Evaluate $\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}e^{-\frac{1}{2}(x^2-xy+y^2)}dx\, dy$ I need to evaluate the following integral:
$$\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}e^{-\frac{1}{2}(x^2-xy+y^2)}dx\, dy$$
I thought of evaluating the iterated integral $\displaystyle\int_{-\infty}^{\infty}dx\int_{-\infty}^{\infty}e^{-\frac{1}{2}(x^2-xy+y^2)}dy$, but because of the presence of $x^2$ and $y^2$ terms, I am not being able to do that. I tried substituting $x=r\cos \theta$ and $y=r\sin \theta$ but in that case I have some confusion regarding the limits of $r$ and $\theta$. Can I get some help?

Recall:
$$x^2-xy+y^2=(x-\frac{1}{2}y)^2+\frac{3}{4}y^2$$
With that you get:
$$\int_{-\infty}^\infty\int_{-\infty}^\infty e^{-\frac{1}{2}(x^2-xy+y^2)}dxdy=\\\int_{-\infty}^\infty\int_{-\infty}^\infty e^{-\frac{1}{2}\left((x-\frac{1}{2}y)^2+\frac{3}{4}y^2\right)}dxdy=\\
\int_{-\infty}^\infty\int_{-\infty}^\infty e^{-\frac{1}{2}\left(x'^2+\frac{3}{4}y^2\right)}dx'dy=\\
\int_{-\infty}^\infty\int_{-\infty}^\infty e^{-\frac{1}{2}x'^2}dx'e^{-\frac{3}{8}y^2}dy=\\
\int_{-\infty}^\infty\sqrt{2\pi}e^{-\frac{3}{8}y^2}dy=\\
\sqrt{2\pi}\sqrt{\frac{8}{3}\pi}=\frac{4}{\sqrt{3}}\pi
$$ 
