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 ?
[Math] Double Integral related to Gaussian Integral.
definite integralsgaussian-integralimproper-integralsintegrationmultivariable-calculus
Best Answer
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: $\det A^{\frac12} = \sqrt{\det A} = \frac{\sqrt 3}2$.