Integrate the functions in Exercises: ^ -1 x 1-x^2 - Vidyadip

Question

Integrate the functions in Exercises:
$\frac{\sin^{-1}\text{x}}{\sqrt{1-\text{x}^2}}$

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Answer

$\text{Let I}=\int\frac{\sin^{-1}\text{x}}{\sqrt{1 - \text{x}^2}}\text{ dx}\ \ \ \ \ \ ...\text{(i)} $Putting $\sin^{-1}\text{x}=\text{t}\ \ \ \Rightarrow\ \ \ \ \frac{1}{\sqrt{1 - \text{x}^2}}=\frac{\text{dt}}{\text{dx}}\ \ \ \ \Rightarrow \ \ \ \ \frac{\text{dx}}{\sqrt{1 - \text{x}^2}}=\text{ dt} $
$\therefore\ \ \ \ \ $From eq. (i), $\text{I}=\int\text{t }\text{ dt}=\frac{\text{t}^2}{2}+\text{c}= \frac{1}{2}\big(\sin^{-1}\text{x}\big)^2+\text{c}$

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