The idea for the integration is to "complete the square" ; I mean, for instance, writing so that (by letting ). You would have to do this exact kind of manipulation to integrate with respect to , and then with respect to . Give it a try.
You can write the exponent in the integral like:
and procede like I did (with more cumbersome constants...). And remember that, for positive , one has . After a few lines, you should have a condition on for the integral to be finite, and the value of the integral.
But actually, I would do it in a slightly different way, more general and with more geometrical insight. On the other hand, I'll need some algebra you may not know yet... The idea is to rotate the basis so that the quadratic form becomes , which is easy to integrate.
Given a symmetric -matrix , I claim that and the integral is infinite if is nonpositive. This follows from the diagonalization of since the change of basis is a rotation ( is symmetric) and from the fact that . I can give more details if anyone needs.
In the present case, , and with , we have for . We have . And is positive iff and , i.e. iff , and .
Under these conditions, I finally get .
I hope this helps; at least I enjoyed it...