Find integral $\int_{R^2}\; \exp (-x^2-xy-y^2)\,dl_2(x,y)$

Question

Find integral $$\int_{R^2} \exp (-x^2-xy-y^2)\,dl_2(x,y)$$

Should I use Fubini theorem and divide it into 2 separate integrals? Still not sure how to do it.

Answer 1

Hint: complete the square in the exponential.

Answer 2

Assuming that $l_2(x,y)$ is the Lebesgue measure on $\mathbb{R}^2$, we have:

$$ I = \int_{\mathbb{R}^2} e^{-(x+y/2)^2-3y^2/4}\, d\mu = \int_{\mathbb{R}^2}e^{-w^2-3y^2/4}\,d\mu=\frac{2}{\sqrt{3}}\int_{\mathbb{R}^2}e^{-(w^2+v^2)}\,d\mu$$ so $I=\color{red}{\frac{2\pi}{\sqrt{3}}}.$