In calculus, the constant of integration), is a constant term added to an antiderivative to indicate that the indefinite integral (i.e., the set), on a connected domain, is only defined up to an additive constant. This constant expresses an ambiguity inherent in the construction of antiderivatives.

is defined on an interval then the set of all is an arbitrary constant (meaning that any although the constant of integration might be sometimes omitted in lists of integrals for simplicity.

Origin

The derivative The constant is a way of expressing that every function with at least one antiderivative will have an infinite number of them.

for every real number x for every real number x.

making the goal to prove that an everywhere differentiable function whose derivative is always zero must be constant:

For any x, the fundamental theorem of calculus vanishes, implying that

is a constant function.

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, so for example, the general form for the integral of 1/x is:

be the Heaviside step function, which is zero for negative values of x and one for non-negative values of x are everywhere continuous and almost everywhere to be the Cantor function

is constant of integration

Significance

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.

can be integrated in at least three different ways.

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 that has the value 400 at x).

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 is a linear operator maps a function to zero if and only if that function is constant. Consequently, the kernel is the space of all constant functions. The process of indefinite integration amounts to finding a pre-image of a given function. There is no canonical pre-image for a given function, but the set of all such pre-images forms a coset. Choosing a constant is the same as choosing an element of the coset. In this context, solving an initial value problem is interpreted as lying in the hyperplane given by the initial conditions.

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