The derivative function «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»f«/mi»«mo mathvariant=¨italic¨»`«/mo»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«mo»=«/mo»«mn»5«/mn»«msup»«mi»x«/mi»«mn»4«/mn»«/msup»«/math» may be recognized as the derivative of the function «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»f«/mi»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«mo»=«/mo»«msup»«mi»x«/mi»«mn»5«/mn»«/msup»«/math».

However, «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»f«/mi»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«mo»=«/mo»«msup»«mi»x«/mi»«mn»5«/mn»«/msup»«/math» is not the only possible function. There are infinitely many other functions with a derivative of «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»f«/mi»«mo mathvariant=¨italic¨»`«/mo»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«mo»=«/mo»«mn»5«/mn»«msup»«mi»x«/mi»«mn»4«/mn»«/msup»«/math», such as

• «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»f«/mi»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«mo»=«/mo»«msup»«mi»x«/mi»«mn»5«/mn»«/msup»«mo»+«/mo»«mn»8«/mn»«/math»,
• «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»f«/mi»«mfenced»«mi»x«/mi»«/mfenced»«mo»=«/mo»«msup»«mi»x«/mi»«mn»5«/mn»«/msup»«mo»§#8722;«/mo»«mn»12«/mn»«/math», and
• «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»f«/mi»«mfenced»«mi»x«/mi»«/mfenced»«mo»=«/mo»«msup»«mi»x«/mi»«mn»5«/mn»«/msup»«mo»§#8722;«/mo»«mo»§#960;«/mo»«/math».

In fact, any function of the form «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»f«/mi»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«mo»=«/mo»«msup»«mi»x«/mi»«mn»5«/mn»«/msup»«mo»+«/mo»«mi»C«/mi»«/math», where «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»C«/mi»«/math» is a constant, will produce the derivative function «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»f«/mi»«mo mathvariant=¨italic¨»`«/mo»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«mo»=«/mo»«mn»5«/mn»«msup»«mi»x«/mi»«mn»4«/mn»«/msup»«/math» since the derivative of a constant is zero.

The original function from which the derivative was obtained is termed the antiderivative or the integral. Antidifferentiation is the reverse operation of differentiation. Like the derivative, the antiderivative may be represented in a variety of ways.

If «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»y«/mi»«mo mathvariant=¨italic¨»`«/mo»«mo»=«/mo»«mi»f«/mi»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«/math» is the derived function, then «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»y«/mi»«mo»=«/mo»«mi»F«/mi»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«/math» is used to denote the antiderivative. Remember any function of the form «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»y«/mi»«mo»=«/mo»«mi»F«/mi»«mfenced»«mi»x«/mi»«/mfenced»«mo»+«/mo»«mi»C«/mi»«/math», where «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»C«/mi»«/math» is a constant, will have the same derivative. When the value of «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»C«/mi»«/math» is unknown, the function «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»y«/mi»«mo»=«/mo»«mi»F«/mi»«mfenced»«mi»x«/mi»«/mfenced»«mo»+«/mo»«mi»C«/mi»«/math» is referred to as the general antiderivative.


The antiderivative, or integral, can also be represented as follows.

The symbol «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#8747;«/mo»«mi»___«/mi»«mi»d«/mi»«mi»x«/mi»«/math» is the inverse of the symbol «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mi»d«/mi»«mrow»«mi»d«/mi»«mi»x«/mi»«/mrow»«/mfrac»«mi»___«/mi»«/math».

The solution for Example 1 could be written as follows.

This integral is referred to as the indefinite integral, since the constant of integration, «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»C«/mi»«/math», is arbitrary.

Because «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»2«/mn»«mi»x«/mi»«/math» is the derivative of «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«/math» and «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»3«/mn»«/math» is the derivative of «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»3«/mn»«mi»x«/mi»«/math», the general antiderivative is «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»F«/mi»«mfenced»«mi»x«/mi»«/mfenced»«mo»=«/mo»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«mn»3«/mn»«mi»x«/mi»«mo»+«/mo»«mi»C«/mi»«/math».

Alternatively, the solution could be expressed as «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#8747;«/mo»«mfenced»«mrow»«mn»2«/mn»«mi»x«/mi»«mo»+«/mo»«mn»3«/mn»«/mrow»«/mfenced»«mi»d«/mi»«mi»x«/mi»«mo»=«/mo»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«mn»3«/mn»«mi»x«/mi»«mo»+«/mo»«mi»C«/mi»«/math».

To check, simply find the derivative of «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»F«/mi»«mfenced»«mi»x«/mi»«/mfenced»«mo»=«/mo»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«mn»3«/mn»«mi»x«/mi»«mo»+«/mo»«mi»C«/mi»«/math», and compare it to «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»f«/mi»«mfenced»«mi»x«/mi»«/mfenced»«mo»=«/mo»«mn»2«/mn»«mi»x«/mi»«mo»+«/mo»«mn»3«/mn»«/math».

By inspection, the antiderivative is as follows.

    
Check by finding the derivative.


Sometimes it is difficult to determine the antiderivative by inspection (just looking and seeing). One method for finding the antiderivative of a polynomial or a basic rational or radical function is to reverse the power rule. The solution is obtained by increasing the exponent on each power by one, and dividing each power by the new exponent.


Use this method to integrate «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#8747;«/mo»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mi mathvariant=¨normal¨» «/mi»«mi»d«/mi»«mi»x«/mi»«/math».