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

Question

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 1

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 $x<b$ one has $f(x)>1$ hence for all $a<b$ one has $$ \int_{a}^{b} f(x) \ge b-a \to +\infty $$ 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...