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!

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$\ds{\int_{0}^{\infty}\int_{0}^{\infty}\expo{-\pars{x + y}^{2}}\,\dd y\,\dd x:\
     {\large ?}}$

\begin{align}
&\color{#66f}{\large%
\int_{0}^{\infty}\int_{0}^{\infty}\expo{-\pars{x + y}^{2}}\,\dd y\,\dd x}
\\[5mm] = &\
\int_{0}^{\infty}\int_{x}^{\infty}\expo{-y^{2}}\,\dd y\,\dd x
=\left.\int_{0}^{\infty}\int_{0}^{\infty}\expo{-y^{2}}\,\dd y\,\dd x
\right\vert_{y\ >\ x}
\\[3mm]&=\left.\int_{0}^{\infty}\expo{-y^{2}}\int_{0}^{\infty}\,\dd x\,\dd y
\right\vert_{x\ <\ y}
=
\int_{0}^{\infty}\expo{-y^{2}}\int_{0}^{y}\,\dd x\,\dd y
=\int_{0}^{\infty}y\expo{-y^{2}}\,\dd y
\\[5mm] = &\
\left.-\,\half\,\expo{-y^{2}}\right\vert_{0}^{\infty} =
\color{#66f}{\Large\half}
\end{align}

