Integration problem: $\int _ {-\infty} ^ {\infty} \frac {e^{-x^2}}{\sqrt{\pi}} e^x\ dx$

Question

I have to integrate

$$\int _ {-\infty} ^ {\infty} \frac {e^{\large-x^2}}{\sqrt{\pi}} e^{\large x}\ dx.$$

I've already done by numerical approximations, like Simpson's rule and Gauss-Hermite, but I need the analytical way to do it, and I'm seriously running out of ideas. If someone could help me, it would be great. Thanks so much! And please, if you do not understand something from the question, just ask me, please. Thanks again!

Answer 1

Let us use that we do know the value if $e^{-x^2}$ is the only thing in the integral.

Then we could complete the square in the exponent and translate the variable.

This is: The $\sqrt{\pi}$ factor is irrelevant; we can take it out of the integral.

Then is only $e^{-x^2+x}=e^{-(x^2-x)}=e^{-[(x-1/2)^2+1/4]}=e^{-1/4}e^{-(x-1/2)^2}$.

The factor $e^{-1/4}$ can be taken out of the integral. And change variable $y=x-1/2$.

You will get the integral of $e^{-x^2}$. Do you know the value of this one and how to get it?

Answer 2

Hint:

$$\int_{-\infty}^{\infty}\frac{e^{-x^{2}}}{\sqrt{\pi}}e^{x}dx=\frac{1}{\sqrt{\pi}}\int_{-\infty}^{\infty}e^{-((x-\frac{1}{2})^{2}-\frac{1}{4})}dx=\frac{1}{\sqrt{}\pi}e^{\frac{1}{4}}\int_{-\infty}^{\infty}e^{-(x-\frac{1}{2})^{2}}dx$$