A clean way to obtain an (analytic or numeric) solution for this integral? A friend and I have been looking at the crazy integral 
$$\iiiint \limits^{\infty}_{-\infty}\exp\left[-(x-t)^2-(x-h)^2-(y+t)^2-(y-h)^2-10\right]\mathrm{d}V$$
and can't come up with a decent method on how to obtain a solution. ($x,y,t,h$ are all variables, not constants) Fubini's theorem would let it split into 4 integrals, but those aren't the cleanest either. I can't think of another method that will work (of course u-sub/parts, etc).
Could residue theorem/contour integration work? Fourier integral? Any hints would be appreciated.
Edit -- changed $y-t$ to $y+t$
 A: According to http://m.wolframalpha.com/input/?i=%28x-t%29%5E2+%2B+%28x-h%29%5E2+%2B+%28y%2Bt%29%5E2+%2B+%28y-h%29%5E2&x=0&y=0
$$(x - t)^2 + (x - h)^2 + (y + t)^2 + (y - h)^2 =2 h^2 + 2 t^2 - 2 h x - 2 t x + 2 x^2 - 2 h y + 2 t y + 2 y^2$$
Therefore, if  $\vec x = (x,y,t, h)$, then
$$e^{-10} \int \exp{-\frac12 (\vec  x \cdot A \cdot \vec x)},$$
Where
$$A = \left(
\begin{array}{cccc}
 4 & 0 & -2 & -2 \\
 0 & 4 & 2 & -2 \\
 -2 & 2 & 4 & 0 \\
 -2 & -2 & 0 & 4 \\
\end{array}
\right)$$
You can check that $A$ is positive definite. The result is
$$e^{-10}\sqrt{\frac{π^4}{\det A}} = e^{-10} \frac{\pi^2}{8}$$
EDIT: The trick is that, if you have a quadratic form $q(\vec x)$ (a polynomial with every term of degree two $q(x_1,x_2,\ldots) = A_{1,1}x_1^2 + A_{1,2}x_1 x_2 + A_{2,2}x_2^2 + A_{1,3}x_1x_3 + \cdots$), and $q$ is positive definite ($q(\vec x)$ is always positive), then there is a linear change of variables $T$ that diagonalises $q$ (it converts it to $x_1^{\prime2}+x_2^{\prime2}+x_3^{\prime2}+\cdots x_n^{\prime2}$). You can prove that $dV = \frac{dV'}{\sqrt{\det{A}}}$. So
$$\int \exp{-q(\vec x)} dV = \int \exp{(-x_1^{\prime2}-x_2^{\prime2}-\cdots x_n^{\prime2})}\frac{dV'}{\sqrt{\det{A}}} =\\
\frac{1}{\sqrt{\det{A}}}
\int \exp{(-x_1^{\prime2})} dx_1 \int \exp{(-x_2^{\prime2})} dx_2 \cdots
\int \exp{(-x_n^{\prime2})} dx_n =\\
\frac{1}{\sqrt{\det{A}}}
\left(\int \exp{(-t^2)} dt \right)^n,
$$
which is the integral of the Gaussian function.
This integrals are actually very useful in probability theory when you want to study the distribution of correlated random variables. 
