高等数学第四章第二节课件.ppt

1、4.2 换元积分法一、第一类换元法二、第二类换元法上页下页铃结束返回首页上页下页铃结束返回首页积分表第二类换元法第二类换元法第一类换元法第一类换元法xxxfd)()(uufd)(基本思路基本思路 设,)()(ufuF)(xu可导,xxxfd)()(CxF)()(d)(xuuuf)()(xuCuF)(dxFxxxfd)()(则有上页下页铃结束返回首页积分表一、第一类换元法下页定理1(换元积分公式)设f(u)具有原函数,且u(x)可导,则有换元公式)()()()(xuduufdxxxf (也称配元法配元法,凑微分法凑微分法)上页下页铃结束返回首页积分表下页一、第一类换元法定理1(换元积分公式)设f

2、(u)具有原函数,且u(x)可导,则有换元公式)()()()()()(xuduufxdxfdxxxfCxFCuFxu)()()(设f(u)具有原函数F(u),则 换元积分过程 )()()()()()(xuduufxdxfdxxxf)()()()()()(xuduufxdxfdxxxfCxFCuFxu)()()()()()()(xuduufdxxxf 上页下页铃结束返回首页积分表CxFCuFduufxdxfdxxxf)()()()()()()(例 1)2(2cos)2(2cos2cos2xxddxxxxdx 例1 Cuudusincos例 2)23(23121)23(23121231xdxdxx

3、xdxx 例2 Cudxu|ln21121Cx|23|ln21例 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)()(222222

4、duexdedxxedxxeuxxx)()(222222duexdedxxedxxeuxxx)()(222222 CeCexu2 下页上页下页铃结束返回首页积分表例 4.)1(121)1(121122222xdxdxxxdxxx例 5.xdxdxxxxdxcoscos1cossintan 例4 Cuduu|ln1 例5 CxCuduu2322321)1(313121Cxxdx|cos|lntan,Cxxdx|sin|lncot 积分公式：)1(121)1(121122222xdxdxxxdxxx)1(121)1(121122222xdxdxxxdxxx CxCuduu2322321)1(313

5、121CxCuduu2322321)1(313121 xdxdxxxxdxcoscos1cossintanxdxdxxxxdxcoscos1cossintan Cuduu|ln1Cuduu|ln1ln|cos x|C CxFCuFduufxdxfdxxxf)()()()()()()(下页上页下页铃结束返回首页积分表CxFCuFduufxdxfdxxxf)()()()()()()(CxFCuFduufxdxfdxxxf)()()()()()()(例6 例 6.axdaxadxaxadxxa22222)(111)(1111Caxaarctan1 积分公式：例7 当a0时,dxaxadxxa222)

6、(1111Caxaxdaxarcsin)(112Caxadxxaarctan1122,Caxdxxaarcsin122 axdaxadxaxadxxa22222)(111)(1111axdaxadxaxadxxa22222)(111)(1111 dxaxadxxa222)(1111Caxaxdaxarcsin)(112dxaxadxxa222)(1111Caxaxdaxarcsin)(112dxaxadxxa222)(1111Caxaxdaxarcsin)(112 下页上页下页铃结束返回首页积分表 例8 CxFCuFduufxdxfdxxxf)()()()()()()(CxFCuFduufxd

7、xfdxxxf)()()()()()()(例 9 dxaxaxadxax)11(211221121dxaxdxaxa )(1)(121axdaxaxdaxa Caxaxa|ln|ln21 Caxaxa|ln21 Caxaxadxax|ln21122 dxaxaxadxax)11(21122 积分公式：下页上页下页铃结束返回首页积分表CxFCuFduufxdxfdxxxf)()()()()()()(CxFCuFduufxdxfdxxxf)()()()()()()(例 10 xxdxxdxxdxln21)ln21(21ln21ln)ln21(Cx|ln21|ln21 例 11 Cexdexdedx

8、xexxxx3333323322 例9 例10 xxdxxdxxdxln21)ln21(21ln21ln)ln21(xxdxxdxxdxln21)ln21(21ln21ln)ln21(Cexdexdedxxexxxx3333323322Cexdexdedxxexxxx3333323322Cexdexdedxxexxxx3333323322 下页上页下页铃结束返回首页积分表含三角函数的积分：例11 例12 例 12 xdxxxdxsinsinsin23xdxcos)cos1(2xxdxdcoscoscos2Cxx3cos31cos例 13 xxdxxdxxsincossincossin4252x

9、dxxsin)sin1(sin222 xdxxxsin)sinsin2(sin642 Cxxx753sin71sin52sin31xdxxxdxsinsinsin23xdxcos)cos1(2xdxxxdxsinsinsin23xdxcos)cos1(2 xxdxdcoscoscos2Cxx3cos31cos xxdxxdxxsincossincossin4252 下页上页下页铃结束返回首页积分表 例13 例14 例 14)2cos(2122cos1cos2xdxdxdxxxdxCxxxxddx2sin412122cos4121例 15 dxxxdx224)(coscosdxx2)2cos1(

10、21dxxx)2cos2cos21(412 dxxx)4cos212cos223(41 Cxxx)4sin812sin23(41 Cxxx4sin3212sin4183 )2cos(2122cos1cos2xdxdxdxxxdx)2cos(2122cos1cos2xdxdxdxxxdx Cxxxxddx2sin412122cos4121 dxxxdx224)(coscosdxx2)2cos1(21dxxxdx224)(coscosdxx2)2cos1(21 下页上页下页铃结束返回首页积分表例 17 dxxxdxsin1csc2cos2tan22cos2sin212xxxddxxx例 16 dx

11、xxxdxx)5cos(cos212cos3cos 例15 例16 Cxx5sin101sin21 CxxCxxxd|cotcsc|ln|2tan|ln2tan2tandxxxxdxx)5cos(cos212cos3cos dxxxdxsin1csc2cos2tan22cos2sin212xxxddxxxdxxxdxsin1csc2cos2tan22cos2sin212xxxddxxxdxxxdxsin1csc2cos2tan22cos2sin212xxxddxxx CxxCxxxd|cotcsc|ln|2tan|ln2tan2tanCxxCxxxd|cotcsc|ln|2tan|ln2tan

12、2tan Cxxxdx|cotcsc|lncsc 积分公式：下页上页下页铃结束返回首页积分表 例17 Cxxxdx|cotcsc|lncsc 例 18 dxxxdx)2csc(secCxx|)2 cot()2 csc(|ln ln|sec xtan x|C dxxxdx)2csc(sec Cxxxdx|tansec|lnsec 积分公式：首页上页下页铃结束返回首页积分表常用的几种配元形式常用的几种配元形式:xbxafd)()1()(bxaf)(dbxa a1xxxfnnd)()2(1)(nxfnxdn1xxxfnd1)()3()(nxfnxdn1nx1万能凑幂法xxxfdcos)(sin)4(

13、)(sin xfxsindxxxfdsin)(cos)5()(cosxfxcosd上页下页铃结束返回首页积分表思考与练习思考与练习1.下列各题求积方法有何不同?xx4d)1(24d)2(xxxxxd4)3(2xxxd4)4(2224d)5(xx24d)6(xxxxx4)4(d22221)(1)d(xx22214)4(dxxxxd441241xx2121xd2)2(4x)2(dx上页下页铃结束返回首页积分表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(d1011xxx101

14、x10d x10110(x10dx101上页下页铃结束返回首页积分表二、第二类换元法定理2 设x(t)是单调的、可导的函数,并且(t)0 又设f(t)(t)具有原函数F(t),则有换元公式其中t1(x)是x(t)的反函数 这是因为,由复合函数和反函数求导法则,)()(1)()()()(1xftfdtdxttfdxdttFxFCxFtFdtttfdxxf)()()()()(1 下页)()(1)()()()(1xftfdtdxttfdxdttFxF)()(1)()()()(1xftfdtdxttfdxdttFxF)()(1)()()()(1xftfdtdxttfdxdttFxF 上页下页铃结束返回

15、首页积分表常用的变换 令)2 2(sinttax,则 tatataxacoscossin12222 令)2 2(tanttax,则 tatataaxsecsectan12222tatataxacoscossin12222 dxacos tdt tatataaxsecsectan12222 dxasec2tdt 下页 令)2 0(secttax,则当 xa 时,tatataaxtantan1sec2222tatataaxtantan1sec2222 dxasec ttan tdt tatataxacoscossin12222,dxacos tdt tatataaxtantan1sec2222,d

16、xasec ttan tdt tatataaxsecsectan12222,dxasec2tdt 上页下页铃结束返回首页积分表Caxaaxaaxa22222arcsin2 tdtatdtatadxxatax22sin22coscoscos 令CxFCtFdtttfdxxftx)()()()()(1)(例19 例 19 求dxxa22(a0)解 tdtatdtatadxxatax22sin22coscoscos 令tdtatdtatadxxatax22sin22coscoscos 令下页Ctta)2sin4121(2Cttatacossin2222Cxaxaxa22221arcsin2 Ctta

17、)2sin4121(2Cttatacossin2222 dttatdta22cos1cos222dttatdta22cos1cos222注 进行变换和逆变换均要根据此图 积分表上页下页铃结束返回首页积分表CxFCtFdtttfdxxftx)()()()()(1)(例20 例 20 求22axdx(a0)解:(C1Clna)Ctttdtdttataaxdxtax|tansec|lnsecsecsec 2tan22令Ctttdtdttataaxdxtax|tansec|lnsecsecsec 2tan22令Ctttdtdttataaxdxtax|tansec|lnsecsecsec 2tan22令

18、下页Ctttdt|tansec|lnsecCaaxxCaaxax2222ln)ln(122)ln(Caxx,Ctttdt|tansec|lnsecCaaxxCaaxax2222ln)ln(积分表上页下页铃结束返回首页积分表12222|ln|lnCaxxCaaxax 例21 例 23 求22axdx(a0)解 当xa 时,Ctttdtdttattaaxdxtax|tansec|lnsectantansec sec22令(C1Clna)12222|ln|lnCaxxCaaxaxCtttdtdttattaaxdxtax|tansec|lnsectantansec sec22令Ctttdtdttatt

19、aaxdxtax|tansec|lnsectantansec sec22令Ctttdtdttattaaxdxtax|tansec|lnsectantansec sec22令Ctttdtdttattaaxdxtax|tansec|lnsectantansec sec22令下页积分表上页下页铃结束返回首页积分表当x0)解 当xa 时,Ctttdtdttattaaxdxtax|tansec|lnsectantansec sec22令(C1Clna)12222|ln|lnCaxxCaaxaxCtttdtdttattaaxdxtax|tansec|lnsectantansec sec22令Ctttdtd

20、ttattaaxdxtax|tansec|lnsectantansec sec22令Ctttdtdttattaaxdxtax|tansec|lnsectantansec sec22令Ctttdtdttattaaxdxtax|tansec|lnsectantansec sec22令积分表上页下页铃结束返回首页积分表原式21)1(22ta221a例22 求.d422xxxa解解:令,1tx 则txtdd21原式ttd12tttad)1(2122,0时当x42112tta Cata2223)1(23当 x 0 时,类似可得同样结果.Cxaxa32223)(23)1(d22ta倒代换上页下页铃结束返回

21、首页积分表补充积分公式 结束Cxxdx|cos|lntan,Cxxdx|sin|lncot,Cxxxdx|tansec|lnsec,Cxxxdx|cotcsc|lncsc,Caxadxxaarctan1122,Caxaxadxax|ln21122,Caxdxxaarcsin122,Caxxaxdx)ln(2222,Caxxaxdx|ln2222 积分表上页下页铃结束返回首页积分表小结小结:1.第二类换元法常见类型第二类换元法常见类型:,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第四节讲上页下页铃结束返回首页积分表(7)分母中因子次数较高时,可试用倒代换倒代换,d)()6(xafx令xat 上页下页铃结束返回首页积分表思考与练习思考与练习1.下列积分应如何换元才使积分简便?xxxd1)1(25xex1d)2()2(d)3(7xxx令21xt令xet1令xt1上页下页铃结束返回首页积分表作业作业P2042 (4),(6),(15),(19),(23),(37),(40),(41),(43)