Question
Integrate the following functions w.r.t. x:
$\frac{\left(\sin ^{-1} x\right)^{\frac{3}{2}}}{\sqrt{1-x^2}}$
✓
Answer
Let $I =\int \frac{\left(\sin ^{-1} x\right)^{\frac{3}{2}}}{\sqrt{1-x^2}} d x$
Put $\sin ^{-1} x=t . \quad \therefore \frac{1}{\sqrt{1-x^2}} d x=d t$
$\therefore I=\int t^{\frac{3}{2}} d t=\frac{t^{\frac{5}{2}}}{5 / 2}+c$
$=\frac{2}{5}\left(\sin ^{-1} x\right)^{\frac{5}{2}}+c$
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