]; \[I=\frac{1}{2}\left( \frac{\pi }{3}-\frac{\pi }{6} \right)=\frac{\pi }{12}\].
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