Question

$\underset{0}{\overset{\frac{\pi }{2}}{\int }}\frac{{\tan }^{7}x}{{\tan }^{7}x+{\cot }^{7}x}dx$ is equal to

• A. $\frac{\pi }{2}$
• B. $\frac{\pi }{4}$
• C. $\frac{\pi }{6}$
• D. $\frac{\pi }{3}$

Answer 1

The correct option is B $\frac{\pi }{4}$

Let
$I=\underset{0}{\overset{\frac{\pi }{2}}{\int }}\frac{{\tan }^{7}x}{{\tan }^{7}x+{\cot }^{7}x}dx$
Using the property $\underset{a}{\overset{b}{\int }}f\left(x\right)dx=\underset{a}{\overset{b}{\int }}f(a+b-x)dx$
$I=\underset{0}{\overset{\frac{\pi }{2}}{\int }}\frac{{\cot }^{7}x}{{\tan }^{7}x+{\cot }^{7}x}dx$

Adding both integrals
$2I=\underset{0}{\overset{\frac{\pi }{2}}{\int }}\frac{{\tan }^{7}x+{\cot }^{7}x}{{\tan }^{7}x+{\cot }^{7}x}dx$
$\Rightarrow 2I=\underset{0}{\overset{\frac{\pi }{2}}{\int }}dx=\frac{\pi }{2}$
$\Rightarrow I=\frac{\pi }{4}$