Constant of integration


In calculus, the constant of integration, often denoted by (or ), is a constant term added to an antiderivative of a function to indicate that the indefinite integral of (i.e., the set of all antiderivatives of ), 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.

More specifically, if a function is defined on an interval, and is an antiderivative of then the set of all antiderivatives of is given by the functions where is an arbitrary constant (meaning that any value of would make a valid antiderivative). For that reason, the indefinite integral is often written as [4] although the constant of integration might be sometimes omitted in lists of integrals for simplicity.



The derivative of any constant function is zero. Once one has found one antiderivative   for a function   adding or subtracting any constant   will give us another antiderivative, because   The constant is a way of expressing that every function with at least one antiderivative will have an infinite number of them.

Let   and   be two everywhere differentiable functions. Suppose that   for every real number x. Then there exists a real number   such that   for every real number x.

To prove this, notice that   So   can be replaced by   and   by the constant function   making the goal to prove that an everywhere differentiable function whose derivative is always zero must be constant:

Choose a real number   and let   For any x, the fundamental theorem of calculus, together with the assumption that the derivative of   vanishes, implying that


thereby showing 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, one can extend this theorem to disconnected domains. For example, there are two constants of integration for  , and infinitely many for  , so for example, the general form for the integral of 1/x is:[5][6]


Second,   and   were assumed to be everywhere differentiable. If   and   are not differentiable at even one point, then the theorem might fail. As an example, let   be the Heaviside step function, which is zero for negative values of x and one for non-negative values of x, and let   Then the derivative of   is zero where it is defined, and the derivative of   is always zero. Yet it's clear that   and   do not differ by a constant, even if it is assumed that   and   are everywhere continuous and almost everywhere differentiable the theorem still fails. As an example, take   to be the Cantor function and again let  

It turns out that adding and subtracting constants is the only flexibility available in finding different antiderivatives of the same function. That is, all antiderivatives are the same up to a constant. To express this fact for   one can write:   where   is constant of integration. 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. For example,   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. A general solution containing the arbitrary constant is often necessary to identify the correct particular solution. For example, to obtain the antiderivative of   that has the value 400 at x = π, then only one value of   will work (in this case  ).

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. The operator   maps a function to zero if and only if that function is constant. Consequently, the kernel of   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.


  1. ^ Stewart, James (2008). Calculus: Early Transcendentals (6th ed.). Brooks/Cole. ISBN 0-495-01166-5.
  2. ^ Larson, Ron; Edwards, Bruce H. (2009). Calculus (9th ed.). Brooks/Cole. ISBN 0-547-16702-4.
  3. ^ "Definition of constant of integration |". Retrieved 2020-08-14.
  4. ^ Weisstein, Eric W. "Constant of Integration". Retrieved 2020-08-14.
  5. ^ "Reader Survey: log|x| + C", Tom Leinster, The n-category Café, March 19, 2012
  6. ^ Banner, Adrian (2007). The calculus lifesaver : all the tools you need to excel at calculus. Princeton [u.a.]: Princeton University Press. p. 380. ISBN 978-0-691-13088-0.