Question

Consider the following definite integral:
$I={\int }_0^1\frac{(si{n}^{-1}x{)}^2}{\sqrt{1-x^2}}dx$
The value of the integral is

• A. $\frac{{\pi }^3}{24}$
• B. $\frac{{\pi }^3}{12}$
• C. $\frac{{\pi }^3}{48}$
• D. $\frac{{\pi }^3}{64}$

Answer 1

The correct option is A $\frac{{\pi }^3}{24}$

$I={\int }_0^1\frac{(si{n}^{-1}x{)}^2}{\sqrt{1-x^2}}dx$
Put $si{n}^{-1}x=t$
$\Rightarrow \frac{dx}{\sqrt{1-x^2}}=dt$
$I={\int }_0^{\pi /2}{t}^{2}dt={\left[\frac{t^3}{3}\right]}_0^{\pi /2}=\frac{{\pi }^3}{24}$