So. In calculus both the derivative and integral are linear operators:

   D[a*f(x)] = a*Df[x]
   D[f(x)+g(x)] = Df(x) + Dg(x)

and the indefinite (without limits) integral is an "antiderivative", right? I.e. y(x) = I(f(x)) is the solution to

   D[y(x)] = f(x)

Here's the problem: there are multiple solutions to the case where f(x) = 0. Indeed, for any constant y(x) you have

   D[y(x)] = 0

This is why you're drilled in engineering classes to always add a + C to your indefinite integral. The solution to an indefinite integral is always a class of functions -- the part without +C continues to be linear, but you have to tag that along.

This is also why Initial Value Problems like

    D[f(x)] = g(x)

always need an initial condition.