An antiderivative of a function f(x) is any function F(x) whose derivative equals f(x), written as the integral of f(x) with respect to x. Since differentiation of a constant equals zero, antiderivatives differ by a constant C, making the general form F(x) + C where C represents all possible vertical translations of the antiderivative. This is the real knowscape from knowhere, not a picture of one. Drag it. Watch what actually changes.
Calculus is one machine run in two directions: differentiation takes a function and hands you its rate of change; integration takes the rate and hands the function back. The Fundamental Theorem is the proof they're inverses — which is why every substitution is just differentiation read backwards.
An antiderivative F(x) reverses differentiation so F'(x) equals f(x), and the general antiderivative includes plus C because constants disappear when differentiating.
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