> Integrating the 0 function gives us back the 0 function so it's a kind of counter-example to the fundamental theorem of calculus. I guess that is one way to

> Integrating the 0 function gives us back the 0 function so it's a kind of counter-example to the fundamental theorem of calculus. I guess that is one way to think about it, but if that is the case you can just differentiate a constant function and you the fundamental theorem of calculus won't return it to you (if you don't like that example just take the Heaviside function). In finite dimensional linear spaces, the derivative operator has a rank 1 kernel, but the integral does not. From that perspective, the derivative destroys information (specifically it destroys constants), the integral operator cannot restore it automatically and the whole + C is meant to identify an equivalence class for the indefinite anti-derivative.