Answer
$$0$$
Work Step by Step
\begin{align*}
\int_{-\infty}^{\infty} 2 x e^{-x^{2}} d x&=\int_{-\infty}^{0} 2 x e^{-x^{2}} d x+\int_{0}^{\infty} 2 x e^{-x^{2}} d x\\
&=\lim _{b \rightarrow-\infty}\left[-e^{-x^{2}}\right]_{b}^{0}+\lim _{c \rightarrow \infty}\left[-e^{-x^{2}}\right]_{0}^{c}\\
&=\lim _{b \rightarrow-\infty}\left[-1-\left(-e^{-b^{2}}\right)\right]+\lim _{c \rightarrow \infty}\left[-e^{-c^{2}}-(-1)\right]\\&=0
\end{align*}
