Evaluate the following integrals:

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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