real analysis - 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?

Answer

$$3x^2+2\sqrt{2} xy + 3y^2
=\begin{bmatrix}x & y \end{bmatrix}
\begin{bmatrix} 3 & \sqrt{2} \\ \sqrt{2} & 3 \end{bmatrix}
\begin{bmatrix}x \\ y \end{bmatrix}$$

so the integrand is
$$\exp(- v^\top \Omega v/2)$$
where $v = \begin{bmatrix}x \\ y \end{bmatrix}$ and $\Omega = 2\begin{bmatrix} 3 & \sqrt{2} \\ \sqrt{2} & 3 \end{bmatrix}$.

By using the density of a $N(0, \Sigma)$ distribution we have
$$\frac{1}{\sqrt{(2 \pi)^2 \det (\Omega^{-1})}} \int_{-\infty}^\infty \int_{-\infty}^\infty \exp(-v^\top \Omega v / 2) \, dx \, dy = 1.$$