I've encountered with the following problem:
Consider the map
$$\int: \Omega_c^n(\Bbb R^n)\to\Bbb R$$
$$\alpha(x)dx^1\land...\land dx^n\mapsto \int_{\Bbb R^n}\alpha(x)dx^1\land...\land dx^n$$
It's trivial that this map$\int$ is a homomorphism,here comes the question:
Is it true that $Ker\int$={exact $n$ form with compact support}?
If $\omega$ is an $(n-1)$-form with compact support $C,$ there exist $p\in\mathbb{R}^n$ and $r>0$ such that $C\subset B(p,r).$ Then, from Stokes' theorem
$$\int_{B(p,r)}d\omega=\int_{\partial B(p,r)}\omega=0,$$ since $\omega|_{\partial B(p,r)}\equiv 0.$ This shows that any exact form with compact support belongs to the kernel.
To show the converse, that is, that any form in the kernel is exact, have a look at Lemma $8.1$ in Hitchin
