1、Chapter 7 Techniques of IntegrationLHospital Rule and Improper Integrals7.1 Basic Integration FormulasTable of Indefinite integrals xd)1(C is a constant)Cxxxnd)2(Cxnn111xxd)3(Cx ln)1(n)ln(xx121d)4(xxCx arctanxxdcos)6(Cx sinxx2cosd)8(xxdsec2Cx tanorCx cotarc21d)5(xxCx arcsinorCx cosarcxxdsin)7(Cx cos
2、xx2sind)9(xxdcsc2Cx cotxxxdtansec)10(Cx secxxxdcotcsc)11(Cxcscxexd)12(Cexxaxd)13(Caaxln2shxxeexCx chxxdch)15(Cx shxxdsh)14(2chxxeexExercise:dxxxxxdxdxxxx222)tan(sec.38.21992.1xdxdxxxxdxxxdxxsec.72373.6123.54cos41.4224/0Exercise:7.2 Integration by PartsIntegration by PartsIf and are differentiable fu
3、nctions,then()()()()()()f x g xfx g xf x g x()()()()()()f x g xdxfx g x dxf x g x dx()f x()g x()()()()()()f x g x dxf x g xdxfx g x dx()()()()()()f x g x dxf x g xfx g x dxThe formula for integration by parts()()()()()()f x g x dxf x g xfx g x dxLet()and()uf xvg xThen the formula for integration by
4、parts becomesudvuvvduxsinx Cxxxcossin xdxcosxxdxsincos,dvxdxcosFindxxdxExample 1SolutionL e t,uxT h e n ,s ind ud xvxudvuvvdu xdxsinx22cosx dx xcosx2 2cosxdxxcosx2 xxd sin2xx cos2Cxxxcos2sin2xcosx2 )sinsin(2xdxxx2sinEvaluatexxdxExample 2Solutioncos(2)xx dxudvuvvduxxdexe dxCexexxdxxexxxedxxex221xdexx
5、xdexex222121dxexexxx222121xEvaluatexe dxExample 3Solution2xx dexdxex2Cxxex222dxexx2xex2xex22xxdexex22dxexexxCexeexxxx222xEvaluatex e dxExample 4Solution xdxlnxExample5 dxxx13223xlnx 3223 dxx32xlnx 3223Cx 9423xlnx 3223332222ln()ln33xxxdxxd2arctan2xxdxxxx2211121arctan2Cxxxxarctan21arctan222arctan2xxdd
6、xxx22112xdxxarctanExample6Cxxx21arctan1212xdxarccosExample 7dxxx21Cx 21xx arccosxx arccosxx2lnxdxln2xx2ln1ln 2dxxxxxxx2lnCxxx2ln2xdx2lnExample 8dxxxx1ln2xx2ln2lnxdxe dcosxx xxdexcossinxe dxxdxexsin Cxxexcossin21Example9xdxexsinxexcosxexcos sin xexxexcossoxdxexsinorxdxexsinsinxxde sin xex cosxxde sin
7、 xexxexcosxdxexsin Cxxexcossin21so,xdxexsinxdxexcos sinxexIntegration by parts can also be usd in connectionWith definite integrals,the formula is()()()()()()bbbaaaf x g x dxf x g xfx g x dxorbbbaaaudvuvvdu()()()()()()()()()()()()()()()()()()bbaabbbaaabbbaaaf x g xdxfx g xf x g x dxf x g xfx g x dxf
8、 x g x dxf x g x dxf x g xfx g x dxExampe.arctan10 xdx10arctan xx.2ln2141021dxxx10arctan xdx4102)1ln(21xExample.ln1exdxx211ln2exdxexx12ln21edxxx1212122e ex1241412e20)sin(xex2e2200sinsinxxexdxxde20cosxxde20cosxdxex20)sin(xex20)cos(xex20sinxdxex12e20sinxdxex20sinxdxex21212eExample20sinxexdxsoThusExamp
9、le Prove the reduction1211sincossinsinnnnnxdxxxxdxnn Where n2 is integerProof1sinsincosnnxdxxdx 11122122sincoscossinsincos(1)cossinsincos(1)(1sin)sinnnnnnnxxxdxxxnxxdxxxnxxdx 12212sincos(1)(1sin)sinsincos(1)sin(1)sinnnnnnxxnxxdxxxnxdxnxdx 12Thereforesincossin(1)sinnnnnxdxxxnxdx 12Thus11sincossinsinnnnnxdxxxxdxnn
