Arbitrary constant of integration
Encyclopedia
In calculus
, the indefinite integral of a given function (i.e., the set of all antiderivative
s of the function) 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
is defined on an interval
and
is an antiderivative of 
, then the set of all antiderivatives of 
is given by the functions 
, where C is an arbitrary constant. The constant of integration is sometimes omitted in lists of integrals for simplicity.
, adding or subtracting a constant C will give us another antiderivative, because 
. The constant is a way of expressing that every function has an infinite number of different antiderivatives.
For example, suppose one wants to find antiderivatives of
. One such antiderivative is 
. Another one is 
. A third is 
. Each of these has derivative 
, so they are all antiderivatives of 
.
It turns out that adding and subtracting constants is the only flexibility we have in finding different antiderivatives of the same function. That is, all antiderivatives are the same up to a constant. To express this fact for cos(x), we write:
Replacing C by a number will produce an antiderivative. By writing C instead of a number, however, a compact description of all the possible antiderivatives of cos(x) is obtained. C is called the constant of integration. It is easily determined that all of these functions are indeed antiderivatives of
:
, the constant will always cancel with itself.
However, trying to set the constant equal to zero doesn't always make sense. For example,
can be integrated in two different ways:
So setting C to zero can still leave a constant. This means that, for a given function, there is no "simplest antiderivative".
Another problem with setting C equal to zero is that sometimes we want to find an antiderivative that has a given value at a given point (as in an initial value problem
). For example, to obtain the antiderivative of
that has the value 100 at x = π, then only one value of C will work (in this case C = 100).
This restriction can be rephrased in the language of differential equations. Finding an indefinite integral of a function
is the same as solving the differential equation 
. Any differential equation will have many solutions, and each constant represents the unique solution of a well-posed initial value problem
. Imposing the condition that our antiderivative takes the value 100 at x = π is an initial condition. Each initial condition corresponds to one and only one value of C, so without C it would be impossible to solve the problem.
There is another justification, coming from abstract algebra
. The space of all (suitable) real-valued functions on the real number
s 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 preimage of a given function. There is no canonical preimage for a given function, but the set of all such preimages 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.
and 
be two everywhere differentiable functions. Suppose that 
for every real number x. Then there exists a real number C such that 
for every real number x.
To prove this, notice that
. So F can be replaced by F-G and G by the constant function 0, making the goal to prove that an everywhere differentiable function whose derivative is always zero must be constant:
Choose a real number a, and let
. For any x, the fundamental theorem of calculus
says that
which implies that
. So F is a constant function.
Two facts are crucial in this proof. First, the real line is connected
. If the real line were not connected, we would not always be able to integrate from our fixed a to any given x. For example, if we 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 function
s, we can extend this theorem to disconnected domains. For example, there are two constants of integration for
and infinitely many for 
Second, F and G were assumed to be everywhere differentiable. If F and G are not differentiable at even one point, the theorem fails. 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 F is zero where it is defined, and the derivative of G is always zero. Yet it's clear that F and G do not differ by a constant.
Even if it is assumed that F and G are everywhere continuous and almost everywhere
differentiable the theorem still fails. As an example, take F to be the Cantor function
and again let G = 0.
Calculus
Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem...
, the indefinite integral of a given function (i.e., the set of all antiderivative
Antiderivative
In calculus, an "anti-derivative", antiderivative, primitive integral or indefinite integralof a function f is a function F whose derivative is equal to f, i.e., F ′ = f...
s of the function) is only defined up to
Up to
In mathematics, the phrase "up to x" means "disregarding a possible difference in x".For instance, when calculating an indefinite integral, one could say that the solution is f "up to addition by a constant," meaning it differs from f, if at all, only by some constant.It indicates that...
an additive constant, the constant of integration. This constant expresses an ambiguity inherent in the construction of antiderivatives. If a function
is defined on an intervalInterval (mathematics)
In mathematics, a interval is a set of real numbers with the property that any number that lies between two numbers in the set is also included in the set. For example, the set of all numbers satisfying is an interval which contains and , as well as all numbers between them...
and
is an antiderivative of , then the set of all antiderivatives of is given by the functions , where C is an arbitrary constant. The constant of integration is sometimes omitted in lists of integrals for simplicity.Origin of the constant
The derivative of any constant function is zero. Once one has found one antiderivative, adding or subtracting a constant C will give us another antiderivative, because . The constant is a way of expressing that every function has an infinite number of different antiderivatives.For example, suppose one wants to find antiderivatives of
. One such antiderivative is . Another one is . A third is . Each of these has derivative , so they are all antiderivatives of .It turns out that adding and subtracting constants is the only flexibility we have in finding different antiderivatives of the same function. That is, all antiderivatives are the same up to a constant. To express this fact for cos(x), we write:
Replacing C by a number will produce an antiderivative. By writing C instead of a number, however, a compact description of all the possible antiderivatives of cos(x) is obtained. C is called the constant of integration. It is easily determined that all of these functions are indeed antiderivatives of
:Necessity of the constant
At first glance it may seem that the constant is unnecessary, since it can be set to zero. Furthermore, when evaluating definite integrals using the fundamental theorem of calculusFundamental theorem of calculus
The first part of the theorem, sometimes called the first fundamental theorem of calculus, shows that an indefinite integration can be reversed by a differentiation...
, the constant will always cancel with itself.
However, trying to set the constant equal to zero doesn't always make sense. For example,
can be integrated in two different ways:So setting C to zero can still leave a constant. This means that, for a given function, there is no "simplest antiderivative".
Another problem with setting C equal to zero is that sometimes we want to find an antiderivative that has a given value at a given point (as in an initial value problem
Initial value problem
In mathematics, in the field of differential equations, an initial value problem is an ordinary differential equation together with a specified value, called the initial condition, of the unknown function at a given point in the domain of the solution...
). For example, to obtain the antiderivative of
that has the value 100 at x = π, then only one value of C will work (in this case C = 100).This restriction can be rephrased in the language of differential equations. Finding an indefinite integral of a function
is the same as solving the differential equation . Any differential equation will have many solutions, and each constant represents the unique solution of a well-posed initial value problemInitial value problem
In mathematics, in the field of differential equations, an initial value problem is an ordinary differential equation together with a specified value, called the initial condition, of the unknown function at a given point in the domain of the solution...
. Imposing the condition that our antiderivative takes the value 100 at x = π is an initial condition. Each initial condition corresponds to one and only one value of C, so without C it would be impossible to solve the problem.
There is another justification, coming from abstract algebra
Abstract algebra
Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...
. The space of all (suitable) real-valued functions on the real number
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
s is a vector space
Vector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
, and the differential operator
Differential operator
In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation, accepting a function and returning another .This article considers only linear operators,...
is a linear operator. The operator maps a function to zero if and only if that function is constant. Consequently, the kernelKernel (algebra)
In the various branches of mathematics that fall under the heading of abstract algebra, the kernel of a homomorphism measures the degree to which the homomorphism fails to be injective. An important special case is the kernel of a matrix, also called the null space.The definition of kernel takes...
of
is the space of all constant functions. The process of indefinite integration amounts to finding a preimage of a given function. There is no canonical preimage for a given function, but the set of all such preimages forms a cosetCoset
In mathematics, if G is a group, and H is a subgroup of G, and g is an element of G, thenA coset is a left or right coset of some subgroup in G...
. Choosing a constant is the same as choosing an element of the coset. In this context, solving an initial value problem
Initial value problem
In mathematics, in the field of differential equations, an initial value problem is an ordinary differential equation together with a specified value, called the initial condition, of the unknown function at a given point in the domain of the solution...
is interpreted as lying in the hyperplane
Hyperplane
A hyperplane is a concept in geometry. It is a generalization of the plane into a different number of dimensions.A hyperplane of an n-dimensional space is a flat subset with dimension n − 1...
given by the initial conditions.
Reason for a constant difference between antiderivatives
This result can be formally stated in this manner: Let and be two everywhere differentiable functions. Suppose that for every real number x. Then there exists a real number C such that for every real number x.To prove this, notice that
. So F can be replaced by F-G and G by the constant function 0, making the goal to prove that an everywhere differentiable function whose derivative is always zero must be constant:Choose a real number a, and let
. For any x, the fundamental theorem of calculusFundamental theorem of calculus
The first part of the theorem, sometimes called the first fundamental theorem of calculus, shows that an indefinite integration can be reversed by a differentiation...
says that
which implies that
. So F is a constant function.Two facts are crucial in this proof. First, the real line is connected
Connected space
In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint nonempty open subsets. Connectedness is one of the principal topological properties that is used to distinguish topological spaces...
. If the real line were not connected, we would not always be able to integrate from our fixed a to any given x. For example, if we 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
Connected space
In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint nonempty open subsets. Connectedness is one of the principal topological properties that is used to distinguish topological spaces...
of the domain. In general, by replacing constants with locally constant function
Locally constant function
In mathematics, a function f from a topological space A to a set B is called locally constant, if for every a in A there exists a neighborhood U of a, such that f is constant on U....
s, we can extend this theorem to disconnected domains. For example, there are two constants of integration for
and infinitely many for Second, F and G were assumed to be everywhere differentiable. If F and G are not differentiable at even one point, the theorem fails. As an example, let
be the Heaviside step functionHeaviside step function
The Heaviside step function, or the unit step function, usually denoted by H , is a discontinuous function whose value is zero for negative argument and one for positive argument....
, which is zero for negative values of x and one for non-negative values of x, and let
. Then the derivative of F is zero where it is defined, and the derivative of G is always zero. Yet it's clear that F and G do not differ by a constant.Even if it is assumed that F and G are everywhere continuous and almost everywhere
Almost everywhere
In measure theory , a property holds almost everywhere if the set of elements for which the property does not hold is a null set, that is, a set of measure zero . In cases where the measure is not complete, it is sufficient that the set is contained within a set of measure zero...
differentiable the theorem still fails. As an example, take F to be the Cantor function
Cantor function
In mathematics, the Cantor function, named after Georg Cantor, is an example of a function that is continuous, but not absolutely continuous. It is also referred to as the Devil's staircase.-Definition:See figure...
and again let G = 0.