Constant of integration
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). For that reason, the indefinite integral is often written as ∫ f (x) d x = F (x) + C, {\textstyle \int f(x)\,dx=F(x)+C,} although the constant of integration might be sometimes omitted in lists of integrals for simplicity.
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