Evaluate the following integrals: e^ ^ -1 x ^2 1-x^2 dx - Vidyadip

Question

Evaluate the following integrals:
$\int\frac{\big\{\text{e}^{\sin^{-1}\text{x}}\big\}^2}{\sqrt{1-\text{x}^2}}\text{dx}$

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Answer

Let $\text{e}^{\sin^{-1}\text{z}}=\text{t}$
Differentiating both sides w.r.t. x,
$\text{e}^{\sin^{-1}\text{x}}\times \frac{1}{\sqrt{1-\text{x}^2}}\text{dx}=\text{dt}$
Now, $\int \frac{(\text{e}^{\sin-1_\text{x}})^2}{\sqrt{1-\text{x}^2}}\text{dx}$
$\int \text{e}^{\sin^{-1}\text{z}}\times \frac{\text{e}^{\sin^{-1}\text{x}}}{\sqrt{1-\text{x}^2}}\text{dx}$
$\int \text{t}.\text{dt}$
$=\frac{\text{t}^2}{2}+\text{C}$
$=\frac{(\text{e}^{\sin^{-1}\text{x}})^2}{2}+\text{C}$

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