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Questions  >>  JEEMAIN and NEET  >>  Mathematics  >>  Class12  >>  Integral Calculus
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Q)

Integrate : $\int \limits_0^{\frac{\pi}{2}} \large\frac{\phi(x)}{\phi (\Large\frac{\pi}{2}-x)+\phi (x)}$$dx$

\[\begin {array} {1 1} (a)\;\frac{\pi}{4} \\ (b)\;\frac{\pi}{2} \\ (c)\;\pi \\ (d)\;\frac{3 \pi}{2} \end {array}\]

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A)
$\int \limits_0^{\frac{\pi}{2}} \large\frac{\phi(x)}{\phi (\Large\frac{\pi}{2}-x)+\phi (x)}$$dx$------(1)
$f(x) \geq f(a-x)$
$\int \limits_0^{\frac{\pi}{2}} \large\frac{\phi(\Large\frac{\pi}{2}-x)}{\phi (x)+\phi (\Large\frac{\pi}{2}-x)}$---------(ii)
By adding (i) and (ii)
=> $2I= \int \limits_0^{\large\frac{\pi}{2}} dx$
=> $I= \large\frac{1}{2} \int \limits _0^{\frac{\pi}{2}} $$dx$
=> $\large\frac{1}{2}\bigg[ \large\frac{\pi}{2}\bigg]$
=> $\large\frac{\pi}{4}$
Hence a is the correct answer.

 

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