Find : $\displaystyle \int \frac{2 x}{\left(x^{2}+1\right)\left(x^{2}-4\right)} \mathrm{d} x$.

$$\begin{aligned} & I=\int \frac{2 x}{\left(x^{2}+1\right)\left(x^{2}-4\right)} \mathrm{d} x \\ & \text { Put } x^{2}=t \Rightarrow 2 x \mathrm{d} x=\mathrm{d} t \\ & I=\int \frac{1}{(t+1)(t-4)} \mathrm{d} t=\frac{1}{5} \int \frac{\mathrm{d} t}{t-4}-\frac{1}{5} \int \frac{\mathrm{d} t}{t+1} \\ & I=\frac{1}{5} \log \left|x^{2}-4\right|-\frac{1}{5} \log \left|x^{2}+1\right|+c \text { or } \frac{1}{5} \log \left|\frac{x^{2}-4}{x^{2}+1}\right|+c \end{aligned} $$