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. 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 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, 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.