How to Integrate It: A Practical Guide to Finding Elementary Integrals PDF
Preview How to Integrate It: A Practical Guide to Finding Elementary Integrals
HowtoIntegrateIt APracticalGuidetoFindingElementaryIntegrals Whiledifferentiatingelementaryfunctionsismerelyaskill,findingtheirintegralsis anart.Thispracticalintroductiontotheartofintegrationgivesreadersthetoolsand confidencetotacklecommonanduncommonintegrals. AfterareviewofthebasicpropertiesoftheRiemannintegral,eachchapteris devotedtoaparticulartechniqueofelementaryintegration.Thoroughexplanations andplentifulworkedexamplespreparethereaderfortheextensiveexercisesattheend ofeachchapter.Theseexercisesincreaseindifficultyfromwarm-upproblems, throughdrillexamples,tochallengingextensionsthatillustratesuchadvancedtopics astheirrationalityof(cid:2) ande,thesolutionoftheBaselproblem,Leibniz’sseries,and Wallis’sproduct. Theauthor’saccessibleandengagingmannerwillappealtoawideaudience, includingstudents,teachers,andself-learners.Itcanserveasacompleteintroduction tofindingelementaryintegrals,orasasupplementarytextforanybeginningcoursein calculus. How to Integrate It A Practical Guide to Finding Elementary Integrals SEÁN M. STEWART OmegadotTuition,Sydney UniversityPrintingHouse,CambridgeCB28BS,UnitedKingdom OneLibertyPlaza,20thFloor,NewYork,NY10006,USA 477WilliamstownRoad,PortMelbourne,VIC3207,Australia 314–321,3rdFloor,Plot3,SplendorForum,JasolaDistrictCentre, NewDelhi-110025,India 79AnsonRoad,#06-04/06,Singapore079906 CambridgeUniversityPressispartoftheUniversityofCambridge. ItfurtherstheUniversity’smissionbydisseminatingknowledgeinthepursuitof education,learning,andresearchatthehighestinternationallevelsofexcellence. www.cambridge.org Informationonthistitle:www.cambridge.org/9781108418812 DOI:10.1017/9781108291507 ©SeánM.Stewart2018 Thispublicationisincopyright.Subjecttostatutoryexception andtotheprovisionsofrelevantcollectivelicensingagreements, noreproductionofanypartmaytakeplacewithoutthewritten permissionofCambridgeUniversityPress. Firstpublished2018 PrintedintheUnitedKingdombyClays,StIvesplc AcataloguerecordforthispublicationisavailablefromtheBritishLibrary ISBN978-1-108-41881-2Hardback ISBN978-1-108-40819-6Paperback CambridgeUniversityPresshasnoresponsibilityforthepersistenceoraccuracy ofURLsforexternalorthird-partyinternetwebsitesreferredtointhispublication, anddoesnotguaranteethatanycontentonsuchwebsitesis,orwillremain, accurateorappropriate. Contents Preface pagevii 1 TheRiemannIntegral 1 2 BasicPropertiesoftheDefiniteIntegral:PartI 20 3 SomeBasicStandardForms 25 4 BasicPropertiesoftheDefiniteIntegral:PartII 36 5 StandardForms 53 6 IntegrationbySubstitution 64 7 IntegrationbyParts 79 8 TrigonometricIntegrals 105 9 HyperbolicIntegrals 119 10 TrigonometricandHyperbolicSubstitutions 134 11 IntegratingRationalFunctionsbyPartialFraction Decomposition 144 12 SixUsefulIntegrals 157 13 InverseHyperbolicFunctionsandIntegralsLeadingto Them 169 14 TangentHalf-AngleSubstitution 178 15 FurtherTrigonometricIntegrals 188 16 FurtherPropertiesforDefiniteIntegrals 209 v vi Contents 17 IntegratingInverseFunctions 233 18 ReductionFormulae 242 19 SomeOtherSpecialTechniquesandSubstitutions 264 20 ImproperIntegrals 274 21 TwoImportantImproperIntegrals 298 AppendixA:PartialFractions 313 AppendixB:AnswerstoSelectedExercises 333 Index 365 Preface Calculus occupies an important place in modern mathematics. At its heart it is the study of continuous change. It forms the foundation of mathematical analysis while the immense wealth of its ideas and usefulness of the tools to haveemergedfromitsdevelopmentmakeitcapableofhandlingawidevariety ofproblemsbothwithinandoutsideofmathematics.Indeed,thesheernumber of applications that the calculus finds means it continues to remain a central component for any serious study of mathematics for future mathematicians, scientists,andengineersalike. Thematerialpresentedinthisvolumedealswithoneofthemajorbranches ofcalculusknownastheintegralcalculus–theotherbeingthedifferential,with thetwobeingintimatelybound.Theintegralcalculusdealswiththenotionof anintegral,itsproperties,andmethodofcalculation.Ourwordfor‘integrate’is derivedfromtheLatinintegratusmeaning‘tomakewhole’.Ascalculusdeals withcontinuouschange,integration,then,isageneralmethodforfindingthe wholechangewhenyouknowalltheintermediate(infinitesimal)changes. AprecursortotheconceptofanintegraldatesbacktotheancientGreeks,to EudoxusinthefourthcenturybceandArchimedesinthethirdcenturybce,and theirworkrelatedtothemethodofexhaustion.Themethodofexhaustionwas usedtocalculateareasofplanefiguresandvolumesofsolidsbasedonapprox- imatingtheobjectunderconsiderationbyexhaustivelypartitioningitintoever smallerpiecesusingthesimplestpossibleplanarfiguresorbodies,suchasrect- anglesorcylinders.Summingitsconstituentpartstogetherthengavethearea or volume of the whole. Integration thus renders something whole by bring- ing together all its parts. Its modern development came much later. Starting inthelateseventeenthcenturywiththeseminalworkofNewtonandLeibniz, it was carried forward in the eighteenth century by Euler and the Bernoulli brothers, Jacob and Johann, and in the nineteenth century most notably by CauchybeforethefirstrigoroustreatmentoftheintegralwasgivenbyRiemann vii viii Preface duringthemiddlepartofthatcentury.Sincethistimemanyothernotionsfor the integral have emerged. In this text we focus exclusively on the first and perhapssimplestofthesenotionstoemerge,thatoftheRiemannintegral. Unlikedifferentiation,whichoncelearntismerelyaskillwhereasetofrules areapplied,asthereisnosystematicprocedureforfindinganintegral,evenfor functionsthatbehave‘nicely’,manylookuponintegrationasan‘art’.Finding anintegraltendstobeacomplicatedaffairinvolvingasearchforpatternsand ishardtodo.Theunavailabilityofamechanicalapproachtointegrationmeans many different techniques for finding integrals of well-behaved functions in termsoffamiliarfunctionshavebeendeveloped.Thismakesintegrationhard andisexactlyhowitisperceivedbymostbeginningstudents.Whenencoun- tering integration for the first time one is often bewildered by the number of differentmethodsthatneedtobeknownbeforetheproblemofintegrationcan besuccessfullytackled. Thisbookisanattemptattakingsomeofthemysteryoutoftheartofinte- gration.Thetextprovidesaself-containedpresentationofthepropertiesofthe definite integral together with many of the familiar, and some not so famil- iar,techniquesthatareavailableforfindingelementaryintegrals.Prerequisites neededfortheproperstudyofthematerialpresentedinthetextareminimal. The reader is expected to be familiar with the differential calculus including theconceptofalimit,continuity,anddifferentiability,togetherwithaworking knowledgeoftherulesofdifferentiation. Thebooktakesthereaderthroughthevariouselementarymethodsthatcan beusedtofind(Riemann)integralstogetherwithintroducinganddeveloping thevariouspropertiesassociatedwiththedefiniteintegral.Thefocusisprimar- ilyonideasandtechniquesusedforintegratingfunctionsandontheproperties associatedwiththedefiniteintegralratherthanonapplications.Bydoingsothe aimistodevelopinthestudenttheskillsandconfidenceneededtoapproach thegeneralproblemofhowtofindanintegralintermsoffamiliarfunctions. Oncethesehavebeendevelopedandthoroughlymastered,thestudentshould beinafarbetterpositiontomoveontothemultitudeofapplicationstheinte- gralfindsforitself.Ofcoursethisisnottosayapplicationsfortheintegralare notto be found inthe text. They are. Whilethe text makes no attempt touse integration to calculate areas or volumes in any schematic way, applications developed through the process of integration that lead to important results in otherareasofmathematicsaregiven.Asexamples,theproofoftheirrationality ofthenumbers(cid:2) andeispresented,asarethesolutionstotheBaselproblem, Leibniz’sseries,andWallis’sproduct. Mostchaptersofthebookarequiteshortandsuccinct.Eachchapterisself- contained and is structured such that after the necessary theory is introduced Preface ix and developed, a range of examples of increasing level of difficulty are pre- sentedshowinghowthetechniqueisusedandworksinpractice.Alongtheway, variousstrategiesandsoundadvicearegiven.Attheconclusionofeachchapter, anextensivesetofexercisesappear.Thetextcanserveasacompleteintroduc- tionandguidetofindingelementaryintegrals.Alternatively,itcanserveasa resourceorsupplementaltextforanybeginningcourseincalculusbyallowing studentstofocusonparticularproblemareastheymightbehavingbyworking throughoneormorerelevantchaptersatatime. Incontemplatingthematerialpresenteditcannotbeoverstatedhowimpor- tantitisforonetoattempttheexerciseslocatedattheendofeachchapter.For thosehopingtobecomefluentintheartofintegration,thisproficiencyisbest gainedthroughperseveranceandhardwork,andinthepracticeofansweringas manydifferentandvariedquestionsaspossible.Indeed,alargeportionofthe textisdevotedtosuchexercisesandproblems;theyareaveryimportantcom- ponentofthebook.Tohelpaidstudentsintheirendeavoursanattempthasbeen made to divide the exercises that appear at the end of the chapters into three types: (i) warm-ups, (ii) practice questions, and (iii) extension questions and challengeproblems.Thewarm-upsarerelativelysimplequestionsdesignedto gentlyeasethestudentintothematerialjustconsidered.Thepracticequestions consolidate knowledge of the material just presented, allowing the student to gainfamiliarityandconfidenceintheworkingsofthetechniqueunderconsid- eration.Finally,theextensionquestionsandchallengeproblemscontainamix ofquestionsthatareeithersimplychallenginginnatureorthatextend,inoften quite unexpected ways, the material just considered. It is hoped many of the questionsfoundinthislastgroupwillnotonlychallengethereaderbutpique theirinterestasmoreadvancedresultsaregraduallyrevealed.Ofcourse,judg- ingtheperceived levelofdifficultyisoftenintheeye ofthebeholder soone mayexpectsomeoverlapbetweenthevariouscategories.Forproblemsconsid- eredmoredifficult,hintsareprovidedalongthewayintheformofinterrelated partsthatitishopedwillhelpguidethestudenttowardsthefinalsolution.In all,wellover1,000problemsrelatingtofindingorevaluatingintegralsorprob- lemsassociatedwithpropertiesforthedefiniteintegralcanbefounddispersed throughouttheend-of-chapterexercisesets. Chapter1introducesformallywhatwemeanbyanintegralintheRiemann sense. The approach taken is one via Darboux sums. The fundamental theo- remofcalculus,whichwedivideintwoparts,isalsogiven.Propertiesforthe definite integral are given in three chapters (Chapters 2, 4, and 16). Sixteen chapters(Chapters3and5–19)arethendevotedtoeitheraparticularmethod thatcanbeusedtofindagivenintegraloraparticularclassofintegrals.Here methods including standard forms, integration by substitution, integration by
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