Confusion related to integral of a Gaussian I am a bit confused about calculating the integral of a Gaussian
$$\int_{-\infty}^{\infty}e^{-x^{2}+bx+c}\:dx=\sqrt{\pi}e^{\frac{b^{2}}{4}+c}$$
Given above is the integral of a Gaussian. The integral of a Gaussian is Gaussian itself. But what is the mean and variance of this Gaussian obtained after integration?

The question is only meaningful if $\Im{b} \ne 0$.  Let's say that, rather, $\Re{b} = 0$ and $b = i B$.  Now you can assign a mean/variance to the resulting Gaussian.  This, BTW, is related to the well-known fact that a Fourier transform of a Gaussian is a Gaussian. 
