Double integral of exponential function

Question

Evaluate $$ \int^{+\infty}_{-\infty}\, \int^{+\infty}_{-\infty}\, e^{-(3x^{2}+2\sqrt{2}xy+3y^2)}dxdy\,. $$

Answer 1

HINT:

$$\int_{-\infty}^\infty\exp\left(-\text{n}\cdot x^2-\text{m}\cdot x\right)\space\text{d}x=\frac{\sqrt{\pi}}{\sqrt{\text{n}}}\cdot\exp\left(\frac{\text{m}^2}{4\cdot\text{n}}\right)\tag1$$

When $\Re\left(\text{n}\right)>0$

Now, write your integral in the following form:

$$\int^{+\infty}_{-\infty}\, \int^{+\infty}_{-\infty}\, e^{-(3x^{2}+2\sqrt{2}xy+3y^2)}dxdy=\int^{+\infty}_{-\infty}e^{-3y^2}\, \int^{+\infty}_{-\infty}\, e^{-3x^{2}-2\sqrt{2}xy}dxdy\tag2$$