I cannot solve the integration

Question

Please help me to solve the integration below:
$$\int (e^{x - 1/x})(1+1/x^2) \mathrm dx$$

Answer 1

Let $u=x-\frac{1}{x}$. Thus, $\mathrm du=1+\frac{1}{x^2} \mathrm dx$. Now the integral is $$\int e^u \mathrm du=e^u+C=e^{x-\frac{1}{x}}+C$$

This is easily checked by differentiation of $e^{x-1/x}.$