Let $M$ be a real, smooth manifold. Let $\omega_1$ and $\omega_2$ be differential 1-forms on $M$, and let $C_1(\mathbb Z,M)$ denote the set of 1-chains with integer coefficients.
If \begin{align} \int_c\omega_1 = \int_c\omega_2 \end{align} for all $c\in C_1(\mathbb Z,M)$, then does $\omega_1=\omega_2$?
Yes. Let $\omega$ be a 1-form on $M$. You need to show that $\omega$ vanishes if its integral along all curves in $M$ vanishes. Fix a point $x\in M$ and $v\in T_x M$ and integrate $\omega$ along a curve starting in $x$ in the direction of $v$. By making the curve shorter and shorter, these integrals give better and better approximations to $\omega(v)$, hence $\omega(v)$ must vanish.