I am looking for a version of Stokes Theorem on manifolds that does not assume compact support of the differential form $\omega$. In fact I have found the following statement in the book Riemannian Geometry And Geometric Analysis by Jost:
Theorem 2.1.6 Let $M$ be an orientable $m$-dimensional differentiable submanifold of the differentiable manifold $N$ with smooth boundary. That is, $\partial M$ is assumed to be an $(m-1)$-dimensional differentiable submanifold of $N$, and let $\omega$ be a smooth $(m-1)$-form on $N$. Then, provided that these integrals exist, for instance if $\overline{M}$ is compact,
$\int_Md\omega = \int_{\partial M}\omega$.
I am having some doubts about this result, and found something which I believe might be a counterexample.
Choose $N=\mathbb{R}$, $\overline{M}= [0,\infty)$, such that $\partial M = \{0\}$. Let $\omega = f$ be a function which is bounded towards infinity, $f(\infty):=\lim_{x\to \infty}f(x)<\infty$. Integrating, we then have
$\int_Md\omega = \int_0^\infty f'dx = f(\infty) - f(0)$, but $\int_{\partial M}\omega = -f(0)$.
This happens because the limit point $\{\infty\}$ is not viewed as a boundary of $\mathbb{R}$. Is this a valid counterexample or did I make some mistake? How could we fix this Theorem to still hold, without assuming compact support?