How to integrate differential form actually. As far as I know, a differential form is a multilinear function mapping from a vector space to a real number. Let's take $\int_c fdx+gdy$ as an example. It is integrating a differential $1$-form while I don't quite get the meaning of the integral process.
Shouldn't the function act on something, like the tangent vector in the tangent space? For instance $dx^i(e_p^i)=1$ however when we are doing integration, we won't add a tangent vector next to $dx$, why?