How do I do this integral? How do I integrate $$\int\limits_{-c}^c \frac{e^{-\gamma\, x^2}}{1-\beta \, x}\, dx $$, under the constraints (if necessary) that $\beta < 1\, /\, c$ and $\beta \ll 1$ ?

HINT:


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*Using the exponential integral:
$$\text{Ei}(x)=\int_{1}^{\infty}\frac{e^{-tx}}{t}\space\text{d}t$$



$$\int\frac{e^{-\gamma x}}{1-\beta x}\space\text{d}x=-\frac{e^{-\frac{\gamma}{\beta}}\text{Ei}\left(\gamma\left(\frac{1}{\beta}-x\right)\right)}{\beta}+\text{C}$$
So with your boundaries we get:
$$\int_{-c}^{c}\frac{e^{-\gamma x}}{1-\beta x}\space\text{d}x=\left[-\frac{e^{-\frac{\gamma}{\beta}}\text{Ei}\left(\gamma\left(\frac{1}{\beta}-x\right)\right)}{\beta}\right]_{-c}^{c}=\frac{e^{-\frac{\gamma}{\beta}}\left(\text{Ei}\left(\frac{\gamma(\beta c+1)}{\beta}\right)-\text{Ei}\left(-\frac{\gamma(\beta c+1)}{\beta}\right)\right)}{\beta}$$
