In my notes of differential topology, I read the following:
If $\omega$ is a $n$-form on $\mathbb{R}^n$, then $\int_Ud\omega=0$ (if the form has compact support).
However, if I differentiate the $1$-form on $\mathbb{R}^2 \omega = y^2dx+xdy$ and then integrate on a generic compact subset of $\mathbb{R}$, I don't get zero. Where am I wrong?