Techniques in Integration

Techniques in Integration - Contents 11 Techniques of...

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Unformatted text preview: Contents 11 Techniques of Integration 168 11.1 Basic Substitution and Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 11.2 Integration by Parts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 11.3 Improper Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 CONTENTS 2 Chapter 11 Techniques of Integration 11.1 Basic Substitution and Formulas Definition The indefinite integral of a function f , written Z f ( x ) dx, is the most general antiderivative of f ; that is, Z f ( x ) dx = F ( x ) + C ( C is any constant) if and only if F ( x ) = f ( x ). Basic Formulas: (Memorize this list) Z u n du = u n +1 n + 1 + C ( n 6 =- 1) Z du u = ln | u | + C, if u 6 = 0 Z e u du = e u + C Z a u du = a u ln a + C Z sin udu =- cos u + C Z cos udu = sin u + C Z sec 2 udu = tan u + C Z csc 2 udu =- cot u + C Z sec u tan udu = sec u + C Z csc u cot udu =- csc u + C Z du √ 1- u 2 = sin- 1 u + C Z du 1 + u 2 = tan- 1 u + C 169 11.1 Basic Substitution and Formulas Z du | u | √ u 2- 1 = sec- 1 u + C, | u | > 1 Z tan udu =- ln | cos u | + C or ln | sec u | + C Z cot udu = ln | sin u | + C Z sec udu = ln | sec u + tan u | + C Z csc udu = ln | csc u- cot u | + C Z [ f ( u ) + g ( u ) du ] = Z f ( u ) du + Z g ( u ) du Z kf ( u ) du = k Z f ( u ) du if k is a constant 1. Z ( x 2 + a ) 2 dx 2. Z ( x- 3) 2 dx 3. Z x 5 2 dx 4. Z aπ dx x 5. Z [( x + 1) 3 + e x ] dx 6. Z x 2 e x + x x 2 dx 7. Z 10sec 2 xdx 8. Z 16(2 x + 1) dx 9....
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This note was uploaded on 09/22/2011 for the course ECON 101 taught by Professor Mr.tull during the Spring '11 term at De La Salle University.

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Techniques in Integration - Contents 11 Techniques of...

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