A function

*F*(

*x*) is called an

**antiderivative** of a function of

*f*(

*x*) if

*F*′(

*x*) =

*f*(

*x*) 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,

*F*(

*x*) =

*x* ^{3},

*G*(

*x*) =

*x* ^{3} + 5, and

*H*(

*x*) =

*x* ^{3} − 2 are all antiderivatives of

*f*(

*x*) = 3

*x* ^{2} because

*F*′(

*x*) =

*G*′(

*x*) =

*H*′(

*x*) =

*f*(

*x*) for all

*x* in the domain of

*f*. It is clear that these functions

*F, G*, and

*H* differ only by some constant value and that the derivative of that constant value is always zero. In other words, if

*F*(

*x*) and

*G*(

*x*) are antiderivatives of

*f*(

*x*) on some interval, then

*F*′(

*x*) =

*G*′(

*x*) and

*F*(

*x*) =

*G*(

*x*) +

*C* for some constant

*C* in the interval. Geometrically, this means that the graphs of

*F*(

*x*) and

*G*(

*x*) are identical except for their vertical position.

The notation used to represent all antiderivatives of a function *f*( *x*) is the **indefinite integral** symbol written , where . The function of *f*( *x*) is called the integrand, and *C* is reffered to as the constant of integration. The expression *F*( *x*) + *C* is called the indefinite integral of *F* with respect to the independent variable *x*. Using the previous example of *F*( *x*) = *x* ^{3} and *f*( *x*) = 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 *f*( *x*) = cos *x*.

**Example 2:** Find the general antiderivative of *f*( *x*) = –8.

- Because the derivative of
*F*( *x*) = −8 *x* is *F*′( *x*) = −8, write