Chapter 8 - Techniques of Integration - 8.6 Strategies for Integration - Exercises - Page 431: 20

$$-(x^2-1)^{-1/2}+C$$

Given $$\int \frac{x d x}{\left(x^{2}-1\right)^{3 / 2}}$$ Let $$ u=x^2 -1\ \ \ \ \ \ \ \ du=2xdx$$ Then \begin{align*} \int \frac{x d x}{\left(x^{2}-1\right)^{3 / 2}}&=\frac{1}{2}\int \frac{ d u}{u^{3 / 2}}\\ &=-u^{-1/2}+C\\ &=-(x^2-1)^{-1/2}+C \end{align*}