Let's recall the definition of integral of a compactly supported smooth $m$-form $\omega$ over an orientable $m$-differential manifold $M$: if $\{(U_\alpha,\phi_\alpha )\}$ is a differentiable atlas, and $\{ \rho_\alpha\}$ is a partition of unity subordinate to $\{U_\alpha\}$, then we define $\int_M \omega := \sum_\alpha \int_{\phi_\alpha (U_\alpha)} (\phi_\alpha ^{-1} )^*(\rho_\alpha \omega )$ where the latter integrals are just usual Lebesgue integrals over open sets of $\mathbb R ^m $. It can be proved that definition does not depend on partition of unity or atlas we choose.
I am wondering if we can define an integral for non-compactly supported form. More specifically: if we have a $m$-form $\omega$, and we choose a countable atlas $\{\phi_n , U_n \}$ of $M$ and a partition of unity $\{\rho_n\}$, then the limit (if it exists) $\lim_{n\to +\infty} \sum_{k=1} ^n \int_{\phi_n (U_n)} (\phi_n ^{-1})^*(\rho_n \omega)$ depends on the choice of the atlas of the partition of unity?
In complete generality, this is not possible. For example, consider the $1$-form $\omega = \sin x\,dx$ on $\mathbb R$. It's possible to find a countable open cover and partition of unity such that half of the terms $\int_{\phi_n(U_n)}(\phi_n^{-1})^*(\rho_n\omega)$ are equal to some constant $c$, and the other half are equal to $-c$. The limit of the sum depends strongly on the order you choose for the summation.
There are various situations in which it is possible to make sense of such an integral. For example, if $M$ is oriented and the form $\omega$ is everywhere nonnegative with respect to the given orientation, then the integral can be interpreted as a sum of nonnegative terms, so it either converges absolutely or diverges to $+\infty$.
Alternatively, if $M$ is endowed with a Riemannian metric, then every $m$-form can be written as a function times the Riemannian volume form, $\omega = f\, dV_g$, and you can interpret $\int_M f\, dV_g$ as the Lebesgue integral with respect to the measure $\mu(E) = \int_E 1\, dV_g$. This allows you to define the Lebesgue spaces $L^p(M)$ consisting of functions $f$ such that $|f|^p$ is integrable, and the Sobolev spaces $W^{k,p}(M)$ consisting of functions whose first $k$ covariant derivatives are in $L^p(M)$.
These constructions are used regularly in the study of PDEs on manifolds.