Question
Evaluate the following integrals:
$\int\frac{\text{e}^{\text{m}\sin^{-1}\text{x}}}{\sqrt{1-\text{x}^2}}\text{ dx}$
$\int\frac{\text{e}^{\text{m}\sin^{-1}\text{x}}}{\sqrt{1-\text{x}^2}}\text{ dx}$
Putting,
$\text{m}\sin^{-1}\text{x}=\text{t}$ and $\frac{\text{dx}}{\sqrt{1-\text{x}^2}}=\frac{\text{dt}}{\text{m}}$ in equation (1),We get,
$\text{I}=\int\text{e}^\text{t}\frac{\text{dt}}{\text{m}}$
$=\frac{1}{\text{m}}\text{e}^\text{t}+\text{C}$
$=\frac{1}{\text{m}}\text{e}^{\text{m}\sin^{-1}\text{x}}+\text{C}$
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| $\text{X}$ | $0$ | $1$ | $2$ | $3$ |
| $\text{P}(\text{X})$ | $\frac{1}{5}$ | $\frac{2}{5}$ | $\frac{1}{5}$ | $\frac{1}{5}$ |
| $\text{Y}$ | $0$ | $1$ | $2$ | $3$ |
| $\text{P}(\text{Y})$ | $\frac{1}{5}$ | $\frac{3}{10}$ | $\frac{2}{5}$ | $\frac{1}{10}$ |