Antiderivatives/Indefinite Integrals

A function Fx) is called an antiderivative of a function of fx) if F′( x) = fx) for all x in the domain of f. Note that the function F is not unique and that an infinite number of antiderivatives could exist for a given function. For example, Fx) = x 3Gx) = x 3 + 5, and Hx) = x 3 − 2 are all antiderivatives of fx) = 3 x 2 because F′( x) = G′( x) = H′( x) = fx) for all x in the domain of f. It is clear that these functions F, G, and Hdiffer only by some constant value and that the derivative of that constant value is always zero. In other words, if Fx) and Gx) are antiderivatives of fx) on some interval, then F′( x) = G′( x) and Fx) = Gx) + C for some constant C in the interval. Geometrically, this means that the graphs of Fx) and Gx) are identical except for their vertical position.

The notation used to represent all antiderivatives of a function fx) is the indefinite integralsymbol written , where . The function of fx) is called the integrand, and C is reffered to as the constant of integration. The expression Fx) + C is called the indefinite integral of F with respect to the independent variable x. Using the previous example of Fx) = x 3 and fx) = 3 x 2, you find that .
The indefinite integral of a function is sometimes called the general antiderivative of the function as well.
Example 1: Find the indefinite integral of fx) = cos x


Example 2: Find the general antiderivative of fx) = –8.
  • Because the derivative of Fx) = −8 x is F′( x) = −8, write