Evaluate $\int_0^{\infty}\int_0^{\infty}e^{-x^2-2xy-y^2}\,dx\,dy$ I would like to compute the following,
$$
\int_0^{\infty}\int_0^{\infty}e^{-x^2-2xy-y^2}\ dx\,dy
$$
It is obvious that we can rewrite the integral above to,
$$
\int_0^{\infty}\int_0^{\infty}e^{-(x+y)^2}\ dx\,dy
$$
so we are ending up with something looking like a gaussian integral. I think that a smart substitution would help but all I tried ended up to be something I am not able to compute...
I really would appreciate any hint.
Thanks in advance!


Theorem : $$ \iint_A f(x,y)\ dx\,dy=\iint_B g(u,v) |J|\ du\,dv, $$
  where $J$ is Jacobian.

Now, using parametric equations $u=x+y$ and $v=x$ then its Jacobian is $-1$. The corresponding regions are $0<x<\infty\;\Rightarrow\;0<v<\infty$ and $0<y<\infty\;\Rightarrow\;0<u-v<\infty$. Hence
\begin{align}
\int_0^{\infty}\int_0^{\infty}e^{\large-(x+y)^2}\ dx\,dy&=\int_{v=0}^\infty\int_{u=v}^\infty e^{\large-u^2}\ du\,dv\\
&=\int_{u=0}^\infty\int_{v=0}^u e^{\large-u^2}\ dv\,du\\
&=\int_{u=0}^\infty u\ e^{\large-u^2}\ du\qquad;\qquad\text{let}\ t=u^2\;\Rightarrow\;dt=2u\ du\\
&=\frac12\int_{t=0}^\infty \ e^{\large-t}\ dt\\
&=\large\color{blue}{\frac12}.
\end{align}
