Question

Evaluate the following integral:

\[\int_0^\frac{\pi}{2} \frac{\tan^7 x}{\tan^7 x + \cot^7 x}dx\]

Answer 1

\[\text{Let I} = \int_0^\frac{\pi}{2} \frac{\tan^7 x}{\tan^7 x + \cot^7 x}dx.....................(1)\]

Then,

\[I = \int_0^\frac{\pi}{2} \frac{\tan^7 \left( \frac{\pi}{2} - x \right)}{\tan^7 \left( \frac{\pi}{2} - x \right) + \cot^7 \left( \frac{\pi}{2} - x \right)}dx ...................\left[ \int_0^a f\left( x \right)dx = \int_0^a f\left( a - x \right)dx \right]\]
\[ = \int_0^\frac{\pi}{2} \frac{\cot^7 x}{\cot^7 x + \tan^7 x}dx .................(2)\]

Adding (1) and (2), we get

\[2I = \int_0^\frac{\pi}{2} \frac{\tan^7 x + \cot^7 x}{\tan^7 x + \cot^7 x}dx\]
\[ \Rightarrow 2I = \int_0^\frac{\pi}{2} dx\]
\[ \Rightarrow 2I = \left.x\right|_0^\frac{\pi}{2} \]
\[ \Rightarrow 2I = \frac{\pi}{2} - 0 = \frac{\pi}{2}\]
\[ \Rightarrow I = \frac{\pi}{4}\]