Double Integrals involving infinity Double Integrals involving infinity 
$$
\int_0^\infty\int_0^\infty xye^{-(x^2+y^2)}\,dx\,dy
$$
my work 
let $$t = x^2 +y^2$$
then $$ dt =2x\,dx$$
since $$ye^{-t}$$
$$[ye^{-t}]$$ from $\infty$ to $0$ 
what next stuck here 

For every $a,b>0$ we have
$$
\int_0^a\int_0^bxye^{-(x^2+y^2)}\,dxdy=\left(\int_0^axe^{-x^2}\,dx\right)\left(\int_0^bye^{-y^2}\,dy\right)=\left[-\frac12e^{-x^2}\right]_0^a\left[-\frac12e^{-y^2}\right]_0^b=\frac{(1-e^{-a^2})(1-e^{-b^2})}{4}.
$$
If you take the limit as $a,b\to\infty$ you get:
$$
\int_0^\infty\int_0^\infty xye^{-(x^2+y^2)}\,dxdy=\lim_{a,b\to\infty}\int_0^a\int_0^bxye^{-(x^2+y^2)}\,dxdy=\lim_{a,b\to\infty}\frac{(1-e^{-a^2})(1-e^{-b^2})}{4}=\frac14.
$$
