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?

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You are not computing "the integral of a Gaussian", whatever that means. You are computing the integral of a function of the real variable $x$, not of a random variable. The result is the quantity on the right which can be regarded as a constant if $b$ and $c$ are known constants or as a function of two variables $b$ and $c$ if $b$ and $c$ are regarded as parameters of the integrand. So, the question you ask is meaningless: there is no mean and no variance because the result is not a random variable. –  Dilip Sarwate Jan 15 '13 at 16:27
Sorry but where is the random variable? –  Did Jan 15 '13 at 17:38

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.
Sorry but I do not get it. And why do you assume that $\Im b\ne0$? This is not necessary to get a convergent integral. –  Did Jan 15 '13 at 19:23