What is "constant of integration"

Constant \Con"stant\, n.

1. That which is not subject to change; that which is invariable.

2. (Math.) A quantity that does not change its value; -- used in countradistinction to variable.

3. (Astron.) A number whose value, when ascertained (as by observation) and substituted in a general mathematical formula expressing an astronomical law, completely determines that law and enables predictions to be made of its effect in particular cases.

4. (Physics) A number expressing some property or condition of a substance or of an instrument of precision; as, the dielectric constant of quartz; the collimation constant of a transit instrument.

5. (Computers) a data structure that does not change during the course of execution of a program. It may be a number, a string, or a more complex data structure; -- contrasted with variable.

   Aberration constant, or Constant of aberration (Astron.), a number which by substitution in the general formula for aberration enables a prediction to be made of the effect of aberration on a star anywhere situated. Its value is 20[sec].47.

   Absolute constant (Math.), one whose value is absolutely the same under all circumstances, as the number 10, or any numeral.

   Arbitrary constant, an undetermined constant in a differential equation having the same value during all changes in the values of the variables.

   Gravitation constant (Physics), the acceleration per unit of time produced by the attraction of a unit of mass at unit distance. When this is known the acceleration produced at any distance can be calculated.

   Solar constant (Astron.), the quantity of heat received by the earth from the sun in a unit of time. It is, on the C. G. S. system, 0.0417 small calories per square centimeter per second.
   --Young. [Webster 1913 Suppl.]

   Constant of integration (Math.), an undetermined constant added to every result of integration.

n. (context calculus English) An unspecified constant term added to a particular antiderivative to make it represent its whole family of antiderivatives, usually denoted with C.

In calculus, the indefinite integral of a given function (i.e., the set of all antiderivatives of the function) on a connected domain is only defined up to an additive constant, the constant of integration. This constant expresses an ambiguity inherent in the construction of antiderivatives. If a function f(x) is defined on an interval and F(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 for C makes F(x) + C a valid antiderivative). The constant of integration is sometimes omitted in lists of integrals for simplicity.