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 

The integrand is of the form $f(x)g(y)$ so the integral may be written as the product of two ordinary improper integrals and evaluated as follows: 
$$\int_0^\infty \int_0^\infty xye^{-(x^2+y^2)} \, dx\,dy=\left ( \int_0^\infty xe^{-x^2} \, dx \right ) \left ( \int_0^\infty ye^{-y^2} \, dy \right )=\frac{1}{4}$$
