Find : $\displaystyle \int \frac{1}{x\left(x^{2}-1\right)} \mathrm{d} x$.

$$\begin{aligned} & I=\int \frac{\mathrm{d} x}{x\left(x^{2}-1\right)}=\int \frac{\mathrm{d} x}{x^{3}\left(1-\dfrac{1}{x^{2}}\right)}=\frac{1}{2} \int \frac{\left(\dfrac{2}{x^{3}}\right) \mathrm{d} x}{\left(1-\dfrac{1}{x^{2}}\right)} \\ & \text { Put }\left(1-\frac{1}{x^{2}}\right)=t \Rightarrow\left(\frac{2}{x^{3}}\right) \mathrm{d} x=\mathrm{d} t \\ & I=\frac{1}{2} \int \frac{\mathrm{d} t}{t}=\frac{1}{2} \log \left|\left(1-\frac{1}{x^{2}}\right)\right|+c \quad \text { OR } \frac{1}{2} \log \left|\left(\frac{x^{2}-1}{x^{2}}\right)\right|+c \end{aligned} $$