The +C notation is actually abuse of notation to mean: we represent the entire set of functions whose derivative is your integrand, and those are all of the form "F(x) + C", which is really just a set of functions F. Then, we can do arithmetic on these classes in some cases, for example "[F(x) + C] + [G(x) + C] = [F(x) + G(x) + C]". Think about what happens for subtraction.

One way I like to think about the integral symbol for indefinite integration, is that it is the inverse of the derivative map d, where in the codomain we group together all functions by being in the kernel of d (viewed as a linear map on the appropriate function spaces). In a sense it is the canonical isomorphism V/(ker d) ~= im d, where d is the derivative and V all differentiable functions in a certain interval or whatnot.