Fundamental Theorem of Calculus for Differential Forms. Conditions?

Question

Let w be a k-form with compact suport in a k-manifold M ( say $\mathbb R^{k}$ to simplify, but I would appreciate generalizations to k-manifolds ) and define $F:= \int_{\gamma} w $, for $\gamma$ a k-submanifold .Under what conditions do we have : $df=w$? This seems equivalent to passinng the differential ( exterior derivative) from the outside-in, i.e., when can we say that $d \int_{\gamma} w= \int_{\gamma} dw$? I am pretty sure this has to see with Stoke's theorem and DeRham's theorem, and whether w is exact, but not sure how. Thanks for any ideas.