I'm working through some old integrals and I found one that's interesting. I can't quite remember how it's proved, so if someone could set me off in the right direction, it would be really helpful. Thanks!
$\int_{\mathbb{R}} e^{-ax^2 + bx} dx = \sqrt{ \dfrac{\pi}{a}} e^{\frac{b^2}{4a}}$
I've tried change of variable, but I'm not sure this is the right approach.
Yes, it is by change of variable. You need to complete the square in the exponent,
$$ -ax^2 + bx = -a\left(x-\frac{b}{2a}\right)^2 + \frac{b^2}{4a} $$
so to make this like the standard integral you make the change $y = x-b/2a$.