Kernel of the integration map $\int : \ker(d)\subset \Omega^n_c(M^n)\to\mathbb{R}$ .

Question

Let $\Omega^n_c(M^n)$ be the set of all compactly supported $n$ forms on a connected, oriented manifold $M$ of dimension $n$. Then there is the map $\int : \ker(d)\subset \Omega^n_c(M^n)\to\mathbb{R}$. I wish to show $\int_M\omega=0$ for any exact form $\omega=d\tau$, say. By stokes' theorem, enough to show that $\int_{\partial M}\tau=0$. I'm not sure why the last integral is zero, any hint will be helpful. Thank you.