How do I evaluate $\int_{-\infty}^{\infty}\int_{-\infty}^{\infty} e^{-(3x^2+2 \sqrt 2 xy+3y^2)} \mathrm dx\,\mathrm dy$? 
Evaluate $$\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \exp\left(-3x^2-2 \sqrt 2 xy - 3y^2\right) \, \mathrm dx\,\mathrm dy$$

I first evaluate
$$\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \exp\left[-3\bigl(x^2+ y^2\bigr)\right] \,\mathrm dx\,\mathrm dy$$
using polar coordinates, which evaluates to $\pi/3$. But I find difficulty to evaluate the double integral  $$\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \exp\left(-2 \sqrt 2 xy\right) \, \mathrm dx\,\mathrm dy$$ Would anybody please help me finding it out?

Hint: Doing the change of variables $x=X+Y$ and $y=X-Y$, your integral becomes$$2\int_{-\infty}^\infty\int_{-\infty}^\infty \exp\left[-\left(2\sqrt2+6\right)X^2-\left(-2\sqrt2+6\right)Y^2\right]\,\mathrm dX\,\mathrm dY.$$
