Existence of $k$-form with nonzero integral Say that $N$ is an oriented, compact, connected manifold without border. If $\operatorname{dim}(N) = k$, does it always exists some $k$-form $\omega$ such that $\int_N \omega \neq 0$?
I know how to proceed in some particular cases (e.g., $S^k$), but I have no idea how to prove the general case,  if it is even true. In the literature I've checked (mostly Lee's Introduction to Smooth Manifolds, 2nd edition, and Guillemin/Pollack Differential Topology) there is nothing, also - but I may have missed something.

Let $(U, \varphi)$ be an oriented local chart of $M$. On $\varphi (U) \subset \mathbb R^k$ you have a $k$-form $\omega_0 = dx^1 \wedge \cdots \wedge dx^k$ and a bump function $b : \varphi(U) \to \mathbb R$ (i.e. a smooth function with compact support in $\varphi(U)$), then
$$\varphi^*(b \omega_0)$$
is a smooth $k$-form on $M$ (by extending to zero outside of $U$). By definition of the integration,
$$ \int_M \omega = \int_{\varphi (U)} b\omega_0 = \int_{\varphi (U)} b(x) dx^1\cdots dx^k.$$
One can choose this to be non-zero (e.g. choose $b\ge 0$ and $b>0$ on an open subset)
