Explanation

(d)
$\int_{}^{}\frac{x^{2} - 2}{x^{3} - \sqrt{x^{2} - 1}}\ dx = \int_{}^{}\frac{dx}{x\sqrt{x^{2} - 1}} - 2\int_{}^{}\frac{dx}{x^{3}\sqrt{x^{2} - 1}}$
$= \sec^{- 1}x - 2\int_{}^{}\frac{\sec\theta\tan\theta}{\sec^{3}\theta\tan\theta}d\theta$
[Putting $x = \sec\theta \Rightarrow dx = \sec\theta\tan\theta\ d\theta$]
$= \sec^{- 1}x - 2\int_{}^{}{\cos^{2}\theta}d\theta$
$= \sec^{- 1}x - \int_{}^{}\left( 1 + \cos{2\theta} \right)d\theta$
$= sec^{- 1}x - \left( \theta + \frac{\sin{2\theta}}{2} \right) + c$
$= \sec^{- 1}x - \sec^{- 1}x - \frac{\sqrt{x^{2} - 1}}{x^{2}} + c$
$= - \frac{\sqrt{x^{2} - 1}}{x^{2}} + c$