Question

${\int }_0^{\pi /2}\frac{1}{1+{\tan }^{3}x}dx$

Answer 1

We have,

$I={\int }_0^{\pi /2}\frac{1}{1+{\tan }^{3}x}dx........\left(1\right)$

Apply property, we get

$I={\int }_0^{\pi /2}\frac{1}{1+{\cot }^{3}x}dxI={\int }_0^{\pi /2}\frac{{\tan }^{3}x}{{\tan }^{3}x+1}dx.........\left(2\right)$

On adding equation (1) and (2), we get

$2I={\int }_0^{\pi /2}dxI=\frac{\pi }{2}I=\frac{\pi }{4}$

Hence, this is the answer.