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Question

# Evaluate using Integration by Parts as a first step.$\int \frac { \sin ^ { - 1 } x } { x ^ { 2 } } d x$

Solution

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We need to evaluate the integral :

$\int \dfrac{\sin^{-1} x}{x^2}\ dx$

We use method of integration by parts for solving this integral

$\begin{gather} \int u\ dv=uv-\int v\ du \end{gather}$

Here, we take $u =\sin^{-1} x$ and $dv= \dfrac{1}{x^2}\ dx$. So $v=\dfrac{x^{-2+1}}{-2+1} = -\dfrac 1x$ and

$\dfrac{d}{dx}\sin^{-1} x =\dfrac{1}{\sqrt{1-x^2}} \implies du = \dfrac{1}{\sqrt{1-x^2}}\ dx$

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