总习题四

总习题四求下列不定积分(其中a,b为常数):1.xxeedx;解Ceedeedxeeeedxxxxxxxxx|11|ln2111122.2.dxxx3)1(;解Cxxdxxdxxdxxx2323)1(12111)1(1)1(1)1(.3.dxxax662(a0);解Caxaxaxdxadxxax||ln61)()()(1313333332323662.4.dxxxxsincos1;解Cxxxxdxxdxxxx|sin|ln)sin(sin1sincos1.5.dxxxlnln;解Cxxxdxxxxxxxxddxxxlnlnlnln1ln1lnlnlnlnlnlnlnlnln.6.dxxxx4sin1cossin;解Cxxdxxdxxdxxxx222244sinarctan21)(sin)(sin1121sinsin1sinsin1cossin.7.xdx4tan;解xxdxxdxxxdxtansintantancossintan22244xdxxxdxxtan)1tan11(tantan1tantan2224cxxxcxxxtantan31tanarctantantan3133.8.xdxxx3sin2sinsin;解xdxxxxdxxx3sin)cos3(cos213sin2sinsinxdxxxdxx3sincos213sin3cos21dxxxxxd)2sin4(sin41)3(cos3cos61Cxxx2cos814cos1613cos1212.9.)4(6xxdx;解Cxxdxxxxxxdx)4ln(241||ln41)41(41)4(6656.10.)0(adxxaxa;解dxxaxdxxaaduxaxadxxaxa2222221Cxaaxa22arcsin.11.)1(xxdx;解CxxCxxxdxxxdx)1ln(2))(1ln(2)(112)1(22.12.xdxx2cos;解xxdxdxxxxxdxx2sin4141)2cos(21cos22Cxxxxxdxxxx2cos812sin41412sin412sin414122.13.bxdxeaxcos;解因为dxbxeabbxeabxdeabxdxeaxaxaxaxsincos1cos1cosdxbxeabbxeabbxeadebxabbxeaaxaxaxaxaxcossincos1sincos12222,所以Cbxeabbxeabaabxdxeaxaxax)sincos1(cos2222Cbxbbxaebaax)sincos(122.14.xedx1;解duuuduuuduueedxxx)1111(112)1ln(11122令.ceecuuxx1111ln|11|ln.15.122xxdx;解Cttdttdtttttxxxdxsincostansectansec1sec1222令Cxx12.16.2/522)(xadx;解tdtatataxxadxcos)cos(1sin)(52/522令tdtadttatan)1(tan1cos112444Ctatatan1tan31434Cxaxaxaxa224322341)(31.17.241xxdx;解tdttttxxxdx2424secsectan1tan1令tdttdtttsinsincossincos4243Ctttdttsin1sin31sin)sin1sin1(324Cxxxx233213)1(.18.dxxxsin;解tdtttdttttxdxxxsin22sinsin2令tdtttttdt2cos2cos2cos222tdtttttttdttsin4sin4cos2sin4cos222Ctttttcos4sin4cos22Cxxxxxcos4sin4cos2.19.dxx)1ln(2;解dxxxxxxdxx22212)1ln()1ln(dxxxx)111(2)1ln(22Cxxxxarctan22)1ln(2.20.dxxx32cossin;解xdxxxxdxxdxxxtan)1tantan(tantancossincossin2232Cxx)1ln(tan21tan2122.21.dxxarctan;解xdxxxxdxx11arctanarctanxdxxx)111(arctanCxxxxarctanarctanCxxxarctan)1(.22.dxxxsincos1;解Cxxxdxdxxxxdxxx|2cot2csc|ln222csc22cos2sin22cos2sincos1.23.dxxx283)1(;解Cxxxdxxdxxx]arctan1[2141)1(141)1(484428283.提示:已知递推公式])()32()([)1(21)(122122222nnnaxdxnaxxnaaxdx.24.dxxxx234811;解dtttttxdxxxxdxxxx234123412322444884811令dtttdtttt)11241(41)23231(412Cttt|1|ln41|2|ln41Cxxx21ln414444.25.416xdx;解dxxxdxxxxdx)4141(81)4)(4(11622224Cxxx)2arctan21|22|ln41(81Cxxx2arctan161|22|ln321.26.dxxxsin1sin;解dxxxxdxxxxdxxx222cossinsinsin1)sin1(sinsin1sinCxxxdxxxxtansec)cos11cossin(22.27.dxxxxcos1sin;解dxxxdxxxdxxxxdxxxx2cossin212cos212cos2sincos1sin222dxxxxd2tan2tanCxxdxxdxxxx2tan2tan2tan2tan.28.dxxxxxex23sincossincos;解xdxxexdxexdxxxxxexxxsectancoscossincossinsin23sinxdexdxexxsecsinsinsinxxxxdeexxdesinsinsinsecsecxdxexexdxexexxxxcossecsecsinsinsinsinCexxexxsinsinsec.29.dxxxxx)(33;解dtttdtttttttxdxxxxx)111(66)()(52362633令CxxCtt66)1(ln1ln6.30.2)1(xedx;解dttttdtttteedxxx)1111(1111)1(222令Cttt1ln)1ln(Ceexxx11)1ln(.31.dxeeeexxxx1243;解)()(1111222243xxxxxxxxxxxxeedeedxeeeedxeeeeCeexx)arctan(Cx)sh2arctan(.32.dxexexx2)1(;解11)1()1()1(22xxxxxexdedexdxexexxxxxxdeeeexdxeex)1(11111xxxxdeeeex)111(1Ceeexxxx)1ln(ln1Ceexexxx)1ln(1.33.dxxx)1(ln22;解dxxxxxxxdxxx])1([ln)1(ln)1(ln222222dxxxxxxxx22221)1ln(2)1(ln22221)1ln(2)1(lnxdxxxxxdxxxxxxxxxx])1[ln(12)1ln(12)1(ln222222dxxxxxxx2)1ln(12)1(ln2222Cxxxxxxx2)1ln(12)1(ln2222.34.dxxx2/32)1(ln;解因为CxxCttdttdtttxdxx2232/321sincossecsec1tan)1(1令所以dxxxxxxxxxxddxxx111ln)1(ln)1(ln2222/32Cxxxxx)1ln(1ln22.35.xdxxarcsin12;解dtttttdtttxxdxx)2cos(21cossinarcsin122令tdttttttt2sin412sin41412sin414122Ctttt2cos812sin41412122241arcsin121)(arcsin41Cxxxxx.36.dxxxx231arccos;解2222231arccos1arccos1arccosxxdxdxxxxxdxxxxdxxxxxxx)arccos(1arccos12222dxxxxxxxxx)11arccos2(1arccos122222dxxxdxxxxxx2222arccos12arccos132322)1(arccos3231arccos1xxdxxxxdxxxxxxxx)1(32arccos)1(3231arccos1232322Cxxxxxxxx3323229232arccos)1(3231arccos1Cxxxxx)6(91arccos)1(131222.37.dxxxsin1cot;解xdxxxdxxdxxxsin)sin11sin1(sin)sin1(sin1sin1cotln|sinx|ln|1sinx|Cln|cscx1|C.38.xxdxcossin3;解xdxxxdxxxxdxxxxdxcotcos1cotcotcossincoscotcossin1cossin223122sin21|tan|lncot21|cot|lncot)cotcot1(CxxCxxxdxx.39.xxdxsin)cos2(;解令2tanxu,则duuuuduuuuuuxxdx)3(11212)112(1sin)cos2(222222Cuuduuduuu||ln31)3ln(31131323122Cxx|2tan32tan|ln313.40.dxxxxxcossincossin