How to evaluate the integral $\int_{-\infty}^{\infty} \frac{e^{-\frac{x^2}{2}}}{1 + e^{-x}} dx$?

Question

Is there a way to evaluate the integral $$\int_{-\infty}^{\infty} \frac{e^{-\frac{x^2}{2}}}{1 + e^{-x}} dx \ ?$$ I've tried changing variables, integral by parts and gaussian integral, but all got stuck. The numerical integral shows that it equals $\frac{\sqrt{2\pi}}{2}$. Any suggestions on how to solve it analytically?

Answer 1

Assuming the question is on the interval $(-\infty,\infty)$, under the variable interchange $x\leftrightarrow -x$ we have

$$I = \int_{-\infty}^\infty \frac{e^{-\frac{x^2}{2}}}{1+e^x}dx = \int_{-\infty}^\infty \frac{e^{-x}\cdot e^{-\frac{x^2}{2}}}{1+e^{-x}}dx$$

Thus

$$I+I = \int_{-\infty}^\infty e^{-\frac{x^2}{2}}dx = \sqrt{2\pi}$$

by the result of the Gaussian integral, therefore the original integral evaluates to $I=\sqrt{\frac{\pi}{2}}$