To evaluate double integral over entire xy-plane

Question

The problem is to evaluate:

$\displaystyle\int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty}e^{-(19x^2+19y^2+2xy)} \,dx\,dy$

Can anybody help me to solve this type problems? Thanks in advance.

Answer 1

$$19(x^2+y^2)+2xy=\frac{19}{2}[(x+y)^2+(x-y)^2]+\frac{1}{2}[(x+y)^2-(x-y)^2]$$ The the give double integral is $$I=\frac{1}{2} \int_{-\infty}^{\infty}\int_{-\infty}^{\infty} e^{-10u^2} e^{-9v^2} du ~ dv =\frac{4}{2} \int_{0}^{\infty}e^{-10u^2} du \int_{0}^{\infty}e^{-9v^2} dv= 2 \sqrt{\frac{\pi}{40}} \frac{\sqrt{\pi}}{6} =\frac{ \pi}{6\sqrt{10}}. $$

Answer 2

Since $Q(x,y)=19x^2+2xy+19y^2$ is a positive definite quadratic form we have $$\iint_{\mathbb{R}^2}\exp(-Q(x,y))\,dx\,dy = \frac{\pi}{\sqrt{19^2-1}}.$$