A) \[\frac{\pi \sqrt{3}}{4}\]
B) \[\frac{\pi }{4\sqrt{2}}\]
C) 0
D) none of these
Correct Answer: C
Solution :
Let \[I=\int_{0}^{\pi /2}{\frac{\frac{\pi }{4}-x}{\sqrt{\sin x}+\cos x}}dx\] ?(i) \[\Rightarrow \]\[I=\int_{0}^{\pi /2}{\frac{\frac{\pi }{4}-\left( \frac{\pi }{2}-x \right)}{\sqrt{\sin \left( \frac{\pi }{2}-x \right)+\cos \left( \frac{\pi }{2}-x \right)}}dx}\] \[\Rightarrow \] \[I=\int_{0}^{\pi /2}{\frac{x-\frac{\pi }{4}}{\sqrt{\cos x+\sin x}}}dx\] ?(ii) On adding Eqs. (i) and (ii), we get \[2I=\int_{0}^{\pi /2}{\frac{0}{\sqrt{\sin x+\cos x}}dx}\] \[\Rightarrow \] \[I=0\]
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