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