Assume we have the $2$-form $\omega = \frac 1 {2 \pi i} \frac {dz \wedge d\overline z}{(1 + z \overline z)^2}$ given on $U = \{[1:z] : z \in \mathbb C \} \subset \mathbb C \mathbb P^1$, where $z: U \to \mathbb C$ is the map $[1:z] \mapsto z$.
How can I integrate this form, i.e. what is $\int_U \omega$ ?