Constant of integration

In calculus [Calculus], the constant of integration, often denoted by \(C\) (or \(c\)), is a constant term added to an antiderivative [Antiderivative] of a function \(f(x)\) to indicate that the indefinite integral of \(f(x)\) (i.e., the set of all antiderivatives of \(f(x)\)), on a connected domain, is only defined up to an additive constant. This constant expresses an ambiguity inherent in the construction of antiderivatives.

More specifically, if a function \(f(x)\) is defined on an interval, and \(F(x)\)(x)} is an antiderivative of \(f(x)\), then the set of all antiderivatives of \(f(x)\) is given by the functions \(F(x) + C\), where \(C\) is an arbitrary constant (meaning that any value of \(C\) would make \(F(x) + C\) a valid antiderivative). For that reason, the indefinite integral is often written as \(\int f(x)\,dx=F(x)+C\), although the constant of integration might be sometimes omitted in lists of integrals for simplicity.

(“Constant of Integration” 2022)

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