  Integral categories were recently developed as a counterpart to differential
categories. In particular, integral categories come equipped with an
integration operator, known as an integral transformation, whose axioms
generalize the basic integration identities from calculus such as integration
by parts. However, the literature on integral categories contains no example
that captures integration of arbitrary smooth functions: the closest are
examples involving integration of polynomial functions. This paper fills in
this gap by developing an example of an integral category whose integral
transformation operates on smooth 1-forms. We also provide an alternative
viewpoint on the differential structure of this key example, investigate
derivations and coderelictions in this context, and prove that free
$C^{\infty}$-rings are Rota-Baxter algebras.
