不定积分公式

Ch4、不定积分§1、不定积分的概念与性质1、原函数与不定积分定义1：若)()(xfxF，则称)(xF为)(xf的原函数。①连续函数一定有原函数；②若)(xF为)(xf的原函数，则CxF)(也为)(xf的原函数；事实上，)()()(''xfxFCxF③)(xf的任意两个原函数仅相差一个常数。事实上，由0)()()()()()('2'1'11xfxfxFxFxFxF，得CxFxF)()(21故CxF)(表示了)(xf的所有原函数，其中)(xF为)(xf的一个原函数。定义2：)(xf的所有原函数称为)(xf的不定积分，记为dxxf)(，积分号，)(xf被积函数，x积分变量。显然CxFdxxf)()(①Ckxkdx②1ln1111CxCxdxx2、基本积分表（共24个基本积分公式）3、不定积分的性质①dxxgdxxfdxxgxf)()()()(②)0()()(kdxxfkdxxkf⑤Cxxxdxxxdxdxxxxcsccotcotcsccsccotcsccsc2⑥Cxxxdxxdxdxxxxxxxdxtancotseccsccossincossincossin22222222⑦Cxxdxxdxxcot1csccot222§2、不定积分的换元法一、第一类换元法（凑微分法）1、baxdadxbaxdbaxfadxbaxf1,1即例1、求不定积分①Cxuduuxxxdxdx)5cos(51sin51555sin515sin②CxCxxdxdxx81777211612117121)21(212121③)20(arctan111222Caxaaxaxdaxadx④)23(arcsin1222Caxaxaxdxadx2、nnnnnndxdxxdxxfndxxxf11,1即例2、求不定积分①CxCxxdxdxxx232121221221221311112111211②Cexdedxexxxx333323131③xddxxCxxdxdxxx111sin11cos1cos122④xddxxCxxdxdxxx21sin2cos2cos3、,tansec,sincos,cossin,,ln12xdxdxxdxdxxdxdxdedxexddxxxx,,arcsin11,arctan11,sectansec222222xaddxxaxxddxxxddxxxdxdxx①)16(seclncoslncoscoscossintanCxCxxxddxxxxdx3②)17(coslnsinlnsinsinsincoscotCxCxxxddxxxxdx③)18(tanseclntansectansectansectansecsecsecCxxxxxxddxxxxxxxdx④)19(cotcsclncotcsccotcsccotcsccotcsccsccscCxxxxxxddxxxxxxxdx⑤Cxxxddxxxlnlnlnlnln1⑥Cxxxdxxdx1tanln1tan1tantan1cos2⑦Ceeeddxeexxxxx1ln111⑧Cexeeeedxxxxxx1ln111⑨Ceededxeexxxxxarctan1122⑩Cexdedxexxxxx2122121211例4、求不定积分①axaxdaxaxdadxaxaxaaxdx)()(21112122)22)(21(ln21Caxaxa②dxxxdxxxxdxxxx2222213113112Cxxxxdxxxdxarctan31ln211311212222③413525221526222152422222xdxxxxxddxxxxdxxxxCxxx21arctan2352ln212④Cxxxxdxdxxxdx2sin412122cos21212122cos1sin24⑤Cxxdxxxxdxx2cos418cos1612sin8sin213cos5sin⑥Cxxxdxxxdxxdxdxxxsinlnlnsinlnsinlnsinlnsinsinsinlnsincossinlncot⑦Cxxxxdxdxdxxxxdxcos1tancoscosseccossin1sin1222⑧44csc214sin2sincosxdxxdxxxdxCxx4cot4cscln21二、第二类换元法1、三角代换例1、dxxa22解：令)cos(sintatax或，则tdtadxtaxacos,cos22原式=ttddtadttatdtata22cos21222cos1coscos22CaxaaxaaxaCtata22222224arcsin22sin42Cxaxaxa22221arcsin21例2、Caxaxaxdxadxarcsin1222解：令taxsin原式=CaxCtdttatdtaarcsincoscos例3、22xadx解：令)cot(tantatax或，则tdtadxtaxa222sec,sec5原式=CaxaaxCtttdttatdta222lntanseclnsecsecsec)24(ln22Caxx例4、42xxdx解：令)cot(tantatax或，则tdtdxtx22sec2,sec24原式=CaxaaxCtttdttatdta222lntanseclnsecsecsec例5、22axdx解：令)csc(sectatax或，则tdttadxtaaxtansec,tan22原式=caaxaxCtttdttatdtta22lntanseclnsectantansec)25(ln22Caxx例6、dxxx92解：令taxsec，则tdttdxtxtansec3,tan392原式=Cttttdttdtttttan31sec3tan3tansec3sec3tan322CxxCxx3arccos393arccos39322小结：)(xf中含有222222axaxxa可考虑用代换taxtaxtaxsectansin2、无理代换6例7、311xdx解：令dttdxtxtx2333,1,1则原式=Ctttdtttdttttdtt1ln231113111313222Cxxx333211ln313123例8、31xxdx解：令dttdxtxtx5666,,则原式=Cttdttdtttttdttarctan611161616222235Cxx66arctan6例9、dxxxx11解：令22212,11,1ttdtdxtxtxx则原式=Ctttdttdtttttdttt11ln212111212121222222Cxxxxxx11ln12例10、xedx1解：令12,1ln,122ttdtdxtxtex则原式CeeCtttdtdttttxx1111ln11ln21212121224、倒代换7例11、46xxdx解：令2676,4111,1tdtdxttxxtx则原式CxxCtttdtdtt4ln24114ln2411414241416666666Cxx4ln241ln416§3、分部积分法分部积分公式：VUUVVUVUVUUV,VdxUdxUVdxVU，故VdUUVUdV（前后相乘）（前后交换）例1、xdxxcosCxxxxdxxxxxdcossinsinsinsin例2、dxxexCexedxexexdexxxxx例3、xdxlnCxxxdxxxxxxxdxxln1lnlnln或解：令textx,ln原式CxxxCetedtetetdetttttln例4、xdxarcsinCxxxxxdxxdxxxxxxxdxx22221arcsin1121arcsin1arcsinarcsinarcsin或解：令txtxsin,arcsin原式CxxxCttttdtttttd21arcsincossinsinsinsin8例5、xdxexsinxdxexxexdexexexdexexdxexexdexxxxxxxxxxsincossincoscossincossincossinsin故Cxxexdxexxcossin21sin例6、dxxx2cosCxxxxdxxxxxdseclntantantantan例7、dxxx21lnCxxxxdxxxxxxdxxxxxxxxx222222211ln11ln1111ln§4、两种典型积分一、有理函数的积分有理函数01110111)()()(bxbxbxbaxaxaxaxQxPxRmmmmnnnn可用待定系数法化为部分分式，然后积分。例1、将6532xxx化为部分分式，并计算dxxxx6532解：3223323236532xxBAxBAxBxAxxxxxx653231BABABA故Cxxxdxxdxdxxxx)3ln(6)2ln(536256532或解：65211656521651152212222xxdxxxxxddxxxxIdxxxxx213121165ln2129Cxxxx