Poincare duality on non closed manifolds

Poincare duality on non closed manifolds

For proving the Poincare duality $H_{\mathrm{dR},p}\cong
H_\mathrm{dR}^{m-p}$ one can use the bilinear form
$B:H_\mathrm{dR}^p\times H_\mathrm{dR}^{m-p}\to\mathbb{R}$ given by
$B([\omega],[\beta]):=\int_M\omega\wedge\beta$. $B$ only depends on the
cohomolgy class $[\omega]$ since if $\omega_1,\omega_2\in[\omega]$ one has
for a closed $(m-p)$-form $\beta$
$$\int_M\omega_1\wedge\beta=\int_M\omega_2\wedge\beta-\int_M\mathrm{d}\left(\alpha\wedge\beta\right)=\int_M\omega_2\wedge\beta$$
if $M$ is closed and orientable.
However, some authors (e.g. Jost) only assume $M$ to be compact and
orientable and still use the equation above in their proofs. I am aware
that there is chomolgy with compact support, but as far as I can see the
equation only holds if $\partial M=\varnothing$, i.e. $M$ is closed. What
am I missing?