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1、積分表4. 2 換元積分法一、第一類換元法二、第二類換元法積分表第二類換元法第二類換元法第一類換元法第一類換元法xxxfd)()(uufd)(基本思路基本思路 設, )()(ufuF)(xu可導,xxxfd)()(CxF)()(d)(xuuuf)()(xuCuF)(dxFxxxfd)()(則有積分表一、第一類換元法v定理1(換元積分公式) 設f(u)具有原函數(shù), 且u(x)可導, 則有換元公式 )()()()(xuduufdxxxf (也稱配元法配元法 , 湊微分法湊微分法)積分表一、第一類換元法v定理1(換元積分公式) 設f(u)具有原函數(shù), 且u(x)可導, 則有換元公式 )()()()(

2、)()(xuduufxdxfdxxxfCxFCuFxu)()()( 設f(u)具有原函數(shù)F(u), 則 v換元積分過程 )()()()()()(xuduufxdxfdxxxf)()()()()()(xuduufxdxfdxxxfCxFCuFxu)()()( )()()()(xuduufdxxxf 積分表CxFCuFduufxdxfdxxxf)()()()()()()(例 1 )2(2cos)2(2cos2cos2xxddxxxxdx 例1 Cuudusincos例 2 )23 (23121)23 (23121231xdxdxxxdxx 例2 Cudxu|ln21121Cx |23 |ln21例

3、 3 duexdedxxedxxeuxxx)()(222222 例3 CeCexu2)2(2cos)2(2cos2cos2xxddxxxxdx)2(2cos)2(2cos2cos2xxddxxxxdx CuudusincosCuudusincossin 2xC )23 (23121)23 (23121231xdxdxxxdxx)23 (23121)23 (23121231xdxdxxxdxxCudxu|ln21121Cx |23 |ln21Cudxu|ln21121Cx |23 |ln21 duexdedxxedxxeuxxx)()(222222duexdedxxedxxeuxxx)()(22

4、2222duexdedxxedxxeuxxx)()(222222 CeCexu2 積分表例 4. )1 (121)1 (121122222xdxdxxxdxxx例 5. xdxdxxxxdxcoscos1cossintan 例4 Cuduu|ln1 例5 CxCuduu2322321)1 (313121Cxxdx|cos|lntan, Cxxdx|sin|lncot 積分公式：)1 (121)1 (121122222xdxdxxxdxxx)1 (121)1 (121122222xdxdxxxdxxx CxCuduu2322321)1 (313121CxCuduu2322321)1 (31312

5、1 xdxdxxxxdxcoscos1cossintanxdxdxxxxdxcoscos1cossintan Cuduu|ln1Cuduu|ln1ln|cos x|C CxFCuFduufxdxfdxxxf)()()()()()()(積分表CxFCuFduufxdxfdxxxf)()()()()()()(CxFCuFduufxdxfdxxxf)()()()()()()( 例6 例 6. axdaxadxaxadxxa22222)(111)(1111Caxaarctan1 積分公式： 例7 當a0時,dxaxadxxa222)(1111Caxaxdaxarcsin)(112Caxadxxaarc

6、tan1122, Caxdxxaarcsin122 axdaxadxaxadxxa22222)(111)(1111axdaxadxaxadxxa22222)(111)(1111 dxaxadxxa222)(1111Caxaxdaxarcsin)(112dxaxadxxa222)(1111Caxaxdaxarcsin)(112dxaxadxxa222)(1111Caxaxdaxarcsin)(112 積分表 例8 CxFCuFduufxdxfdxxxf)()()()()()()(CxFCuFduufxdxfdxxxf)()()()()()()(例 9 dxaxaxadxax)11(2112211

7、21dxaxdxaxa )(1)(121axdaxaxdaxa Caxaxa|ln|ln21 Caxaxa|ln21 Caxaxadxax|ln21122 dxaxaxadxax)11(21122 積分公式：積分表CxFCuFduufxdxfdxxxf)()()()()()()(CxFCuFduufxdxfdxxxf)()()()()()()(例 10 xxdxxdxxdxln21)ln21 (21ln21ln)ln21 (Cx |ln21 |ln21 例 11 Cexdexdedxxexxxx3333323322 例9 例10 xxdxxdxxdxln21)ln21 (21ln21ln)ln

8、21 (xxdxxdxxdxln21)ln21 (21ln21ln)ln21 ( Cexdexdedxxexxxx3333323322Cexdexdedxxexxxx3333323322Cexdexdedxxexxxx3333323322 積分表含三角函數(shù)的積分： 例11 例12 例 12 xdxxxdxsinsinsin23xdxcos)cos1 (2xxdxdcoscoscos2Cxx3cos31cos例 13 xxdxxdxxsincossincossin4252xdxxsin)sin1 (sin222 xdxxxsin)sinsin2(sin642 Cxxx753sin71sin52s

9、in31xdxxxdxsinsinsin23xdxcos)cos1 (2xdxxxdxsinsinsin23xdxcos)cos1 (2 xxdxdcoscoscos2Cxx3cos31cos xxdxxdxxsincossincossin4252 積分表 例13 例14 例 14 )2cos(2122cos1cos2xdxdxdxxxdxCxxxxddx2sin412122cos4121例 15 dxxxdx224)(coscosdxx2)2cos1 (21dxxx)2cos2cos21 (412 dxxx)4cos212cos223(41 Cxxx)4sin812sin23(41 Cxxx

10、4sin3212sin4183 )2cos(2122cos1cos2xdxdxdxxxdx)2cos(2122cos1cos2xdxdxdxxxdx Cxxxxddx2sin412122cos4121 dxxxdx224)(coscosdxx2)2cos1 (21dxxxdx224)(coscosdxx2)2cos1 (21 積分表例 17 dxxxdxsin1csc2cos2tan22cos2sin212xxxddxxx例 16 dxxxxdxx)5cos(cos212cos3cos 例15 例16 Cxx5sin101sin21 CxxCxxxd|cotcsc|ln|2tan|ln2tan

11、2tandxxxxdxx)5cos(cos212cos3cos dxxxdxsin1csc2cos2tan22cos2sin212xxxddxxxdxxxdxsin1csc2cos2tan22cos2sin212xxxddxxxdxxxdxsin1csc2cos2tan22cos2sin212xxxddxxx CxxCxxxd|cotcsc|ln|2tan|ln2tan2tanCxxCxxxd|cotcsc|ln|2tan|ln2tan2tan Cxxxdx|cotcsc|lncsc 積分公式：積分表 例17 Cxxxdx|cotcsc|lncsc 例 18 dxxxdx)2csc(secCx

12、x| )2 cot()2 csc(|ln ln|sec xtan x|C dxxxdx)2csc(sec Cxxxdx|tansec|lnsec 積分公式：積分表常用的幾種配元形式常用的幾種配元形式: xbxafd)() 1 ( )(bxaf)(dbxa a1xxxfnnd)()2(1)(nxfnxdn1xxxfnd1)()3()(nxfnxdn1nx1萬能湊冪法xxxfdcos)(sin)4()(sin xfxsindxxxfdsin)(cos)5()(cosxfxcosd積分表思考與練習思考與練習1. 下列各題求積方法有何不同? xx4d) 1 (24d)2(xxxxxd4)3(2xxxd

13、4)4(2224d)5(xx24d)6(xxxxx4)4(d22221)(1)d(xx22214)4(dxxxxd441241xx2121xd2)2(4x)2(dx積分表xxxd) 1(1102. 求.) 1(d10 xxx提示提示:法法1法法2法法3 ) 1(d10 xxx10)x ) 1(d10 xxx) 1(1010 xx ) 1(d10 xxx)1 (d1011xxx101x10d x10110(x10dx101積分表二、第二類換元法v定理2 設x(t)是單調的、可導的函數(shù), 并且(t)0 又設f (t)(t)具有原函數(shù)F(t), 則有換元公式其中t1(x)是x(t)的反函數(shù) 這是因為

14、, 由復合函數(shù)和反函數(shù)求導法則,)()(1)()()()(1xftfdtdxttfdxdttFxFCxFtFdtttfdxxf)()()()()(1 )()(1)()()()(1xftfdtdxttfdxdttFxF)()(1)()()()(1xftfdtdxttfdxdttFxF)()(1)()()()(1xftfdtdxttfdxdttFxF 積分表v常用的變換 令)2 2 ( sinttax, 則 tatataxacoscossin12222 令)2 2 ( tanttax, 則 tatataaxsecsectan12222tatataxacoscossin12222 dxacos td

15、t tatataaxsecsectan12222 dxasec2tdt 令)2 0( secttax, 則當 xa 時, tatataaxtantan1sec2222tatataaxtantan1sec2222 dxasec ttan tdt tatataxacoscossin12222, dxacos tdt tatataaxtantan1sec2222, dxasec ttan tdt tatataaxsecsectan12222, dxasec2tdt 積分表Caxaaxaaxa22222arcsin2 tdtatdtatadxxatax22sin22coscoscos 令CxFCtFd

16、tttfdxxftx)()()()( )(1)( 例19 例 19 求dxxa22(a0) 解 tdtatdtatadxxatax22sin22coscoscos 令tdtatdtatadxxatax22sin22coscoscos 令Ctta)2sin4121(2Cttatacossin2222Cxaxaxa22221arcsin2 Ctta)2sin4121(2Cttatacossin2222 dttatdta22cos1cos222dttatdta22cos1cos222注 進行變換和逆變換均要根據(jù)此圖 積分表積分表CxFCtFdtttfdxxftx)()()()( )(1)( 例20

17、例 20 求22axdx(a0) 解: (C1Clna) Ctttdtdttataaxdxtax|tansec|lnsecsecsec 2tan22令Ctttdtdttataaxdxtax|tansec|lnsecsecsec 2tan22令Ctttdtdttataaxdxtax|tansec|lnsecsecsec 2tan22令Ctttdt|tansec|lnsecCaaxxCaaxax2222ln)ln(122)ln(Caxx, Ctttdt|tansec|lnsecCaaxxCaaxax2222ln)ln(積分表積分表12222|ln|lnCaxxCaaxax 例21 例 23 求22

18、axdx(a0) 解 當xa 時, Ctttdtdttattaaxdxtax|tansec|lnsectantansec sec22令(C1Clna) 12222|ln|lnCaxxCaaxaxCtttdtdttattaaxdxtax|tansec|lnsectantansec sec22令Ctttdtdttattaaxdxtax|tansec|lnsectantansec sec22令Ctttdtdttattaaxdxtax|tansec|lnsectantansec sec22令Ctttdtdttattaaxdxtax|tansec|lnsectantansec sec22令積分表積分表當

19、x0) 解 當xa 時, Ctttdtdttattaaxdxtax|tansec|lnsectantansec sec22令(C1Clna) 12222|ln|lnCaxxCaaxaxCtttdtdttattaaxdxtax|tansec|lnsectantansec sec22令Ctttdtdttattaaxdxtax|tansec|lnsectantansec sec22令Ctttdtdttattaaxdxtax|tansec|lnsectantansec sec22令Ctttdtdttattaaxdxtax|tansec|lnsectantansec sec22令積分表積分表原式21)

20、1(22ta221a例22 求.d422xxxa解解: 令,1tx 則txtdd21原式ttd12tttad) 1(2122,0時當x42112tta Cata2223) 1(23當 x 0 時, 類似可得同樣結果 .Cxaxa32223)(23) 1(d22ta倒代換積分表v補充積分公式 Cxxdx|cos|lntan,Cxxdx|sin|lncot,Cxxxdx|tansec|lnsec,Cxxxdx|cotcsc|lncsc, Caxadxxaarctan1122,Caxaxadxax|ln21122, Caxdxxaarcsin122,Caxxaxdx)ln(2222, Caxxaxdx|ln2222 積分表積分表小結小結:1. 第二類換元法常見類型第二類換元法常見類型: ,d),() 1 (xbaxxfn令nbxat,d),()2(xxfndxcbxa令ndxcbxat,d),()3(22xxaxf令taxsin或taxcos,d),()4(22xxaxf令taxtan,d),()5(22xaxxf令taxsec第四節(jié)講積分表(7) 分母中因子次數(shù)較高時, 可試用倒代換倒代換 ,d)()6(xafx令xat 積分表思考與練習思考與練習1. 下列積分應如何換元才使積分簡便 ?xxxd1) 1 (25xex1d)2( )2(d)3(7xxx令21xt令xet1令xt1