Evaluate the following integral:

re-writing the given equation as

$\int \frac{1+\frac{1}{x^{2}}}{x^{2}-1+\frac{1}{x^{2}}} d x$

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

Substituting $t$ as $x-\frac{1}{x}$

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

$\int \frac{d t}{t^{2}+1}$

Using identity $\int \frac{1}{\mathrm{x}^{2}+1} \mathrm{dx}=\arctan (\mathrm{x})$

$\arctan t+c$

Substituting tas $\mathrm{x}-\frac{1}{\mathrm{x}}$

$\arctan \left(x-\frac{1}{x}\right)+c$