I'm trying to connect the idea of covectors to integration.

From what I understand, given a basis $\{e\_1, \dots, e\_n\}$ of a vector space $V$, there exists a dual basis of covectors $\{f^1, \dots, f^n\}$ of the dual space $V^\*$ such that $\langle f^i, e\_j\rangle = {\delta^i}\_j$.

A line integral in $\Bbb R^n$ has the form $$\int\_\gamma \vec f \cdot d\vec r = \int\_\gamma f\_{x^1}dx^1 + f\_{x^2}dx^2 + \cdots + f\_{x^n}dx^n$$

Where are the covectors in the above expression? Does $dx^i$ really mean $\langle f^i, x^ie\_i\rangle$? That doesn't seem right, because isn't $dx^i$ supposed to be some kind of derivative (the exterior derivative)?