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?