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Integration by substitution

2003 AP MC

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? ? ? ? ? ? ? ? ? ? ? ? ? ? M AREA - COMPLETE THIS AREA AT EVERY EXAMINATION. ~The AP"'ADVANCEDTo maintain lhe security 01 lhe exam and the vaftdily 01 my AP grade, I will allow no one other than myseff to see the multiple-ohoice queslicns and wiD seal the appropriate section when asked to do so. t will not discuss these questions .. .. College PLACEMENT PLACE AN AP ? NUMBER LABEL _ Board PROGRAM" Iwith anyone at any time after the completion of the multipl_ section. I em awam 01 and agree 10 the Program'. ~ policies and procecilJteS as outlined in the 2003 Buffetin for AP Stt.tdImts and Parents. OR WRITE YOUR AP NUMBER Answer Sheet for May 2003, Form 3ZBP HERE AT EVERY EXAMINATION. PAGE 1 ,...,. W Plintexamination name:.______________________

AP Calc Study Guide

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Copyright 1996,1997 Elaine Cheong All Rights Reserved Study Guide for the Advanced Placement Calculus AB Examination By Elaine Cheong 1 Table of Contents INTRODUCTION 2 TOPICS TO STUDY 3 ? Elementary Functions 3 ? Limits 5 ? Differential Calculus 7 ? Integral Calculus 12 SOME USEFUL FORMULAS 16 CALCULATOR TIPS AND PROGRAMS 17 BOOK REVIEW OF AVAILABLE STUDY GUIDES 19 ACKNOWLEDGEMENTS 19 2 Introduction Advanced Placement1 is a program of college-level courses and examinations that gives high school students the opportunity to receive advanced placement and/or credit in college. The Advanced Placement Calculus AB Exam tests students on introductory differential and integral calculus, covering a full-year college mathematics course.

AP Calculus Cheat Sheet

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Calculus Cheat Sheet Visit http://tutorial.math.lamar.edu for a complete set of Calculus notes. ? 2005 Paul Dawkins Limits Definitions Precise Definition : We say ( )limx a f x L? = if for every 0e > there is a 0d > such that whenever 0 x a d< - < then ( )f x L e- < . ?Working? Definition : We say ( )limx a f x L? = if we can make ( )f x as close to L as we want by taking x sufficiently close to a (on either side of a) without letting x a= . Right hand limit : ( )limx a f x L+? = . This has the same definition as the limit except it requires x a> . Left hand limit : ( )limx a f x L-? = . This has the same definition as the limit except it requires x a< . Limit at Infinity : We say ( )limx f x L?? = if we

Ch8 SG

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317 CHAPTER 8 Principles of Integral Evaluation EXERCISE SET 8.1 1. u = 3? 2x, du = ?2dx, ? 1 2 ? u3 du = ?1 8 u4 + C = ?1 8 (3? 2x)4 + C 2. u = 4 + 9x, du = 9dx, 1 9 ? u1/2 du = 2 3 ? 9u 3/2 + C = 2 27 (4 + 9x)3/2 + C 3. u = x2, du = 2xdx, 1 2 ? sec2 u du = 1 2 tanu+ C = 1 2 tan(x2) + C 4. u = x2, du = 2xdx, 2 ? tanu du = ?2 ln | cosu |+ C = ?2 ln | cos(x2)|+ C 5. u = 2 + cos 3x, du = ?3 sin 3xdx, ? 1 3 ? du u = ?1 3 ln |u|+ C = ?1 3 ln(2 + cos 3x) + C 6. u = 3x 2 , du = 3 2 dx, 2 3 ? du 4 + 4u2 = 1 6 ? du 1 + u2 = 1 6 tan?1 u+ C = 1 6 tan?1(3x/2) + C 7. u = ex, du = exdx, ? sinhu du = coshu+ C = cosh ex + C 8. u = lnx, du = 1 x dx, ? secu tanu du = secu+ C = sec(lnx) + C 9. u = cotx, du = ? csc2 xdx, ? ? eu du = ?eu + C = ?ecot x + C
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