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...