4.2第一换元积分法

第二节换元积分法)()]([)]([:xxfxf求导复合法则dxxxfxf)()]([)]([:两边积分)(:.凑微分法第一换元法公式一dxxxfdxxg)()()()()()(xxfxgxdx3cos.1例)(cosxdx3331.sincx331dxx8)12(.2例dxxx)()(1212218)()(1212218xdxcx912219)(.)(cx912181dxxx)3(3cos31udufxu)()(dxxx21.3例)1(12122xdxcx23213221.131232cxdxxx41.4例)(112124xdx.arctan212cxdxxx2ln.5例)ln(ln2xdx.ln313cxxdxxcossin.63例xdxsinsin3.sin414cx:凑微分常用的)(baxdadx1)()()(baxdabxdxdxdx222212121)()()(baxdakxdkdxxkkk111111)ln()ln()(lnxbadbxadxddxx11)()(bededdxexxx)(sin)(sincosbxdxddxx)(cos)(cossinbxdxddxx2211xdxdxx,以上方法不必硬背!而在于熟练运用课堂练习例8.求.dsec6xx解:原式=xdxx222sec)1(tanxtandxxxtand)1tan2(tan24x5tan51x3tan32xtanC.dsecsec24xxx例9求三角函数的不定积分当n为偶数时应先降次后再积分；结论:一般地,对形如这样的不定积分:当n为奇数时应先凑微分再积分；对形如这样的不定积分应先积化和差后再积分.dxxa221.10例axdaaxa211.arcsincaxdxxa221.11例axdaaxa2211.arctancaxa1.11,10加入基本积分表例例84.122xxxd例4222)()(xxd.2)2(arctan2110cx例.,的适当的函数运算是必要在积分过程中dxxtan.13例dxxxcossinxxdcoscos)(.cosln新公式cx)(.sinlncot新公式cxdxx.,可用微分法检验积分是否正确22.14axdx例dxaxaxa1121dxaxdxaxa1121caxaxalnln21)(.ln21新公式caxaxa22xadx22axdx)(.ln21新公式caxaxaxdxxcossin1dxxxtansec2xxdtantan.tanlncxdxxcsc.16例dxxsin12221xdxxcossin15例.15例.cotcscsincoscossinsintanxxxxxxxx12222222.tanlncx2.cotcsclncsccxxdxx)(新公式dxxsecxdx22csccxx22cotcscln.tanseclncxx)(新公式.cotcsclncsccxxdxx)(新公式dxxsec.tanseclncxxdxxxxxxtansec)tan(secsecxxdxxxxtansec)tansec(sec2xxxxdtansec)tan(sec例17..1dxex解法1xeeexxxd1)1(xdxxee1)1(dxCex)1ln(解法2xeexxd1xxee1)1(dCex)1ln()]1(ln[)1ln(xxxeee两法结果一样dxxx12321.18例dxxxxxxx123212321232dxxx41232dxxdxx12413241)()(121281323281xdxxdx.cxx331212132121dxxcos11.19例dxxx211coscosdxxxx221sincossindxxxxcotcsccsc2.csccotcxxdxxx2arcsin41.202例dxxx24121arcsindxxa221)(arcsin公式cax221xdxarcsinarcsin.arcsinlncx222cos2cos1:xdxxdx另解...sec222xxd.)(,cos)(sin.2122xfxxf求设例,sin.2xu记解.1)(:uuf代入上式得uduuf)()(1.cuu22.)(cxxxf22例22求.)1(arctandxxxx解.)(1arctan22xdxx原.1arctan22duuuxu令)(arctanarctan2uduCu2arctanCx2arctan)203P(:补充的积分公式.coslntan.cxdxx16.tanseclnsec.cxxdxx18.cotcsclncsc.cxxdxx19.arcsin.caxxadx2220.ln.caxaxaaxdx212222见下一节.ln.242222caxxaxxd.ln.cxaxaaxadx212322.arctan.caxaxadx12122.sinlncot.cxdxx17思考与练习1.下列各题求积方法有何不同?xx4d)1(xxxd4)3(2xxxd4)4(2224d)5(xx24d)6(xxx