A) \[\frac{\pi }{4}\]
B) \[\frac{\pi }{2}\]
C) \[\pi \]
D) \[\frac{3\pi }{2}\]
Correct Answer: A
Solution :
Let\[I=\int_{0}^{\pi /2}{\frac{\phi (x)}{\phi (x)+\phi \left( \frac{\pi }{2}-x \right)}}dx\] ?.(i) Replace\[x\]by\[\left( \frac{\pi }{2}-x \right)\]in Eq. (i), we get \[I=\int_{0}^{\pi /2}{\frac{\phi \left( \frac{\pi }{2}-x \right)}{\phi \left( \frac{\pi }{2}-x \right)+\phi (x)}}dx\] ?.(ii) On adding Eqs. (i) and (ii), we get \[2I=\int_{0}^{\pi /2}{\frac{\phi (x)+\phi \left( \frac{\pi }{2}-x \right)}{\phi (x)+\phi \left( \frac{\pi }{2}-x \right)}}dx\] \[\Rightarrow \] \[2I=\int_{0}^{\pi /2}{1\,dx}\] \[\Rightarrow \] \[2I=\frac{\pi }{2}\] \[\Rightarrow \] \[I=\frac{\pi }{4}\]
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