Let $M$ be a n-dimensional closed smooth manifold, $\eta_{k}$ be the k-form on $M$, $X$ be the smooth vector field on $M$.Let $\eta=\eta_{0}+\eta_{1}+\cdots+\eta_{n-1}+\eta_{n}$ be a ($d-i_{X}$)-closed form, then we have $$i_{X}\eta_{n}=d\eta_{n-2},$$ the relation imply that $\eta_{n}$ is exact outside the set $M_{0}$ of zeros of the vector field $X$. How to get the result?

Edit: I read this result in the chapter 7 of "Heat Kernels and Dirac Operators", page 203.