讲稿综合详解45 dxtanh1 ud.pdfVIP

INTEGRATION

Thefollowingisabriefdiscussionofsomeintegrationtechniques.Amorecompletediscussioncanbefoundinanumberofgood

textbooks.However,thepurposeofthisintroductionissimplytodiscussafewoftheimportanttechniqueswhichmaybeused,in

conjunctionwiththeintegraltablewhichfollows,tointegrateparticularfunctions.

Nomatterhowextensivetheintegraltable,itisafairlyuncommonoccurrencetofindinthetabletheexactintegraldesired.

Usuallysomeformoftransformationwillhavetobemade.Thesimplesttypeoftransformation,andyetthemostgeneral,is

substitution.Simpleformsofsubstitution,suchasy=ax,areemployedalmostunconsciouslybyexperiencedusersofintegral

tables.Othersubstitutionsmayrequiremorethought.Insomesectionsofthetables,appropriatesubstitutionsaresuggestedfor

integralsthataresimilarto,butnotexactlylike,integralsinthetable.Findingtherightsubstitutionislargelyamatterofintuition

andexperience.

Severalprecautionsmustbeobservedwhenusingsubstitutions:

1.Besuretomakethesubstitutioninthedxterm,aswellaseverywhereelseintheintegral.

2.Besurethatthefunctionsubstitutedisone-to-oneandcontinuous.Ifthisisnotthecase,theintegralmustberestrictedin

suchawayastomakeittrue.Seetheexamplefollowing.

3.Withdefiniteintegrals,thelimitsshouldalsobeexpressedintermsofthenewdependentvariable.Withindefiniteintegrals,

itisnecessarytoperformthereversesubstitutiontoobtaintheanswerintermsoftheoriginaltvariable.This

mayalsobedonefordefiniteintegrals,butitisusuallyeasiertochangethelimits.

Example:

�√x4dx