Question

The value of${\int }_0^1\left[\frac{x^3}{1+x^8}\right]dx$ is

• A. $\frac{\mathrm{\pi }}{8}$
• B. $\frac{\pi }{4}$
• C. $\frac{\pi }{16}$
• D. $\frac{\pi }{6}$
• E. $\frac{\pi }{12}$

Answer 1

The correct option is C

$\frac{\pi }{16}$

Explanation for the correct answer.

Evaluate the integral.

$I={\int }_0^1\frac{x^3}{1+x^8}dx$

Let $x^4=z$

$4x^{3}dx=dz$

$$\begin{gathered} =\frac{1}{4}{\int }_0^1\frac{dz}{1+z^2} \\ =\frac{1}{4}{\left[{\tan }^{-1}z\right]}_0^1 \\ =\frac{1}{4}\left[\frac{\mathrm{\pi }}{4}-0\right] \\ =\frac{\mathrm{\pi }}{16} \end{gathered}$$

Hence, option C is correct .