Evaluate the following integrals:

$\int \frac{x \sin ^{-1} x}{\left(1-x^{2}\right) \sqrt{1-x^{2}}} d x$

we can put $\sin ^{-1} x=t ; d x /\left(1-x^{2}\right)^{1 / 2}=d t ;\left(1-x^{2}\right)=\cos ^{2} t$ and $x=\sin t$.

$\int \frac{t \sin t}{\cos ^{2} t} d t=\int t$ tant sect $d t$

By by parts,

$\int t$ tant $\sec t d t=t \sec t-\int \operatorname{sect} d t \ldots \ldots$

$\because \int \operatorname{sect} \tan t d t=\int \frac{\sin t}{\cos ^{2} t} d t$

$=\mathrm{t} \sec \mathrm{t}-\log (\tan \mathrm{t}+\sec \mathrm{t})+\mathrm{C}^{\prime}$

Put cost $=u$

$-\sin t d t=d u$

$=\sin ^{-1} x \sec \left(\sin ^{-1} x\right)-\log \left(\tan \left(\sin ^{-1} x\right)+\sec \left(\sin ^{-1} x\right)\right)+c^{\prime} \int-u^{-2} d u$

$=-\left(-\mathrm{u}^{-1}\right)+\mathrm{C}$

$=\mathrm{sec} \mathrm{t}+\mathrm{C}$