Answer
The integral converges and is equal to: $0$
Work Step by Step
First, we will compute the integral for the negative interval
$\int_{-\infty}^{\infty} xe^{-x^2} \ dx= \int_{-\infty}^0 xe^{x^2} \ dx \\=\lim\limits_{R \to \infty} \int_{R}^0 x e^{-x^2} \ dx \\=\lim\limits_{R \to \infty} (-\dfrac{1}{2}e^{-R^2}) \\=0$
Next, we will compute the integral for the positive interval
$\int_{0}^{\infty} e^{-x^2} \ dx= \lim\limits_{R \to \infty}\int_{0}^{\infty} x e^{-x^2} \ dx \\=\lim\limits_{R \to \infty} (-\dfrac{1}{2}e^{-R^2}) \\=0$
Hence, the integral converges and yields the result:
$\int_{-\infty}^{\infty} xe^{-x^2} \ dx=0+0=0$
