Question

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

• A. $\frac{\pi }{2}$
• B. $\frac{\pi }{4}$
• C. $\pi$
• D. none of these

Answer 1

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

$I={\int }_0^{\pi /2}\frac{1}{1+{\tan }^{3}x}dx$
Using the property : ${\int }_a^{b}f\left(x\right)dx={\int }_a^{b}f(a+b-x)dx$
$I={\int }_0^{\pi /2}\frac{{\tan }^{3}x}{{\tan }^{3}x+1}dx$
Adding both ,we get
$2I={\int }_0^{\pi /2}\frac{{\tan }^{3}x+1}{{\tan }^{3}x+1}dx=\frac{\pi }{2}$
$I=\frac{\pi }{4}$