Evaluate the following integral without using the error function

Question

Evaluate $\int_0^{\infty } e^{-2 a x-x^2} dx$ without using error function

Answer 1

Why so adamant about no error function? Completing the square in the exponential you have $$ \begin{align} \int_0^{\infty} e^{-x^2 - 2ax} dx = \sqrt{\pi}e^{a^2}\int_0^{\infty}\frac{1}{\sqrt{\pi}}e^{-(x+a)^2}dx. \end{align} $$ So the second term is just the Gaussian with location $-a$ and scale $1$, so there really isn't any closed form and the question just reduces to deriving approximations on the tails of Gaussians, which is another question entirely but see something like https://arxiv.org/abs/1012.2063