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UNIT2

# UNIT2 - Unit 2 The Antiderivative Denition 2.1 F(x is said...

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Unit 2: The Antiderivative Definition 2.1. F ( x ) is said to be an antiderivative of f ( x ) on an interval if F ( x ) = f ( x ) or equivalently d dx ( F ( x )) = f ( x ) for every value of x on the interval. Example 1 . Find an antiderivative of x 3 Solution: The exponent on x is 3, so in searching for an antiderivative (thinking of power rule in reverse) it makes sense to begin our search with x 4 . Testing this hypothesis we see that d dx ( x 4 ) = 4 x 3 We require x 3 which we observe to be 1 4 (4 x 3 ). With this in mind we see that d dx ( 1 4 ( x 4 )) = x 3 Definition 2.2. Notice that we defined an antiderivative as opposed to the antiderivative of a function. The reason for this can be demonstrated quite easily as follows: First we see for example that d dx ( 1 4 ( x 4 )) = x 3 1

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and that d dx ( 1 4 ( x 4 ) + 3) = x 3 + 0 = x 3 and that in general d dx ( 1 4 ( x 4 ) + C ) = x 3 , where C is any arbitrary constant 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. Example 2 . Find the general antiderivative, F ( x ), of f ( x ) = x 4 Solution: After some thought (see example 1) we see that 1 5 ( x 5 ) is an antiderivative of x 4 . In light of definition 2 we then have F ( x ) = 1 5 ( x 5 ) + C is the general antiderivative of x 4 . We now introduce some general rules and notation: The general antiderivative of f ( x ) will be denoted f ( x ) dx where the sybol is called the integral sign, f ( x ) dx is called the integrand , and the presence of dx signifies that x is the variable of integration , i.e. we are integrating with respect to x. We will now refer to the operation of finding an antiderivative as integration . Example 3 . ( a ) x 3 dx = 1 4 x 4 + C . 2
( b ) x 4 dx = 1 5 x 5 + C . Rules of Integration x n dx = 1 n +1 ( x n +1 ) + C for n not equal to -1 . This is known as the Power Rule for integration. The special case when n = - 1 is described by x - 1 dx = 1 x dx = ln | x | + C . The use of these two rules can be easily demonstrated. Example 4 . Solve the following integral x 17 dx Solution: Since the power on x is not - 1 we may apply the power rule with n = 17 to obtain x 1 7 dx = 1 18 x 1 8 + C Example 5 . Solve the following integral x 3 dx 3

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Solution: Once again we invoke the power rule, this time with n = 3 2 . x 3 dx = x 3 2 dx = 1 3 2 + 1 x 3 2 +1 = 1 5 2 x 5 2 = 2 5 x 5 2 The following theorems will be of great importance for solving integrals.
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