Question

${\int }_0^{1}xsi{n}^{-1}xdx=$

• A. $\frac{\pi }{8}$
• B. $-\frac{\pi }{8}$
• C. $\frac{\pi }{4}$
• D. $\frac{\pi }{2}$

Answer 1

The correct option is A $\frac{\pi }{8}$

${\int }_0^{1}xsi{n}^{-1}xdx={\left[\frac{x^2}{2}si{n}^{-1}x\right]}_0^1-{\int }_0^1\frac{x^2}{2}.\frac{1}{\sqrt{1-x^2}}dx=\frac{1}{2}.\frac{\pi }{2}-\frac{1}{2}{\int }_0^1(\frac{1}{\sqrt{1-x^2}}-\sqrt{1-x^2})dx$
$=\frac{\pi }{4}-\frac{1}{2}{[si{n}^{-1}x-\frac{x}{2}\sqrt{1-x^2}-\frac{1}{2}si{n}^{-1}x]}_0^1=\frac{\pi }{2}-\frac{1}{2}[\frac{\pi }{2}-\frac{1}{2}.\frac{\pi }{2}]=\frac{\pi }{4}-\frac{\pi }{8}=\frac{\pi }{8}$