Constant of integration in the context of Connected set

In calculus, the constant of integration, often denoted by ${\displaystyle C}$ (or ${\displaystyle c}$), is a constant term added to an antiderivative of a function ${\displaystyle f(x)}$ to indicate that the indefinite integral of ${\displaystyle f(x)}$ (i.e., the set of all antiderivatives of ${\displaystyle f(x)}$), 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 ${\displaystyle f(x)}$ is defined on an interval, and ${\displaystyle F(x)}$ is an antiderivative of ${\displaystyle f(x),}$ then the set of all antiderivatives of ${\displaystyle f(x)}$ is given by the functions ${\displaystyle F(x)+C,}$ where ${\displaystyle C}$ is an arbitrary constant (meaning that any value of ${\displaystyle C}$ would make ${\displaystyle F(x)+C}$ a valid antiderivative). For that reason, the indefinite integral is often written as ${\textstyle \int f(x)\,dx=F(x)+C,}$ although the constant of integration might be sometimes omitted in lists of integrals for simplicity.