Question

Evaluate the following: $\int \frac{xsi{n}^{-1}x}{{(1-x^2)}^{\frac{3}{2}}}dx$

Answer 1

$I=\int \frac{xsi{n}^{-1}x}{(1-x^2{)}^{\frac{3}{2}}}dx$

Put $x=sint\Rightarrow t=si{n}^{-1}x$

$\therefore dx=costdt$

$=\int \frac{sint.si{n}^{-1}\left(sint\right)}{(1-si{n}^{2}t{)}^{\frac{3}{2}}}.costdt$

$=\int \frac{sint.t.cost}{(co{s}^{2}t{)}^{\frac{3}{2}}}dt$

$=\int t\frac{sintcosd}{co{s}^{3}t}dt$

$=\int t\frac{sint}{cost}.\frac{1}{cost}dt$

$=\int ttant.sectdt$

$=t\int tantsectdt-\int 1.sectdt$

$=tsect-ln|sect+tant|+C$

$=si{n}^{-1}xsec\left(si{n}^{-1}x\right)-ln\left|sec\right(si{n}^{-1}x)+tan(si{n}^{-1}x\left)\right|+C$

$=si{n}^{-1}x.\frac{1}{\sqrt{1-x^2}}-ln|\frac{1}{\sqrt{1-x^2}}+\frac{x}{\sqrt{1-x^2}}|+C$

$=si{n}^{-1}x.\frac{1}{\sqrt{1-x^2}}-ln\left|\frac{1+x}{\sqrt{1-x^2}}\right|+C$