Does this integral converge $\int_{-\infty}^{\infty} \frac{e^{-x}}{1+x^2}\,dx$

Question

I need to check whether this integral converges or not $\int^{\infty}_{-\infty} \frac{e^{-x}}{1+x^2}\,dx$

I substituted $y=-x$ then this integral transformed to $\int^{\infty}_{-\infty} \frac{e^{y}}{1+y^2}\,dy$ , then I thought of dividing it into two parts from $-\infty$ to $0$ and then from $0$ to $\infty$, in the first case I think area will be finite but in the second case it's not since $e^x$ grows rapidly, so it diverges!

Answer 1

Hint : You first calculate $$\int_a^b \frac{e^{-x}}{1+x^2}dx$$

and then send $b\rightarrow \infty$ and $a\rightarrow -\infty$.

If this limit exists then, your integral converges.

Answer 2

Hint: Check $\lim_{x\to\pm\infty}\frac{e^{-x}}{1+x^2}$. Are both zero? If not, then this implies that the integral does not converge (since the integrand is strictly positive, we can't have cancellation as in the $\sin(x^2)$ case).