= Solution
Give $\partial X$ the <outward-normal-first boundary orientation>. The <Generalized Stokes theorem> states that, for every compactly supported $(n-1)$-form $\omega$,
$$
\int_Xd\omega=\int_{\partial X}F^*\omega.
$$
Choose an oriented coordinate cover by charts into $\mathbb R^n$ or the half-space $\mathbb H^n=\{x^1\geq0\}$, and choose a <partition of unity> $(\rho_i)$ subordinate to it. Since the family is locally finite and $\omega$ has <compact support>, only finitely many $\rho_i\omega$ are nonzero. It is therefore legitimate to write both integrals as finite sums and prove the identity for a form supported in one chart.
In an interior chart the integral of an exact compactly supported top form is zero by the <fundamental theorem of calculus>. In a boundary chart write
$$
\omega=\sum_{j=1}^n(-1)^{j-1}a_j\,dx^1\wedge\cdots\wedge\widehat{dx^j}\wedge\cdots\wedge dx^n.
$$
Integrating $d\omega=(\sum_j\partial_ja_j)dx^1\wedge\cdots\wedge dx^n$ coordinate by coordinate kills every tangential derivative. The normal derivative leaves precisely the restriction to $x^1=0$, with the sign selected by the outward-normal-first convention. This is $\int_{\partial X}F^*\omega$, proving the theorem.
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