Non-Integrable Function

Question

I am wondering if the function below is Integrable:

$$\frac{\exp{(-\frac{1}{2}(u-2)^2-2u^2)}}{u-2}$$

When I work it out on computer, the integral is finite from -Inf to Inf. But clearly it has a pole at u=2. Is this pole integrable? If yes, what kind of coordinate transform is required?

Any help is appreciated!

Answer 1

A pole is never integrable. However, the Cauchy principal value integral does exist. With Maple's help, I get

$$ -\pi e^{-8} \text{erfi}\left(\frac{4}{5} \sqrt{10}\right) $$