Question

Integrate the following function
$x{\sin }^{-1}x$

Answer 1

${\int }^{}x{\sin }^{-1}xdx$

$={\sin }^{-1}x\times \frac{x^2}{2}-{\int }^{}\frac{1}{\sqrt{1-x^2}}\times \frac{x^2}{2}dx$

$=\frac{x^2}{2}{\sin }^{-1}x+\frac{1}{2}{\int }^{}\frac{-x^2}{\sqrt{1-x^2}}dx$

$=\frac{x^2}{2}{\sin }^{-1}x+\frac{1}{2}{\int }^{}\frac{1-x^2+1}{\sqrt{1-x^2}}dx$

$=\frac{x^2}{2}{\sin }^{-1}x+\frac{1}{2}{\int }^{}\left[\sqrt{1-x^2}+\frac{1}{\sqrt{1-x^2}}\right]dx$

$=\frac{x^2}{2}{\sin }^{-1}x+\frac{1}{2}\left[\frac{x\sqrt{1-x^2}}{2}+\frac{1}{2}{\sin }^{-1}x\right]+\frac{1}{2}{\sin }^{-1}x+c$

$=\frac{x^2}{2}{\sin }^{-1}x+\frac{x}{4}\sqrt{1-x^2}+\frac{1}{4}{\sin }^{-1}x+\frac{1}{2}{\sin }^{-1}x+c$

$=\left(\frac{4x^2+3}{4}\right){\sin }^{-1}x+\frac{x}{4}\sqrt{1-x^2}+c$