In calculus, the constant of integration, often denoted by
), is a constant term added to an antiderivative of a function
), on a connected domain, is only defined up to an additive constant.
[1][2][3] This constant expresses an ambiguity inherent in the construction of antiderivatives.
is an arbitrary constant (meaning that any value of
For that reason, the indefinite integral is often written as
The constant is a way of expressing that every function with at least one antiderivative will have an infinite number of them.
making the goal to prove that an everywhere differentiable function whose derivative is always zero must be constant: Choose a real number
For any x, the fundamental theorem of calculus, together with the assumption that the derivative of
Two facts are crucial in this proof.
First, the real line is connected.
If the real line were not connected, one would not always be able to integrate from our fixed a to any given x.
For example, if one were to ask for functions defined on the union of intervals [0,1] and [2,3], and if a were 0, then it would not be possible to integrate from 0 to 3, because the function is not defined between 1 and 2.
Here, there will be two constants, one for each connected component of the domain.
In general, by replacing constants with locally constant functions, one can extend this theorem to disconnected domains.
, so for example, the general form for the integral of 1/x is:[5][6]
are not differentiable at even one point, then the theorem might fail.
are everywhere continuous and almost everywhere differentiable the theorem still fails.
It turns out that adding and subtracting constants is the only flexibility available in finding different antiderivatives of the same function.
It is easily determined that all of the following functions are antiderivatives of
The inclusion of the constant of integration is necessitated in some, but not all circumstances.
For instance, when evaluating definite integrals using the fundamental theorem of calculus, the constant of integration can be ignored as it will always cancel with itself.
However, different methods of computation of indefinite integrals can result in multiple resulting antiderivatives, each implicitly containing different constants of integration, and no particular option may be considered simplest.
Additionally, omission of the constant, or setting it to zero, may make it prohibitive to deal with a number of problems, such as those with initial value conditions.
The constant of integration also implicitly or explicitly appears in the language of differential equations.
Almost all differential equations will have many solutions, and each constant represents the unique solution of a well-posed initial value problem.
An additional justification comes from abstract algebra.
The space of all (suitable) real-valued functions on the real numbers is a vector space, and the differential operator
The process of indefinite integration amounts to finding a pre-image of a given function.
In this context, solving an initial value problem is interpreted as lying in the hyperplane given by the initial conditions.