Solving Indefinite Integral of Normal Equation

[Erienion]

I've been trying to integrate the following function but have gotten somewhat stuck doing it. The answer i managed to produce gave some bogan answers.

the integral in question is

\int e^\frac{-(x-\mu)^2}{(2\sigma)^2}

where \mu and \sigma are constants.

its part of the normal equation and I've been trying to write a program to do some calculations with it.

 

[Tide]

The integral can be expressed in terms of the error function, erf(x). Unfortunately, there is no elementary form.

 

[quasar987]

As far as I can see, by setting y = x-\mu /2\sigma, we get the famous e^{-y^2} which doesn't have a primitive. You can however develop e^{-y^2} as a Taylor serie and integrate term by term. You get the (convergant) serie of general term

a_n=\frac{(-1)^n x^{2n+1}}{(2n + 1)n!}