Determine convergence/divergence of $$\iint xye^{-xy}\,dx\,dy$$ for $x,y \geqslant 0$ i.e. in the first quadrant.
I have managed to show that $xye^{-xy} \to 0$ in the first quadrant but other than that not gotten very much far unfortunately.
One thought I had was to use polar coordinates and for some radius $r_0$ approximate $xye^{-xy} \thicksim e^{-xy} $. I would then want to investigate $$ \iint e^{-xy} \,dx\,dy$$ in the first quadrant from radius $r_0$ to infinity (the angle lies between zero and $\pi/2$). However that integral was no more easier than the previous one.
Thoughts?
Polar coordinates work, making the shift we get:
$${1\over 2}\int_0^{\pi/2}\int_0^\infty r^3\sin(2\theta)e^{-r^2\sin(2\theta)/2}\,dr\,d\theta$$
Now make the $u$-substitution $u={r^2\over 2}\,du=rdr$
$$\int_0^{\pi/2}\sin(2\theta)\int_0^\infty ue^{-u\sin(2\theta)}\,du\,d\theta$$
Integrating by parts, we get the $u$ terms integrate to:
$$\bigg(-\csc^2(2\theta)e^{-u\sin(2\theta)}-u\csc(2\theta)e^{-u\sin(2\theta)}\bigg|_0^\infty=-\csc^2(2\theta).$$
hence our integral reduces to
$$-\int_0^{\pi/2}\sin(2\theta)\left(\csc^2(2\theta)\right)\,d\theta=\int_0^{\pi/2}\csc(2\theta)\,d\theta$$
And $\csc(2\theta)$ is not Riemann integrable on that interval (it is unbounded).