The literature, at least textbooks, seems to be very scarce on the topic of integrating bundle-valued differential forms. So I wonder where can I read on the topic? I want to see usual theorems, like change of variables formula, Stokes' theorem, ideally fiber integration.
I understand that such integration requires addition of elements in different fibers, so as I understand some additional structure is required, like connection. I'm mostly interested right now in integrating vector-valued $\Omega^n(M,TM)$ and k-form valued $\Omega^n(M,\Lambda^k T^*M)$ differential forms.