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