Question

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

• A. $0$
• B. $1$
• C. $\frac{\pi }{4}$
• D. $\frac{\pi }{2}$

Answer 1

Answer: (C)

$\frac{\pi }{4}$

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

or, $I={\int }_0^{\frac{\pi }{2}}\frac{dx}{1+{\tan }^3(0+\frac{\pi }{2}-x)}$

or, $I={\int }_0^{\frac{\pi }{2}}\frac{dx}{1+{\cot }^{3}x}$

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

Now adding $\left(1\right)$ and $\left(2\right)$ we get,

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

$2I={\int }_0^{\frac{\pi }{2}}dx$

$\Rightarrow 2I=\frac{\pi }{2}$

$\Rightarrow I=\frac{\pi }{4}$