Math151TurboParts

# Math151TurboParts - Tabular Integration by Parts David...

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Tabular Integration by Parts David Horowitz, Golden West College, Huntington Beach, CA 92647 The College Mathematics Journal, September 1990, Volume 21, Number 4, pp. 307–311. Only a few contemporary calculus textbooks provide even a cursory presentation of tabular integration by parts [see for example, G. B. Thomas and R. L. Finney, Calculus and Analytic Geometry, Addison-Wesley, Reading, MA 1988]. This is unfortunate because tabular integration by parts is not only a valuable tool for finding integrals but can also be applied to more advanced topics including the derivations of some important theorems in analysis. The technique of tabular integration allows one to perform successive integrations by parts on integrals of the form (1) without becoming bogged down in tedious algebraic details [V. N. Murty, Integration by parts, Two-Year College Mathematics Journal 11 (1980) 90-94]. There are several ways to illustrate this method, one of which is diagrammed in Table 1. (We assume throughout that F and G are “smooth” enough to allow repeated differentation and integration, respectively.) Table 1 ____________________________________________________ Column #1 Column #2 ____________________________________________________ G :: .. - - - - ____________________________________________________ In column #1 list and its successive derivatives. To each of the entries in this column, alternately adjoin plus and minus signs. In column #2 list and its successive antiderivatives. (The notation denotes the n th antiderivative of G . Do not include an additive constant of integration when finding each antiderivative.) Form successive terms by multiplying each entry in column #1 by the entry in column #2 that lies just below it. The integral (1) is equal to the sum of these terms. If is a F s x d G s 2 n d G s t d F s t d G s 2 n 2 1 d s 2 1 d n 1 1 F s n 1 1 d G s 2 n d s 2 1 d n F s n d G s 2 3 d 2 F s 3 d G s 2 2 d 1 F s 2 d G s 2 1 d 2 F s 1 d 1 F E F s t d G s t d dt

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polynomial, then there will be only a finite number of terms to sum. Otherwise the
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## This note was uploaded on 09/08/2011 for the course MATH 151 taught by Professor Bray during the Spring '07 term at Mesa CC.

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Math151TurboParts - Tabular Integration by Parts David...

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