Given two form $\omega$, find $\mu$ such that $d\mu=\omega$?

Question

I am reading a textbook on differential forms and there is a question of the form: given a $k$-form, $\omega$, on say $\mathbb{R^3}$, find $\mu$ such that $d\mu=\omega$. For example, if $\omega= (x_{1}^2 +x_{2}^2)dx_1 \wedge dx_2$, what is $\mu$?

I am looking for a systematic method to solve these types of questions. Thanks

Answer 1

$d\mu=\omega$ implies that $d^2\mu=d\omega=0$. The Poincare Lemma implies that $\omega$ is exact, if you look at a standard proof of this lemma, it constructs $\mu$ such that $d\mu=\omega$ given a closed form $\omega$.

https://en.wikipedia.org/wiki/Closed_and_exact_differential_forms#Poincar%C3%A9_lemma