Let $M$ be a closed (smooth) $2$-manifold, let $\omega$ be a (smooth) $1$-form on $M$, and let $V$ be a (smooth) vector field on $M$. I am trying to show that if $i_V d\omega=df$ for some $f\in C^\infty(M)$, where $i_V$ is the interior multiplication by $V$, then $$ \int_M fd\omega=\int_M i_V \omega ~d\omega$$
A consequence of the assumptions is : $L_V \omega=i_V d\omega+di_V \omega=df+di_V \omega=d(f+i_V \omega)$, so the Lie derivative $L_V\omega$ of $\omega$ by $V$ is exact. (The first equality is Cartan's magic formula) In particular, $L_V d\omega$ is zero since $L_V d\omega =dL_V \omega$.
Actually I'm not sure that this is a relevant information. Any hints?