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Unformatted text preview: Unit 2: The Antiderivative Definition 2.1. F ( x ) is said to be an antiderivative of f ( x ) on a particular interval if F prime ( x ) = f ( x ) (or equivalently, if d dx ( F ( x )) = f ( x )) for every value of x in the interval. To antidifferentiate a function, we need to answer the question What function has this function as its derivative?. Example 1 . Find an antiderivative of x 3 . Solution: The exponent on x is 3, so in searching for an antiderivative (think- ing of power rule in reverse) it makes sense to begin our search with x 4 . Test- ing this hypothesis we see that d dx ( x 4 ) = 4 x 3 . We observe that we require a function whose antiderivative is x 3 , which can also be expressed as 1 4 (4 x 3 ). That is, we see that x 3 = 1 4 ( d dx ( x 4 ) ) = d dx parenleftBig x 4 4 parenrightBig , which means that x 4 4 is an antiderivative of x 3 . Notice that we defined an antiderivative as opposed to the antiderivative of a function. This is because a function has many antiderivatives. For in- stance, we have seen that d dx parenleftBig x 4 4 parenrightBig = x 3 , i.e., that x 4 4 is an antiderivative of x 3 . But it is also true that, for instance, d dx parenleftBig x 4 4 + 3 parenrightBig = x 3 + 0 = x 3 , so we see that x 4 4 + 3 is also an antiderivative of x 3 . In general, we have d dx ( x 4 4 + C ) = x 3 , where C is any constant, so x 4 4 + C is an antiderivative of x 3 for any value of C . For this reason, we now define the general antiderivative of a function f ( x ) to be the sum of an antiderivative of f ( x ) and an arbitrary constant C . 1 Definition 2.2. If F ( x ) is any antiderivative of f ( x ), then the general an- tiderivative of f ( x ) is given by F ( x ) + C , where C is an arbitrary (i.e., unspecified) constant. For instance, the general antiderivative of x 3 is x 4 4 + C . Example 2 . Find the general antiderivative, F ( x ), of f ( x ) = 2 x 4 . Solution: After some thought (see example 1) we see that 2 x 5 5 is an antideriva- tive of 2 x 4 . That is, d dx parenleftBig 2 x 5 5 parenrightBig = 2 5 d dx ( x 5 ) = 2 5 (5 x 4 ) = 2 x 4 . So the general antiderivative of f ( x ) = 2 x 4 is F ( x ) = 2 x 5 5 + C . We have special notation for the general antiderivative of a function. Definition 2.3. The general antiderivative of f ( x ) is denoted by integraldisplay f ( x ) dx which is called the indefinite integral of f ( x ). The symbol integraltext is called an integral sign, the function f ( x ) is called the integrand , and the presence of dx signifies that x is the variable of integration , that is we are integrating (antidifferentiating) with respect to x . We refer to the operation of finding an antiderivative as integration ....
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