Let I be an open interval of and  f:I  a real function.

A functionF:I  is called an antiderivative or a primitive of f if F is differentiableMathworldPlanetmathPlanetmath and its derivativePlanetmathPlanetmath is equal to f, i.e.

F(x)=f(x)for allxI.

Note that there are an infiniteMathworldPlanetmath number of antiderivatives for any function f since any constant can be added or subtracted from any valid antiderivative to yield another equally valid antiderivative.

To account for this, we express the general antiderivative, or indefinite integral, as follows:


where C is an arbitrary constant called the constant of integration. The dx portion means “with respect to x”, because after all, our functions F and f are functions of x.

There is no loss in generality with this notation since in fact all antiderivatives of f take this form as the following theorem demonstrates:

Theorem. Let F,G be two antiderivatives of a given function f defined on an open interval I. Then F-G=𝑐𝑜𝑛𝑠𝑡.

Proof. Since  F(x)=f(x)  and  G(x)=f(x),  we have  F(x)-G(x)=0  on the whole I.  Thus, by the fundamental theorem of integral calculus,  F(x)-G(x)=const. 

This is no longer true if the domain of the function f is not an open interval (is not connectedPlanetmathPlanetmath). For that scenario, the following more general result holds:

Theorem. Let UR be an open set (not necessarily an interval). Suppose F,G are antiderivatives of a given function f:UR. Then F-G is constant in each connected componentMathworldPlanetmathPlanetmath of U (each interval in U).

For example, consider the function  f:{0}  given by f(x)=1x. Notice that the domain of f is not an interval, but the union of the disjoint intervals  (-, 0)  and  (0,+). Then, all the antiderivatives of f take the form


0.1 Remarks

  • For complex functions, the definition of antiderivative is exactly the same and the above results also hold (one just needs to consider “connected open subsets” instead of “open intervals”).

Title antiderivative
Canonical name Antiderivative
Date of creation 2013-03-22 12:14:55
Last modified on 2013-03-22 12:14:55
Owner asteroid (17536)
Last modified by asteroid (17536)
Numerical id 21
Author asteroid (17536)
Entry type Definition
Classification msc 26A36
Synonym general antiderivative
Synonym indefinite integral
Synonym primitive
Related topic AntiderivativeOfComplexFunction
Defines constant of integration