Constant of integration

Constant expressing ambiguity from indefinite integrals

In calculus, the constant of integration, often denoted by C {\displaystyle C} (or c {\displaystyle c}), is a constant term added to an antiderivative of a function f ( x) {\displaystyle f(x)} to indicate that the indefinite integral of f ( x) {\displaystyle f(x)} (i.e., the set of all antiderivatives of f ( x) {\displaystyle 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) {\displaystyle f(x)} is defined on an interval, and F ( x) {\displaystyle F(x)} is an antiderivative of f ( x) , {\displaystyle f(x),} then the set of all antiderivatives of f ( x) {\displaystyle f(x)} is given by the functions F ( x) + C , {\displaystyle F(x)+C,} where C {\displaystyle C} is an arbitrary constant (meaning that any value of C {\displaystyle C} would make F ( x) + C {\displaystyle F(x)+C} a valid antiderivative).

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