There is a neat little trick to doing this.
First notice that the integrand is an even function. So
Now I am going to integrate because it is simpler than the original.
Call the value of the integral
where x and y are independent variables.
Now, define an xy plane. We are integrating over the whole of the xy plane. I am going to choose to do this with polar, rather than rectangular, coordinates. So define:
in the usual manner.
Then and . So...
The integration is trivial:
Substitute , then , so