Constant of integration
In calculus, the constant of integration, often denoted by formula_1, is a constant term added to an antiderivative of a function formula_2 to indicate that the indefinite integral of formula_2 (i.e., the set of all antiderivatives of formula_2), on a connected domain, is only defined up to an additive constant. This constant expresses an ambiguity inherent in the construction of antiderivatives. More specifically, if a function formula_2 is defined on an interval, and formula_6 is an antiderivative of formula_2, then the set of "all" antiderivatives of formula_2 is given by the functions formula_9, where formula_1 is an  arbitrary constant (meaning that "any" value of formula_1 would make formula_9 a valid antiderivative). For that reason, the indefinite integral is often written as formula_13, although the constant of integration might be sometimes omitted in lists of integrals for simplicity. Origin. The derivative of any constant function is zero. Once one has found one antiderivative formula_6 for a function formula_2, adding or subtracting any constant formula_1 will give us another antiderivative, because formula_17. The constant is a way of expressing that every function with at least one antiderivative will have an infinite number of them. Let formula_18 and formula_19 be  two everywhere differentiable functions. Suppose that formula_20 for every real number "x". Then there exists a real number formula_1 such that formula_22 for every real number "x". To prove this, notice that formula_23. So formula_24 can be replaced by formula_25, and formula_26 by the constant function formula_27, making the goal to prove that an everywhere differentiable function whose derivative is always zero must be constant: Choose a real number formula_28, and let formula_29. For any "x", the fundamental theorem of calculus, together with the assumption that the derivative of formula_24 vanishes, implies that thereby showing that formula_24 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 functions, we can extend this theorem to disconnected domains. For example, there are two constants of integration for formula_33, and infinitely many for formula_34, so for example, the general form for the integral of 1/"x" is: Second, formula_24 and formula_26 were assumed to be everywhere differentiable. If formula_24 and formula_26 are not differentiable at even one point, then the theorem might fail. As an example, let formula_6 be the Heaviside step function, which is zero for negative values of "x" and one for non-negative values of "x", and let formula_41. Then the derivative of formula_24  is zero where it is defined, and the derivative of formula_26 is always zero. Yet it's clear that formula_24 and formula_26 do not differ by a constant, even if it is assumed that formula_24 and formula_26 are everywhere continuous and almost everywhere differentiable the theorem still fails. As an example, take formula_24 to be the Cantor function and again let formula_49. For example, suppose one wants to find antiderivatives of formula_50. One such antiderivative is formula_51. Another one is formula_52. A third is formula_53. Each of these has derivative formula_50, so they are all antiderivatives of formula_50. 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 formula_50, we write: Replacing formula_1 by a number will produce an antiderivative. By writing formula_1 instead of a number, however, a compact description of all the possible antiderivatives of formula_50 is obtained. formula_1 is called the constant of integration. It is easily determined that all of these functions are indeed antiderivatives of formula_50: Necessity. 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 calculus, the constant will always cancel with itself. However, trying to set the constant to zero does not always make sense. For example, formula_64 can be integrated in at least three different ways: So setting formula_1 to zero can still leave a constant. This means that, for a given function, there is no "simplest antiderivative". Another problem with setting formula_1 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 formula_50 that has the value 100 at "x" = π, then only one value of formula_1 will work (in this case formula_70). This restriction can be rephrased in the language of differential equations. Finding an indefinite integral of a function formula_2 is the same as solving the differential equation formula_72. 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 formula_1, so without formula_1 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 numbers is a vector space, and the differential operator formula_75 is a linear operator. The operator formula_75 maps a function to zero if and only if that function is constant. Consequently, the kernel of formula_75 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.