Question:
Evaluate the following integrals:
$\int \frac{\left(\sin ^{-1} x\right)^{3}}{\sqrt{1-x^{2}}} d x$
Solution:
Assume $\sin ^{-1} x=t$
$d\left(\sin ^{-1} x\right)=d t$
$\Rightarrow \frac{\mathrm{dx}}{\sqrt{1-\mathrm{x}^{2}}}=\mathrm{dt}$
$\therefore$ Substituting $t$ and dt in given equation we get
$\Rightarrow \int \mathrm{t}^{3} \mathrm{dt}$
$\Rightarrow \frac{t^{4}}{4}+c$
But $t=\sin ^{-1} x$
$\Rightarrow \frac{\left(\sin ^{-1} x\right)^{4}}{4}+c$
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