I suspect that this exact integral has been asked about before, but finding it among the sea of questions title "difficult integral" is daunting. I am trying to compute
$$\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}e^{-\frac{x^{2}}{2}+x}dx$$
This integral showed up when I found myself needing to compute the expected value of $e^{X}$ where $X$ is a standard normal random variable. Wolfram says that the integral evaluates to $\sqrt{e}$, but I haven't been able to get it to work out. Any help would be appreciated.