# Integral of $\int \frac{1}{x+b}\exp(-(x\pm b)^2)dx$

Can anyone see a solution for varitations of

$$\int \frac{1}{x+b}\exp(-(x\pm b)^2)dx$$

I recognize that the term $$1/(x+b)$$ could be rewritten for the region $$|x|

$$\int \sum_{i=1}^\infty \frac{(-x)^i}{b^i} \exp(-(x \pm b)^2) \textrm{d}x$$

The series can be modified to an arbitrarily large convergence domain by changing the center of the expansion

$$\int \sum_{i=1}^\infty \frac{(x+L)^i}{(L -b)^i} \exp(-(x \pm b)^2) \textrm{d}x$$

But I can't go much further with this approach.

I expect (not required) to have some solution of the form $$Q_1(b,x) \textrm{erfc}(x \pm b) + Q_2(b,x) e^{-(x\pm b)^2}$$ where $$Q_1(x),Q_2(x)$$, or a series in terms of $$Q_k(x) \gamma_k(x)$$ (incomplete gamma function), and the $$Q$$ functions are rational. Valid for specific intervals that will cover the positive real domain. Or simply an exact series that converges quikly for large $$b$$.

Thank you

• For the upper sign, take $\sqrt{t}=x+b$.
– Gary
Apr 4, 2022 at 8:28
• @Gary Thank you, I hope for the lower sign there is anything like that hehe
– Bob
Apr 4, 2022 at 8:33
• I do not want to be pessimistic but, for the lower sign, as Dante Alighieri wrote 700 years ago, for the antiderivative : "O you who enter here give up all hope" Apr 4, 2022 at 10:10
• For the definite integral $\int_0^\infty$, there is a simple solution. Apr 4, 2022 at 11:54