First of all sorry for my English, I'm not used to communicate with this language. I want to ask something about a thing that I discovered while studying physics (AKA applied mathematics).
There is this strange operator $\int$ that integrates, this not so curly "S" that means "infinite sum of infinitesimal quantity/numbers", but not only it does "a sum", but it also takes advantage of the "infinitesimal environment" to "linearize" every kind of figure. Ok? (I don't even know if I'm explaining this in the right way...)
So this guy is kind of an analytic and a geometric operators mixed together, and here comes my question: What is the result of an integration?
Is the result of an integration still a mathematical law? Is it a parametric equation? What's the analytical difference between the two, omitting all the rest we can say about integrals and the entire math?
Usually I keep hearing that "the derivative lowers the grade and the integral raises", but this is not at all about the grade. Also, the grade is only part of an analytical dissertation about a law or a resolution of a problem.
I hope that I'm explaining my question in the right way, thanks for all the replies.
PS. I also think that if I understood this better, I could really understand what the integration constant means.