2017年江苏省专转本高数一不定积分

第三章不定积分本章主要知识点：不定积分的意义，基本公式不定积分的三种基本方法杂例一、不定积分的意义、基本公式不定积分基本特点是基本公式较多，灵活善变，复习此章节主要诀窍在于：基本公式熟练，基本题型运算快捷，有一定题量的训练。1．性质()()fxdxfx()()dfxdxfxdxCxFxdF)()(()()fxdxfxC()(1)()nnfdxfxC2．基本公式（1）11(1)1nnxdxxcnn，cxdxx||ln1（2）caadxaxxln，cedxexx（3）cxxdxcossin，cxxdxsincos，-79-cxxdxtansec2，cxxdxcotcsc2（4）caxdxxaarcsin122，（5）cxaxaadxxa||ln21122（6）221ln||dxxxacxa（7）caxaxaarctan1122二、不定积分的三种基本方法1．凑微分法（第一类交换法）基本原理：()(())xdxdxdx。一些常见的固定类型)()(1)(baxdbaxfadxbaxf1()()xxxxfeedxfede222)(21)(dxxfdxxxfnnnndxxfndxxfx)(1)(1xdxfdxxfxln)(ln)(ln1xdxfdxxxfcos)(cos)(cossinxdxfdxxxfsin)(sin)(sincos21111fdxfdxxxx-80-xdxfdxxxftan)(tan)(tansec2tansec(sec)(sec)secxxfxdxfxdx等等。例3.1．22007(21)xxdx解：原式=2200722200811(21)(21)(21)48032xdxxc例3.2．3sin13sin13sin111cos(3sin1)33xxxxedxedxec例3.3．23sin(57)xxdx解：原式331sin(57)3xdx331sin(57)(57)15xdx31cos(57)15xC例3.4．dxxxx1ln2ln1解：原式xdxxln1ln2lnduuuxu1211221lnduu)1211(2111ln2124uuCCxx1ln2ln41ln21例3.5．44xdxx解：原式=2222)(2121dxxCx2arctan412例3.6．221cos(2tan1)dxxx解：原式222sec1tan12tan12tanxdxdxxx1arctan(2tan)2xC例3.7．2sin(2)xdxx解：原式21112sin22(1cos2)sin(2)224xdxuxuduuuC-81-=1sin(4)4xxC例3.8．dxexex解：原式xxxexexeeedxedeeC例3.9．dxxxx2233解：利用综合除法知12127222323xxxxxx原式Cxxxxdxxxx2ln12731)21272(232例3.10．dxxxxx13236解：原式42222(1)1xxxxdxx53222211111(1)253211xxxxdxdxxx5322111ln(1)2arctan532xxxxxxC例3.11．,sin1dxxdxxcos1解：221sincos11coslnsinsin1cos21cosxdxxdxdxCxxxxdxxcos122cossin11sinlncos1sin21sinxdxxdxCxxx注：此例对于三角函数相当重要，请熟练掌握。*例3.12．dxxcos21解：原式dxxxx)cos2)(cos2(cos2xxddxxdxxx222sin3sincos42cos4cos2-82-222sec1sinarctan()4sec133xxdxx2tan1sin2arctan()4tan333dxxx222tan1sinarctan()(2tan)(3)33dxxx1tan1sinarctan(2)arctan()3333xxxC例3.13．sinsincosxdxxx解：原式=1(sincossincos)2sincosxxxxdxxx=11(cossin)22sincosdxxdxxx=11lnsincos22xxxc例3.14．cos2sin3cosxdxxx解：令()2sin3cosfxxx，则()2cos3sin,fxxx32cos()()1313xfxfx原式＝32()()321313ln|2sin3cos|()1313fxfxdxxxxCfx例3.15．2212sincosdxxx解：原式＝222sec11tanarctan(2tan)2tan12tan12xdxdxxCxx例3.16．xdx4tan解：原式=dxxxx]1)1(tantan[tan224=dxxdxdxxx222sec)tan1(tan=2tantantanxdxxxc=31tantan3xxxc例3.17．dxxxx22322-83-解：原式=dxxx1)1(5)1(22=222(1)15(1)1(1)1dxdxxx=cxxx)1arctan(5)22ln(2例3.18．232xdxxx解：原式=2114(1)xdxx=2221(1)1(1)24(1)4(1)dxdxxx=-214(1)arcsin2xxc例3.19．21xedx解：原式=dxeeexxx22211=2221xxdexe=cexx)1ln(22例3.20．dxxx)23)(121（解：原式=dxxxxx)23)(12()23(2)12(3=dxxdxx122233=cxx12ln23ln例3.21．dxxx)1()1(12解：原式=dxxxxx)1()1(11212=dxxx211211121=1111ln2141xcxx例3.22．dxex112解：原式=211xxdxee=dxeexx21=2)(1xxede=cexarcsin例3.23．33sectanxxdx-84-解：原式＝22425311sectansec(secsec)secsecsec53xxdxxxdxxxC例3.24．324arctan1xxdxx解：原式＝3223224241tan11tan(arctan)arctan()2128xdxxdxxCx2．直接交换法a）题型dxbaxf)(方法：令baxt，abtx)(2，2()()faxbdxtftdta例3.25．dxx11解：令2,txxt，原式=tdtt211=122tdtdt=ctt1ln22=cxx)1ln(22例3.26．1213dxxx解：令1,12txxt原式=22222211112(1)24(1)3(1)3(1)3ttdtdtdtdtttttt=211111ln(24)arctan()ln(213)arctan()3333txttCxxC例3.27．dxxx31解：原式665236xtxttdttt=dttt163=dtttt)111(62=ctttt)1ln(63223=cxxxx)1ln(632663-85-例3.28．dxex11解：原式21ln(1)xtextdtttt1212=dtt1122=ctt11ln=ceexx)1111ln(b)题型dxbaxf)(222()faxdx变换taxsin22()faxdx变换taxtan22()fxadx变换taxsec例3.29．dxxx29解：令3sinxt，原式3cos3cos3sinttdtt=dtttsinsin132=tdtdttsin3sin13=31cosln3cos21costtCt=22291393ln23913xxCx例3.30．4211dxxx解：令tanxt，原式232444seccos1sinsintansecsinsintttdtdtdttttt=31csccsc3ttC例3.31．324xdxx-86-解：令2secxt，原式348sec2tansec8sec2tantttdttdtt=2388(1tan)tan8tantan3tdtttC（还原略）例3.32．32211dxx解：令tanxt，原式2321seccossinsec1xtdttdttcctx例3.33．21(2)22dxxxx解：令1tanxt，原式＝221sincossincossincosttdtdttttt＝22cossin12cos12sindtdttt＝112cos112sinln||ln||2212cos2212sinttCtt（还原略）。3．分部积分法公式：udvuvvdu四种基本题型a）题型1xmPxedx例3.34．2(21)xxedx解：原式=222111(21)(21)2222xxxxdexeedx=221(21)2xxxeeC例3.35．21xedx-87-解：原式21txttttetdttdeteeC=212121xxxeeC例3.36．2432xxxedx解：24242221()222xxxxxeded22xu4422112222xxuuuedueudee=4224222221122222xxxxuuueeeCxeeeC题型2()cosmPxxdx或()sinmPxxdx例3.37．3sin(21)xxdx解：原式=333cos(21)cos(21)cos(21)222xdxxxxdx=33cos(21)sin(21)24xxxC例3.38．2cosxxdx解:原式=21cos21sin2244xxxdxxdx211sinsin2444xxxxdx=211sin2cos(2)448xxxxC例3.39．2cosxdxx解：原式＝tantantantanln|cos|xdxxxxdxxxxC例3.40．sin(1)xdx解：原式sin(1)22cos(1)2cos(1)2cos(1)txttdttdttttdt2cos(1)2sin(1)2cos(1)2sin(1)tttCxxxC-88-题型3cosxexdx或sinxexdx例3.41．2cos3xexdx解：设2222113cos3cos3co