Lect30 - Antiderivatives Any function F(x with the property...

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92.131 Lecture 30 1 of 10 Ronald Brent © 2009 All rights reserved. Antiderivatives Any function ) ( x F with the property that ) ( ) ( x f x F = on an interval I is called an antiderivative of ) ( x f on the interval I . Examples: a) 2 3 6 ) ( x x x x F + + = is an antiderivative of x x x x f 2 3 6 ) ( 2 5 + + = , since ) ( ) 2 3 6 ( ) ( ) ( 2 5 2 3 6 x f x x x x x x dx d x F = + + = + + = b) 1 2 3 ) ( 4 + + = t t t D is an antiderivative of t t t d 2 1 6 ) ( 3 + = , since ) ( 2 1 6 1 2 3 ) ( 3 4 t d t t t t dt d t D = + = + + = Note: We usually associate the uppercase-letter function as antiderivative of corresponding lowercase-letter function.
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92.131 Lecture 30 2 of 10 Ronald Brent © 2009 All rights reserved. Theorem 1: If ) ( x F is an antiderivative of ) ( x f on an interval I , and C is any constant, then C x F + ) ( is the most general antiderivative of ) ( x f on I . Finding Formulas for Antiderivatives Let C k x x F k + + = + 1 ) ( 1 , where 1 k is any real number, and C is an arbitrary constant. Then, ) ( x F is an antiderivative for the function k x x f = ) (. Every antiderivative of ) ( x f , where 1 k has this form.
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92.131 Lecture 30 3 of 10 Ronald Brent © 2009 All rights reserved.
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Lect30 - Antiderivatives Any function F(x with the property...

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