Let $M$ be an oriented $m$-manifold without boundary, $X\in \mathfrak X(M)$ be a vector field, and $\omega \in \Omega^m_c(M)$ be a compactly supported top-degree differential form.
Question: Do we always have $$\int_M \mathcal L_X\omega=0,$$ where $\mathcal L_X$ is the Lie derivative?
Motivation: The integration operation has a diffeomorphism invariance, i.e., if $F:M\to M$ is an orientation-preserving diffeomorphism, then $$\int_M \omega = \int_M F^*\omega.$$ Now, suppose that the diffeomorphism $F$ is infinitesimal, i.e., $F$ maps each $p\in M$ to a point that is infinitesimally separated from $p$. Such infinitesimal diffeomorphism can be heuristically represented by a vector field $X\in \mathfrak X(M)$, which is an 'arrow' from $p$ to $F(p)$.
Furthermore, the infinitesimal version of the pullback $F^*\omega$ is the Lie derivative $\mathcal L_X\omega$. Hence, I suspect that the infinitesimal statement of the diffeomorphism invariance of integral should be $$\int_M \mathcal L_X\omega=0.$$