Double Integral related to Gaussian Integral. We know that $\int_{-\infty}^{\infty} e^{-x^2}dx=\sqrt{\pi}.$  Using this , how can you evaluate $\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} e^{-(x^2+y^2+xy)}dxdy= ?$ Are there any standard tricks for integrals which are related to the gaussian integral ?

The matrix $A:=\pmatrix{1&\frac12\\\frac12&1}$ is positive definite with square root
$$A^{\frac12} = \frac14 \pmatrix{\sqrt 2 + \sqrt 6 & -\sqrt2 + \sqrt 6\\ -\sqrt 2 + \sqrt 6 & \sqrt 2 + \sqrt 6}$$
And your integral with $x := \pmatrix{x_1\\x_2}$ is
$$\int_{\mathbb R^2} e^{-x^T A x} \ \mathrm dx$$
Now substitute $u = A^{\frac12} x$ and use a known integral.
Remark: $\det A^{\frac12} = \sqrt{\det A} = \frac{\sqrt 3}2$.
