$\int^{\infty}_{-\infty} \int^{\infty}_{-\infty} e^{-\left(2x^2+2xy+2y^2\right)} dx\,dy\,$ 
I need to evaluate $$\displaystyle\int^{\infty}_{-\infty} \int^{\infty}_{-\infty} e^{-\left(2x^2+2xy+2y^2\right)} dx\,dy\,$$

I think I'll need $\displaystyle\int^{\infty}_{-\infty} e^{-x^2} dx\,=\sqrt{\pi}$ . But I'm not able to apply it properly.

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\begin{align}&\overbrace{\color{#66f}{\large%
\int^{\infty}_{-\infty}\int^{\infty}_{-\infty}
\exp\pars{-\bracks{2x^{2} + 2xy + 2y^{2}}}\,\dd x\,\dd y}}
^{\ds{\dsc{x}=\dsc{\mu + \nu}}\,,\ \dsc{y}=\dsc{\mu - \nu}}
\\[5mm]&=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}
\exp\pars{-2\bracks{3\mu^{2} + \nu^{2}}}\,\ \overbrace{%
\verts{\partiald{\pars{x,y}}{\pars{\mu,\nu}}}}^{\ds{=\ \dsc{2}}}\,\dd\mu\,\dd\nu
\\[5mm]&=2\int_{-\infty}^{\infty}\exp\pars{-6\mu^{2}}\,\dd\mu
\int_{-\infty}^{\infty}\exp\pars{-2\nu^{2}}\,\dd\nu
\\[5mm]&=2
\bracks{{1 \over \root{6}}\ \overbrace{\int_{-\infty}^{\infty}\exp\pars{-\mu^{2}}\,\dd\mu}^{\dsc{\root{\pi}}}}
\bracks{{1 \over \root{2}}\ \overbrace{\int_{-\infty}^{\infty}\exp\pars{-\nu^{2}}\,\dd\mu}^{\dsc{\root{\pi}}}}
={\pi \over \root{3}}=\color{#66f}{\large{\root{3} \over 3}\,\pi}
\end{align}
