Find: x ^ -1 x 1 - x^ 2 dx. - Vidyadip

Question

Find: $\int \frac{x \sin^{-1} x}{\sqrt{1 - x^{2}}} \text{d}x.$

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Answer

$\text{I} = \int \frac{\text{x} \sin^{-1} \text{x}}{\sqrt{1 - \text{x}^{2}}} \text{ dx}$
$\text{put} \sin^{-1} \text{x = t} \Rightarrow \frac{\text{dx}}{\sqrt{1 - \text{x}^{2}}} = \text{dt}$
$= \int \text{t.} \sin \text{t dt}$
$ = \text{ - t} \cos \text{t} + \sin \text{t + c}$
$= -\sqrt{1 - \text{x}^{2}} \sin^{-1} \text{x + x + c}$

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