Viewing integration as a $\mathbb{Z}$ action on function spaces Note: This question has been edited thanks to a comment by Chris Culter. Thank you for this comment.

The question:

I was fooling around and thought of a perspective which i thought to be quite curious:

Consider the space $X = C^\infty(\mathbb{R})/\{\textrm{polynomials}\}$, quotiented via addition (quotient is a quotient of abelian groups and not rings). Given $\bar{f} \in X$, we may lift it to a $C^\infty$ function $f: \mathbb{R} \to \mathbb{R}$, we define $(\int)^n \bar{f}$ to be: 

(1) For $n$ positive, the integral $\overline{\int \int ... \int f}$, ($n$ times).

(2) For $n \le 0$, $\overline{\textrm{the $n$'th derivative of $f$}}$. 

One can check that this is well defined on the quotient space, and:

Then, one can easily see that $\int^{m+n}\bar{f} = \int^n (\int^m \bar{f})$, and $\int^0 \bar{f} = \bar{f}$. Thus, we see that $n \mapsto \int^n$ defines a $\mathbb{Z}$ action on $X$.

Questions: 

(1) Has anybody come across this perspective? Is this perspective useful?

(2) One question one might ask about this is what are the orbits (and finite orbits). For example, $\{ \sin, \cos, -\sin, -\cos \}$ is an interesting orbit of size $4$. Can anyone think of some nice results along these lines?
$\mathbb{Z}$ is amenable so we have:
Fixed-point property. Any action of the group by continuous affine transformations on a compact convex subset of a (separable) locally convex topological vector space has a fixed point. For locally compact abelian groups, this property is satisfied as a result of the Markov–Kakutani fixed-point theorem.
This might be interesting
$\mathbb{Z}$ is amenable so we have:
Fixed-point property. Any action of the group by continuous affine transformations on a compact convex subset of a (separable) locally convex topological vector space has a fixed point. For locally compact abelian groups, this property is satisfied as a result of the Markov–Kakutani fixed-point theorem.
This might be interesting