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Evaluate the following integrals:

Question:

Evaluate the following integrals:

$\int \frac{\left\{e^{\sin ^{-1} x}\right\}^{2}}{\sqrt{1-x^{2}}} d x$

Solution:

Assume $\sin ^{-1} x=t$

$\Rightarrow \mathrm{d}\left(\sin ^{-1} \mathrm{x}\right)=\mathrm{dt}$

$\Rightarrow \frac{\mathrm{dx}}{\sqrt{1-\mathrm{x}^{2}}}=\mathrm{dt}$

$\therefore$ Substituting $t$ and dt in the given equation we get

$\Rightarrow \int e^{t^{2}} d t$

$\Rightarrow \int e^{2 t} \cdot d t$

$\Rightarrow \frac{\mathrm{e}^{2 t}}{2}+\mathrm{c}$

But $t=\sin ^{-1} x$

$\Rightarrow \frac{e^{2\left(\sin ^{-1} x\right)}}{2}+c$

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