I am trying to make sense of 'linearity' of indefinite integrals.
Let us restrict to the 1-dimensional case. My point is that $\int 0\, dx = C \in \mathbb{R}$, so I cannot really say that $\int$ is a linear operator. Indeed, linearity of $f \colon V \rightarrow W$ ($V,W$ vector spaces) implies $f(0) = 0$.
In order to define $\int \colon V \rightarrow W$ in a good way, I should introduce an equivalence relation $\sim$ on $W$ saying that two elements are in the same equivalence class if they coincide up to a constant. So $\int$ becomes linear as operator $\int \colon V \rightarrow W_{/\sim}$. Assume a primitive of $f$ is $F$. Then $$\int f(x)\,dx = [F(x)] \in W_{/\sim}$$ or, as usual, $F(x)+C, C \in \mathbb{R}$. So I would say that the result of an indefinite integral is actually a coset in some quotient vector space. This even solves the problem that $\int \colon V \rightarrow W $ is not a well defined function.
My question is: is this a good way to think, or is there a better one? I have never seen such a thing, neither in a course of Analysis, nor in the books I have read. I am wondering why. It seems a very natural thing to do when introducing indefinite integrals.