Integrate the function: ^ - 1 x 1 - x^2 - Vidyadip

Question

Integrate the function: $\frac{{{{\sin }^{ - 1}}x}}{{\sqrt {1 - {x^2}} }}$

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Answer

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

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