Antiderivatives and the constant C

Differentiation takes a function and returns its rate of change. Integration asks the reverse question: which function F has the given f as its derivative? Any such F is an antiderivative (or primitive) of f.

Antiderivatives are never unique. The derivative of a constant is zero, so if F′ = f then (F + C)′ = f for every real C. On an interval, any two antiderivatives of the same f differ only by a constant.

The whole family is the indefinite integral: ∫f(x) dx = F(x) + C. Here f(x) is the integrand, x the variable of integration and C the constant of integration.

Geometrically the graphs of F(x) + C are copies of one curve slid up or down. Above any chosen x they all have the same slope, so their tangents there are parallel.

One known point on the curve fixes C. Example: if F′(x) = 4x³ − 6 and F(0) = 3, then F(x) = x⁴ − 6x + C and F(0) = 3 gives C = 3, so F(x) = x⁴ − 6x + 3.

Differentiation and integration undo each other up to a constant: the derivative of ∫f(x) dx is f(x), while ∫f′(x) dx is f(x) + C.

A derivative has a meaning at a single point. An indefinite integral does not: it describes a family of functions over an interval.