real analysis - Test the convergence of the integral $int_{-infty}^{infty}frac{e^{-x}}{1+x^2}.$

Test the convergence of the following integral $$\int_{-\infty}^{\infty}\frac{e^{-x}}{1+x^2}.$$I can not find the indefinite integral of the integrand so that we can check at the limits $-\infty$ and $\infty$ Also I can not apply any theorem about convergence , like Ables test, Dirichlet's test...etc... Can anyone help me?

Answer

Follow @Solitary comment. Let $f(x)$ be the integrand function.
Notice that
$$

\lim_{x\to -\infty} f(x) = +\infty
$$
hence there is some $b<0$ such that for all $x1$ hence for all $a$$
\int_{a}^{b} f(x) \ge b-a \to +\infty
$$
as $a\to -\infty$.

Hence
$$

\int_{-\infty}^0 f(x) = +\infty.
$$
This assumes that you are speaking of improper integrals. If you are speaking of Lebesgue integrals the solution is even simpler...