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Evaluate the following integrals:


Evaluate the following integrals:

$\int \frac{x^{2} \sin ^{-1} x}{\left(1-x^{2}\right)^{3 / 2}} d x$


Let I $=\int \frac{x^{2} \sin ^{-1} x}{\left(1-x^{2}\right)^{3 / 2}} d x$

$\sin ^{-1} x=t$

$\frac{1}{\sqrt{1-x^{2}}} d x=d t$

$I=\int \frac{\sin ^{2} t \times t d t}{1-\sin ^{2} t}$

$=\int \frac{\operatorname{tsin}^{2} t}{\cos ^{2} t} d t$

$=\int t \tan ^{2} t d t$

$=\int t\left(\sec ^{2} t-1\right) d t$

Using integration by parts,

$=\int \operatorname{tsec}^{2} t d t-\int t d t$

$=t \int \sec ^{2} t d t-\int \frac{d}{d t} t \int \sec ^{2} t d t-\frac{t^{2}}{2}$

We know that, $\int \sec ^{2} t d t=\tan t$

$=\operatorname{ttan} t-\int \tan t d t-\frac{t^{2}}{2}$

$=\operatorname{ttan} t-\log |\sec t|-\frac{t^{2}}{2}+c$

$I=\frac{x}{\sqrt{1-x^{2}}} \sin ^{-1} x+\log \left|1-x^{2}\right|-\frac{1}{2}\left(\sin ^{-1} x\right)^{2}+c$

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