Hello,
Originally Posted by
Spec D is the positive quadrant.
$\displaystyle \int \int_D \frac{xydxdy}{(1+x^2+y^2)^3}$
Switching to polar coordinates doesn't help much.
Using polar coordinates:
$\displaystyle \begin{aligned}\iint_D \frac{xy}{(1+x^2+y^2)^3}\,\mathrm{d}x\,\mathrm{d}y
&=\int_0^{\frac{\pi}{2}}\int_0^{\infty} \frac{\rho \cos \theta \rho \sin \theta}{(1+\rho^2)^3}\,\rho\,\mathrm{d}\rho\,\math rm{d}\theta\\
&=\int_0^{\frac{\pi}{2}}\cos \theta \sin\theta \,\mathrm{d}\theta
\int_0^{\infty}\rho^2 \frac{\rho}{(1+\rho^2)^3}\,\mathrm{d}\rho
\end{aligned}$
To compute $\displaystyle \int_0^{\infty}\rho^2 \frac{\rho}{(1+\rho^2)^3}\,\mathrm{d}\rho
$ you can try integration by parts (let $\displaystyle u'=\frac{\rho}{(1+\rho^2)^3}$ and $\displaystyle v=\rho ^2$).