Question

Evaluate
$\int \frac{x^2{\sin }^{-1}x}{(1-x^2{)}^{3/2}}dx$

Answer 1

According to question,

Let, $si{n}^{-1}x=t$

=> $\frac{1}{\sqrt{1-x^2}}dx=dt$ And, $x^2=si{n}^{2}t$

Substituting these values, we get

$⟹\int \frac{t.si{n}^{2}t}{1-si{n}^{2}t}dt$

$⟹\int \frac{t.si{n}^{2}t}{co{s}^{2}t}dt$

$⟹\int t.ta{n}^{2}tdt$

Solving By parts,

=> $t\int ta{n}^{2}t.dt-\int \int ta{n}^{2}tdt$

=>$t.tant-t^2-\int (tant-t)dt$

=> $t.tant-t^2+ln|cost|+\frac{t^2}{2}+C$

Replace value of ${}^{\prime }{t}^{\prime }$,

=> $si{n}^{-1}x.tan\left(si{n}^{-1}x\right)+ln|cos\left(si{n}^{-1}x\right)|-\frac{(si{n}^{-1}x{)}^2}{2}+C$