The value of the integral $\int \frac{2 x}{\left(x^2+1\right)\left(x^2+2\right)} d x$ is:

$\log \left|\frac{x^2+1}{x^2+2}\right|+C$

The correct answer is Option (3) → $\log \left|\frac{x^2+1}{x^2+2}\right|+C$

$\int \frac{2 x}{\left(x^2+1\right)\left(x^2+2\right)} d x$

$=\int 2x\frac{((x^2+2)-(x^2+1))}{(x^2+1)(x^2+2)}dx$

$=\int\frac{2x}{x^2+1}-\frac{2x}{x^2+2}dx=\log|x^2+1|-\log|x^2+2|+c$

$=\log \left|\frac{x^2+1}{x^2+2}\right|+C$