Experimental Physics graduate student looking for some assistance with an integral. I'm attempting to find a new solution to a beam propagation integral, and it has me evaluating an integral of a form
$$\int_{b}^{\infty} e^{Ar^{2} + Br}r dr$$
Where $b$ is some real valued scalar.
I've found solutions to equations that are very similar , but are for lower limits of integration that go to infinity. Apologies if this is something very basic and I'm missing something, I haven't done real calculus in a couple of years.
Hint
Start completing the square $$Ar^2+Br=A\left(r+\frac{B}{2 A}\right)^2-\frac {B^2}{4A }$$ Now, let $r=x-\frac{B}{2 A}$ to make $$\int r\,e^{Ar^2+Br}\,dr=\frac{e^{-\frac {B^2}{4A } } }{2A }\int (B-2Ax)\,e^{A x^2}\,dx$$ which are very simple.
Solve for $x$, go back to $r$ and use bounds (hoping that $A<0$)