Elegant Meaning of an Integral

For the formalists out there I have been troubled by the meaning of,

$\displaystyle \int 2x dx=x^2+C$

Because what is that supposed to represent?

It is more appropriate to think of an integral as a **set** of functions rather than a function.

So here is my version, tell me if you like it.

Let $\displaystyle \mathcal{R}$ represent the set of all real functions that are differenciable on some open interval. Let $\displaystyle R$ represent the set of all functions. *)

We can view the sets $\displaystyle \mathcal{R}$ and $\displaystyle R$ as groups under functional addition.

Then we can define the *derivative* as a group homomorphism,

$\displaystyle d:\mathcal{R}\to R$

I think the beauty in this is defining what an *integral* is. Given any function $\displaystyle f\in R$ we define,

$\displaystyle \int f = d^{-1}[f]$

*)Since I am not a set theorist (noob :mad: ) I cannot tell whether such a set can exist.

Note: It was not necessary for me to mention a group homomorphism I just was thinking whether the kernel of the derivative will be of any use. I do not see of one.