Question

\[\int\limits_0^{\pi/2} \frac{1}{1 + \cot^7 x} dx\]

Answer 1

\[Let, I = \int_0^\frac{\pi}{2} \frac{1}{1 + co t^7 x} d x ..............(1)\]

\[ = \int_0^\frac{\pi}{2} \frac{1}{1 + co t^7 \left( \frac{\pi}{2} - x \right)} d x\]

\[ = \int_0^\frac{\pi}{2} \frac{1}{1 + \tan^7 x} d x ..............(2)\]

Adding (1) and (2)

\[2I = \int_0^\frac{\pi}{2} \frac{1}{1 + co t^7 x} + \frac{1}{1 + \tan^7 x} d x \]

\[ = \int_0^\frac{\pi}{2} \frac{2 + co t^7 x + \tan^7 x}{\left( 1 + co t^7 x \right)\left( 1 + \tan^7 x \right)}dx\]

\[ = \int_0^\frac{\pi}{2} \frac{2 + co t^7 x + \tan^7 x}{2 + co t^7 x + \tan^7 x}dx\]

\[ = \int_0^\frac{\pi}{2} dx\]

\[ = \left[ x \right]_0^\frac{\pi}{2} \]

\[ = \frac{\pi}{2}\]

\[Hence, I = \frac{\pi}{4}\]