[研究生入学考试]D4习题课ppt课件

1、高等数学Mathematic is the Queen of Science 高等数学Mathematic is the Queen of Science 习题课一、一、 求不定积分的根本方法求不定积分的根本方法二、几种特殊类型的积分二、几种特殊类型的积分不定积分的计算方法 第四章 高等数学Mathematic is the Queen of Science 高等数学Mathematic is the Queen of Science 一、一、 求不定积分的根本方法求不定积分的根本方法1. 直接积分法直接积分法经过简单变形, 利用根本积分公式和运算法那么2. 换元积分法换元积分法xxfd)(

2、第一类换元法第一类换元法tttfd)()( 第二类换元法(留意常见的换元积分类型) (代换: )(tx求不定积分的方法 .高等数学Mathematic is the Queen of Science 高等数学Mathematic is the Queen of Science dxxxf21)1(1xdxf1)1(2dxxxf1)(xdxf)(23dxxxf21)(arcsin)(arcsin)(arcsinxdxf4dxxxf21)(arctan)(arctan)(arctanxdxf5xdx2sinxd2sinxd2cos高等数学Mathematic is the Queen of Sci

3、ence 高等数学Mathematic is the Queen of Science 7dxxx2ln1xxdln89xdx1)1ln1 (xxd1ln1dxx)11 (2)1(xxd10dxx)11 (2)1(xxd6dxx)ln1 ()ln(xxd11dxx211)1ln(2xxd高等数学Mathematic is the Queen of Science 高等数学Mathematic is the Queen of Science 3. 3. 分部积分法分部积分法vuvu d运用原那么:1) 由dv易求出 v ;2)vud比uvd好求 .普通阅历: 按“反, 对, 幂,三 ,指 的顺序

4、,排前者取为 u , 排后者取为.vuvd高等数学Mathematic is the Queen of Science 高等数学Mathematic is the Queen of Science 例例1. 1. .d4932xxxxx解解:原式xxxxxd233222xxxd)(1)(23232xx2323232)(1)(dln1xaaaxxdlndCx3ln2ln)arctan(32高等数学Mathematic is the Queen of Science 高等数学Mathematic is the Queen of Science dxxxx42cossin3xxdxdxxx322ta

5、nsectan3xdxxxxtan) 1(sectan23Cxxxxcoslntan21tan23例例2 2高等数学Mathematic is the Queen of Science 高等数学Mathematic is the Queen of Science 例例3. 3. .darctanxeexx解解:xearctan原式xedxxeearctanxexeexxd12xxeearctanxeeexxxd1)1 (222xxeearctanxCex)1 (ln221高等数学Mathematic is the Queen of Science 高等数学Mathematic is the Q

6、ueen of Science 例例4. 4. .dcos1sinxxxx解法解法1 :原式2d2cos2xxx2tandxxln(1 cos )xtan2ln cosln 1 cos22xxxxC分部积分(1cos )1cosdxx高等数学Mathematic is the Queen of Science 高等数学Mathematic is the Queen of Science 例例4. 4. .dcos1sinxxxx解法解法2 :原式xxxxxd2cos22cos2sin222tandxxxxd2tanCxx2tan分部积分高等数学Mathematic is the Queen o

7、f Science 高等数学Mathematic is the Queen of Science 23/ 2ln(1)xxdxx21ln1xdx 221111lnxxdxxCxxxx)ln(ln221111例例5 522ln111xdxxxx 高等数学Mathematic is the Queen of Science 高等数学Mathematic is the Queen of Science 例例6. 6. .d1xx解解: 设1)(xxF1x,1x1x,1x那么)(xF1,1221xCxx1,2221xCxx)(xF延续 , , ) 1 ()1 ()1 (FFF得21211121CC22

8、1121CC记作Cxxd1)(xF1,21221xCxx1,21221xCxx,) 1(221Cx,) 1(221Cx利用 高等数学Mathematic is the Queen of Science 高等数学Mathematic is the Queen of Science 例例7 7试求)(xF)(lim0 xFx.xdex解：解：被积函数中含有绝对值符号， 故分段积分dxexdxex,1Cexdxex,2Cex其中 为恣意常数21,CC由原函数的定义 , 可知 延续 , 得)(xF)0(F11 C)(lim0 xFxlim20Cexx21 C212CC0 x0 x)(xF0 x,1Ce

9、x0 x,21Cex高等数学Mathematic is the Queen of Science 高等数学Mathematic is the Queen of Science 例例8. 8. 设设,)(2xyxy解解: 令, tyx求积分.d31xyxxyxy2)(即txy,123ttx,12tty而ttttxd) 1()3(d2222 1原式ttttd) 1()3(2222123tt132tttttd12Ct1ln221Cyx1)(ln221高等数学Mathematic is the Queen of Science 高等数学Mathematic is the Queen of Scienc

10、e 解解.)1ln(arctan2dxxxx求dxxx)1ln(2)1 ()1ln(2122xdx.21)1ln()1 (21222Cxxx)1ln()1(arctan21222xxxxd原式xxxxarctan)1ln()1(21222dxxxx1)1ln(21222例例9 9高等数学Mathematic is the Queen of Science 高等数学Mathematic is the Queen of Science .2)1ln(23)1ln()1(arctan212222Cxxxxxxxxxxxarctan)1ln()1(21222dxxxx1)1ln(21222高等数学Ma

11、thematic is the Queen of Science 高等数学Mathematic is the Queen of Science dxxxx11ln11102例Cxxxxdxx11411111212lnlnln高等数学Mathematic is the Queen of Science 高等数学Mathematic is the Queen of Science 例例11.11.设 解解:)(xF为)(xf的原函数,时时当当0 x,2sin)()(2xxFxf有有且,1)0(F,0)(xF求. )(xf由题设, )()(xfxF那么,2sin)()(2xxFxF故xxFxFd)(

12、)(xxd2sin2xxd24cos1即CxxxF4sin)(412,1)0(F, 1)0(2FC0)(xF, 因此14sin)(41xxxF故)()(xFxf14sin2sin412xxx又高等数学Mathematic is the Queen of Science 高等数学Mathematic is the Queen of Science 二、几种特殊类型的积分二、几种特殊类型的积分1. 普通积分方法普通积分方法有理函数分解多项式及部分分式之和指数函数有理式指数代换三角函数有理式万能代换简单无理函数三角代换根式代换高等数学Mathematic is the Queen of Scienc

13、e 高等数学Mathematic is the Queen of Science 2. 2. 需求留意的问题需求留意的问题(1) 普通方法不一定是最简便的方法 ,(2) 初等函数的原函数不一定是初等函数 ,要留意综合运用各种根本积分法, 简便计算 . 因此不一定都能积出.例如例如 , ,d2xex,dsinxxx,dsin2xx,dln1xx,1d4 xx,d13xx, ) 10(dsin122kxxk高等数学Mathematic is the Queen of Science 高等数学Mathematic is the Queen of Science 例例1. 1. 求求.1d632xxx

14、eeex解解: 令令,6xet 那么,ln6tx txtdd6原式原式ttttt)1 (d623tttt) 1)(1(d621331362ttttt dtln61ln3t) 1ln(232tCt arctan3Ceeexxxx636arctan3) 1ln() 1ln(323高等数学Mathematic is the Queen of Science 高等数学Mathematic is the Queen of Science 例例2.2. 求不定积分.dsin)cos2(1xxx解解: )cos(xu 令令原式 uuud) 1)(2(12) 1)(2(12uuuA21uB1uC31A61B2

15、1C2ln31u1ln61uCu1ln21)2ln(cos31x)cos1ln(61xCx) 1ln(cos21xxxxdsin)cos2(sin2高等数学Mathematic is the Queen of Science 高等数学Mathematic is the Queen of Science 例例3 3I12xxdx解法解法1 12tttdI当1x时，令,20secttxsec tansec tanttdtIttct cx1arccos当1x时，令, 1ttx12tttdct1arccoscx1arccos高等数学Mathematic is the Queen of Science

16、高等数学Mathematic is the Queen of Science 解法解法2 令,1tx 21 tdtIcx1arccos,12dttdxct arccos, 10 t21 tdtIcx1arccos,12dttdx1arcsinct , 01t高等数学Mathematic is the Queen of Science 高等数学Mathematic is the Queen of Science I342) 1() 1(xxdx解：解：令,113txx3211xt 232) 1(6ttdtdxI322)11() 1(xxxdxtdI23,1213txct 23,113cxx23例

17、例4 4321,1tx高等数学Mathematic is the Queen of Science 高等数学Mathematic is the Queen of Science 例例5. 5. 求求.d15)1ln(22xxxx解解:215)1ln(2xx原式5)1ln(d2xx21xxxxxd)1 (2121dxx325)1ln(2xxC23分析分析: 5)1ln(d2xx高等数学Mathematic is the Queen of Science 高等数学Mathematic is the Queen of Science 例例6 6： 设,2ln) 1(222xxxf且xxfln)(求(

18、 ).x dx解：解：1111ln22xx) 1(2xf1)(1)(ln)(xxxfxlnxxx1)(1)(2( )11xx xdx)(xdx)121 (Cxx1ln2高等数学Mathematic is the Queen of Science 高等数学Mathematic is the Queen of Science 例例7 7 知知,1)()(xfxxf求).(xf解解,1)()(xfxxf),(1 )(xfxxf解得221)(xxxxfdxxxxxf221)(dxxxx)1111(22.arctan)1ln(212cxxx高等数学Mathematic is the Queen of S

19、cience 高等数学Mathematic is the Queen of Science dxxx53cossin1提示：xx53cossin1xxxx5322cossincossinxxxx335cossin1cossin11cossin22xxxx5cossinxx3cossin2xxcossin13xx5cossinxx3cossin2x2sin6xx3sincos例例8 8高等数学Mathematic is the Queen of Science 高等数学Mathematic is the Queen of Science 例例9 9：求：求sin cossincosxxdxxx提

20、示提示: 原式原式21dxxxxxcossin1)cos(sin221dxxx)cos(sin221)4sin()4(xxdsin()sincossin cos444xxx高等数学Mathematic is the Queen of Science 高等数学Mathematic is the Queen of Science 例例10 10 求求dxxxxx23cossincosxesin提示: 原式xdxexsinsinxesinxdxxsectandxexsinxxesinxexsinsec dxexecxsxsincos 高等数学Mathematic is the Queen of Sc

21、ience 高等数学Mathematic is the Queen of Science 作业作业P221 1, 4, 6 , 9 ,14, 17 , 21 ,24, 26 , 30 , 38 , 39, 40 高等数学Mathematic is the Queen of Science 高等数学Mathematic is the Queen of Science 例例12.12.求解：解：,sincossin1xdxbxaxI由于.sincoscos2xdxbxaxI12IbIaxdxbxaxbxasincossincos1Cx12IabIxdxbxaxaxbsincossincos)sin

22、cos(xbxad2sincoslnCxbxa1I2ICxbxaabxba)sincosln(122Cxbxabaxba)sincosln(122高等数学Mathematic is the Queen of Science 高等数学Mathematic is the Queen of Science 例例6. 6. 求求.d)2(23xexxx解解: 取取,23xxuxev2)4(23 xx132xx660)(ku)4(kvxe2xe221xe241xe281xe2161xe2 原式)2(321 xx) 13(241xx681Cxxxex)7264(232816161CxxaxaexPxkndcossin)(阐明阐明: 此法特别适用于此法特别适用于如下类型的积分: 高等数学Mathematic is the Queen of Science 高等数学Mathematic is the Queen of Science 例例8. 8. 求求.942xxd解法解法1. 查积分表查积分表令,2xu 那么原式P349 公式 37Cuu33ln3122Cxx239