I want to see an explicit example of integral de Rham cohomology to gain some intuition that how it work really. More precisely (if I am not mistaken)
A $k$-form $\omega$ is integral (i.e. $[\omega]\in\mathbf{H}^k_{\text{dR}}(M,\Bbb Z)$) iff its integral over all (smooth) singular $k$-cycles is an integer; i.e. $$\forall\ \Sigma\subset M,\ \partial\Sigma=0, \text{and compact}\qquad\int_\Sigma\omega\in\Bbb Z.$$
I need an example that have more choices for $\Sigma$, not like sphere that all $\Sigma$ (I think all of them are $k$-spheres) are homotopic to each other. Obviously not tricky example using Stokes' theorem.