# Closed form for integral of an error function

My question is similar to that posted here.

I have the following integral that I want to determine in a closed form. My uncertainty arises due to the addition term within the Error function:

$$\mathcal{I} = \int_{-L}^L e^{-\beta (x - B)^2}\cdot \text{Erf}\left[\alpha \, (A - x)\right]\,dx.$$

I have simplified the notation by removing constants. $\alpha$ is a complex number whereas $A$ and $B$ are positive real numbers.

I would like to see the steps in the evaluation since I have other similar integrals to evaluate (varying powers of x in the exponential terms etc). Thanks.

• Have you tried differentiating with regard to A inside the integral sign ? – Lucian May 19 '15 at 1:41
• Could you put your suggestion as a solution so that I can see the steps? I used the substitution $z = x - B$ and then used the Binomial expansion on the $(A - z - B)^{2n +1}$ term which arises from the expansion of the Erf term. This then recasts the integral in a similar form to that here. – Sid May 19 '15 at 12:37