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lec21 - 21 Indenite Integrals P K Lamm Lecture Notes(15:10...

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21. Indefinite Integrals P. K. Lamm 11/02/11 (15:10) p. 1 / 7 Lecture Notes: 21. Indefinite Integrals These classnotes are intended to be supplementary to the textbook and are necessarily limited by the time allotted for classes. For full and precise statements of definitions and theorems, as well as material covering other topics and examples, please consult the textbook. 1. Antiderivatives Definition: A function F ( x ) is called an antiderivative of the function f ( x ) if F satisfies F 0 ( x ) = f ( x ) for all x in the domain of f . Example 1.1: We know (sin x ) 0 = cos x, all x, so F ( x ) = sin x is an antiderivative of f ( x ) = cos x . Example 1.2: We know (3 x 2 + x ) 0 = 6 x + 1 , all x, so F ( x ) = 3 x 2 + x is an antiderivative of f ( x ) = 6 x + 1. Also note that (3 x 2 + x + 5) 0 = 6 x + 1 , all x, so F ( x ) = 3 x 2 + x + 5 is also an antiderivative of f ( x ) = 6 x + 1. In fact, for any constant C , F ( x ) = 3 x 2 + x + C is an antiderivative of f ( x ) = 6 x + 1 because (3 x 2 + x + C ) 0 = 6 x + 1 , all x. From the Mean Value Theorem we have the following: Theorem: If F ( x ) is an antiderivative of f ( x ), then all antiderivatives of f ( x ) are given by G ( x ) = F ( x ) + C, for C an arbitrary constant. Then for F ( x ) any antiderivative of f ( x ), the function F ( x ) + C gives the most general formula for all antiderivatives of f ( x ). Definition: The set of all antiderivatives of f is called the indefinite integral of f , written Z f ( x ) dx. 1
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21. Indefinite Integrals P. K. Lamm 11/02/11 (15:10) p. 2 / 7 Thus if F ( x ) is any antiderivative of f , Z f ( x ) dx = F ( x ) + C, for C an arbitrary constant.
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