Is f(x)dx in integral a differential?

Question

Is $dx$ in $\int f(x)dx$ a differential of the identity function? In Zorich analysis, it is said that if $dF = f(x)dx$, then $\int dF=\int f(x)dx$. This only makes sense if $dx$ actually represents a differential, so that $f(x)dx$ is actually a function that maps each point into a linear function (i.e $\mathbb{R}\to(\mathbb{R}\to\mathbb{R})$). So in some sense we are integrating against not a function from $\mathbb{R}$ to $\mathbb{R}$, but a function from $\mathbb{R}$ to $\mathbb{R}\to\mathbb{R}$ (or in particular to the set of linear functions), otherwise th $\int dF$ and the equality $\int dF = \int f(x)dx$ won't make sense. Is this true or am I mising something here.