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Q. $\int\limits_{0}^{\pi/2} \frac{\tan^{7} x}{\cot^{7 } x + \tan^{7 } x} dx$ is equal to

KCETKCET 2017Integrals

Solution:

Let $I=\int \limits_{0}^{\pi / 2} \frac{\tan ^{7} x}{\cot ^{7} x+\tan ^{7} x} d x\,\,\,\,\dots(i)$
$\Rightarrow I=\int\limits_{0}^{\pi / 2} \frac{\tan ^{7}(\pi / 2-x)}{\cot ^{7}(\pi / 2-x)+\tan ^{7}(\pi / 2-x)} d x$
$\Rightarrow I=\int\limits_{0}^{\pi / 2} \frac{\cot ^{7} x}{\tan ^{7} x+\cot ^{7} x} d x\,\,\,\,\,\dots(ii)$
On adding Eqs. (i) and (ii), we get
$2 I=\int\limits_{0}^{\pi / 2} \frac{\tan ^{7} x+\cot ^{7} x}{\tan ^{7} x+\cot ^{7} x} d x$
$=\int\limits_{0}^{\pi / 2} d x=[x]_{0}^{\pi / 2}=\frac{\pi}{2}$
$\therefore \, I=\frac{\pi}{4}$