Answer
See below.
Work Step by Step
We need to verify both sides of the expression. In order to do this, we will differentiate both sides.
$\dfrac{d}{dx} ( \int x coth^{-1} x dx)=\dfrac{d}{dx} ( \dfrac{x^2-1}{2} coth^{-1} x +\dfrac{x}{2} + C)$
or, $ x coth^{-1} x = \dfrac{x^2 -1}{2} (\dfrac{1}{1-x^2})+(coth^{-1} x) (x)+\dfrac{1}{2}+(0)$
or, $x coth^{-1} x =x coth^{-1} x $
Hence, the result has been verified.