What is the difference between a parametric equation and a mathematic law? 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.
 A: Given the definition of mathematical law and parametric equation listed by @user10389, the result of an integration is a parametric equation. The definite integration operator essentially gives you the measure bounded by the function you're integrating, so it is redefining the measure in terms of the function.
As far as the integration constant, you may find it helpful to think of the indefinite integration operator as a halfway step- it tells you something like what the potential energy of the thing you're integrating is. The integration constant just says that the only thing that's important is the difference between potentials, not their actual value. 
A: I am unaware of any formal mathematical way to define a concept of "law", though it may be successfully used in the reference of some other science as physics or philosophy.
Informally, one can model or describe such laws of "nature" by using equations. For example, the process of finding the solution to differential equation, which would be a function, is called integration. This is similar to a more simple situation, when some functions that can be expressed polynomially $(a_n x^n + a_{n-1} x^{n-1} + ..)$ or analytically, once assigned to some value on the right hand - form respective equations, which can be solved as well.
The axiomatics of formal languages is another example.
A: The integral of $f$ from $a$ to $b$ gives the area between the graph of the function $f$ and the horizontal axis in the interval from $a$ to $b$.
It is an important theorem that this integral can be expressed by $F(b)-F(a)$ for any function that happens to have the property $F'=f$.
Notice that these properties do not change when you add a constant to $F$.
If $b$ is a variable $t$ and $a$ is a constant, say 0, then the constant of integration keeps track of the fact that you should subtract $F(a)$, that is, you have to make sure that you choose your constant so that your integral gives 0 for $t=0$, as it clearly should.
