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 ?

Hint
What about writing first $$x^2+x\,y+y^2=\big(x+\frac y2\big)^2+\frac 34 y^ 2$$ So, $$\int e^{-(x^2+xy+y^2)}\,dx=e^{-\frac 34 y^ 2}\int e^{-(x+\frac y2)^2}\,dx=e^{-\frac 34 y^ 2}\int e^{-z^2}\,dz$$ Use the bounds and the known $\int_{-\infty}^{\infty} e^{-z^2}\,dz=\sqrt{\pi}$ to continue.
I am sure that you can take from here.
