# Fundamental solution of Heat Equation, evaluating integral

I'm trying to show that that for a function $f(x)=\exp(-x^2)$, the fundamental solution of the heat equation reduces to a given solution $u(x,t)$.

But, I am stuck on the following part, that is showing:

$\int_{-\infty}^{\infty}e^{-s^2(1+4t)}e^{4xs\sqrt{t}}\,ds = e^{4tx^2\pi/(1+4t)}$$\sqrt{\pi} \over\sqrt{1 + 4t}$

If this can be shown, the the rest of the problem is straightforward.

Any help would be much appreciated. Thanks!

• As done hundred times before somewhere on this site: complete the square and use standard gaussian integral afterwards – tired Nov 29 '16 at 19:01
• @tired Sorry about that! And, also I had tried completing the square, but it made it more complicated i felt. After i did the square, how would it help make the integral easier to solve? Thanks – KidMe Nov 29 '16 at 19:03
• @tired Don't mind, but your name suits after your comment! :-D – Qwerty Nov 29 '16 at 19:08
• Hint: How does a change $x\rightarrow x+a$ changes the range of integration if it is initially $(-\infty,\infty)$ – tired Nov 29 '16 at 19:09
• @KidMe correct. with this result anything should be clear i think – tired Nov 29 '16 at 19:22