Hello! I'm attempting to compute the normalization constant N for the following function $\displaystyle f(x)=N e^{-(x-x_0)/2k^{2})}$. This is generally done by evaluating $\displaystyle \int |f(x)|^{2}dx=1$, where the limits of integration are -inf to +inf.

I can get this right up until the point where I need to make a u substitution.

$\displaystyle 1=\int |N e^{-(x-x_0)/2k^{2})}|^{2}dx$

$\displaystyle 1=\int N^2 e^{-(x-x_0)^2/k^2} dx$

$\displaystyle 1=N^2 \int e^{-(x-x_0)^2/k^2} dx$

making the substitution $\displaystyle u=x-x_0, du=dx$

$\displaystyle 1=N^2 \int e^{-u^2/k^2}du$

This integral is only slightly more clear to me. It looks like a Gaussian integral, but, how do I go about solving it? Any help is much appreciated.