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  4. ∫ limits0(π/2) ( sin n θ/ sin n θ+ cos n θ) d θ is equal to

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Q. $\int\limits_{0}^{\frac{\pi}{2}} \frac{\sin ^{n} \theta}{\sin ^{n} \theta+\cos ^{n} \theta} d \theta$ is equal to

ManipalManipal 2018

Solution:

Let $I=\int\limits_{0}^{\frac{\pi}{2}} \frac{\sin ^{n} \theta}{\sin ^{n} \theta+\cos ^{n} \theta} d \theta$
$=\int\limits_{0}^{\frac{\pi}{2}} \frac{\sin ^{n}\left(\frac{\pi}{2}-\theta\right)}{\sin ^{n}\left(\frac{\pi}{2}-\theta\right)+\cos ^{n}\left(\frac{\pi}{2}-\theta\right)} d \theta$
$=\int\limits_{0}^{\frac{\pi}{2}} \frac{\cos ^{n} \theta}{\cos ^{n} \theta+\sin ^{n} \theta} d \theta \ldots (ii)$
On adding Eqs. (i) and (ii), we get
$2 I=\int\limits_{0}^{\frac{\pi}{2}} \frac{\sin ^{n} \theta+\cos ^{n} \theta}{\cos ^{n} \theta+\sin ^{n} \theta} d \theta=\int\limits_{0}^{\pi / 2} d \theta=\frac{\pi}{2}$
$\Rightarrow I=\frac{\pi}{4}$