[研究生入學(xué)考試]D4習(xí)題課ppt課件

1、高等數(shù)學(xué)Mathematic is the Queen of Science 高等數(shù)學(xué)Mathematic is the Queen of Science 習(xí)題課一、一、 求不定積分的根本方法求不定積分的根本方法二、幾種特殊類型的積分二、幾種特殊類型的積分不定積分的計(jì)算方法 第四章 高等數(shù)學(xué)Mathematic is the Queen of Science 高等數(shù)學(xué)Mathematic is the Queen of Science 一、一、 求不定積分的根本方法求不定積分的根本方法1. 直接積分法直接積分法經(jīng)過簡(jiǎn)單變形, 利用根本積分公式和運(yùn)算法那么2. 換元積分法換元積分法xxfd)(

2、第一類換元法第一類換元法tttfd)()( 第二類換元法(留意常見的換元積分類型) (代換: )(tx求不定積分的方法 .高等數(shù)學(xué)Mathematic is the Queen of Science 高等數(shù)學(xué)Mathematic is the Queen of Science dxxxf21)1(1xdxf1)1(2dxxxf1)(xdxf)(23dxxxf21)(arcsin)(arcsin)(arcsinxdxf4dxxxf21)(arctan)(arctan)(arctanxdxf5xdx2sinxd2sinxd2cos高等數(shù)學(xué)Mathematic is the Queen of Sci

3、ence 高等數(shù)學(xué)Mathematic is the Queen of Science 7dxxx2ln1xxdln89xdx1)1ln1 (xxd1ln1dxx)11 (2)1(xxd10dxx)11 (2)1(xxd6dxx)ln1 ()ln(xxd11dxx211)1ln(2xxd高等數(shù)學(xué)Mathematic is the Queen of Science 高等數(shù)學(xué)Mathematic is the Queen of Science 3. 3. 分部積分法分部積分法vuvu d運(yùn)用原那么:1) 由dv易求出 v ;2)vud比uvd好求 .普通閱歷: 按“反, 對(duì), 冪,三 ,指 的順序

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

5、nsectan3xdxxxxtan) 1(sectan23Cxxxxcoslntan21tan23例例2 2高等數(shù)學(xué)Mathematic is the Queen of Science 高等數(shù)學(xué)Mathematic is the Queen of Science 例例3. 3. .darctanxeexx解解:xearctan原式xedxxeearctanxexeexxd12xxeearctanxeeexxxd1)1 (222xxeearctanxCex)1 (ln221高等數(shù)學(xué)Mathematic is the Queen of Science 高等數(shù)學(xué)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分部積分(1cos )1cosdxx高等數(shù)學(xué)Mathematic is the Queen of Science 高等數(shù)學(xué)Mathematic is the Queen of Science 例例4. 4. .dcos1sinxxxx解法解法2 :原式xxxxxd2cos22cos2sin222tandxxxxd2tanCxx2tan分部積分高等數(shù)學(xué)Mathematic is the Queen o

7、f Science 高等數(shù)學(xué)Mathematic is the Queen of Science 23/ 2ln(1)xxdxx21ln1xdx 221111lnxxdxxCxxxx)ln(ln221111例例5 522ln111xdxxxx 高等數(shù)學(xué)Mathematic is the Queen of Science 高等數(shù)學(xué)Mathematic is the Queen of Science 例例6. 6. .d1xx解解: 設(shè)1)(xxF1x,1x1x,1x那么)(xF1,1221xCxx1,2221xCxx)(xF延續(xù) , , ) 1 ()1 ()1 (FFF得21211121CC22

8、1121CC記作Cxxd1)(xF1,21221xCxx1,21221xCxx,) 1(221Cx,) 1(221Cx利用 高等數(shù)學(xué)Mathematic is the Queen of Science 高等數(shù)學(xué)Mathematic is the Queen of Science 例例7 7試求)(xF)(lim0 xFx.xdex解：解：被積函數(shù)中含有絕對(duì)值符號(hào)， 故分段積分dxexdxex,1Cexdxex,2Cex其中 為恣意常數(shù)21,CC由原函數(shù)的定義 , 可知 延續(xù) , 得)(xF)0(F11 C)(lim0 xFxlim20Cexx21 C212CC0 x0 x)(xF0 x,1Ce

9、x0 x,21Cex高等數(shù)學(xué)Mathematic is the Queen of Science 高等數(shù)學(xué)Mathematic is the Queen of Science 例例8. 8. 設(shè)設(shè),)(2xyxy解解: 令, tyx求積分.d31xyxxyxy2)(即txy,123ttx,12tty而ttttxd) 1()3(d2222 1原式ttttd) 1()3(2222123tt132tttttd12Ct1ln221Cyx1)(ln221高等數(shù)學(xué)Mathematic is the Queen of Science 高等數(shù)學(xué)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高等數(shù)學(xué)Mathematic is the Queen of Science 高等數(shù)學(xué)Mathematic is the Queen of Science .2)1ln(23)1ln()1(arctan212222Cxxxxxxxxxxxarctan)1ln()1(21222dxxxx1)1ln(21222高等數(shù)學(xué)Ma

11、thematic is the Queen of Science 高等數(shù)學(xué)Mathematic is the Queen of Science dxxxx11ln11102例Cxxxxdxx11411111212lnlnln高等數(shù)學(xué)Mathematic is the Queen of Science 高等數(shù)學(xué)Mathematic is the Queen of Science 例例11.11.設(shè) 解解:)(xF為)(xf的原函數(shù),時(shí)時(shí)當(dāng)當(dāng)0 x,2sin)()(2xxFxf有有且,1)0(F,0)(xF求. )(xf由題設(shè), )()(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又高等數(shù)學(xué)Mathematic is the Queen of Science 高等數(shù)學(xué)Mathematic is the Queen of Science 二、幾種特殊類型的積分二、幾種特殊類型的積分1. 普通積分方法普通積分方法有理函數(shù)分解多項(xiàng)式及部分分式之和指數(shù)函數(shù)有理式指數(shù)代換三角函數(shù)有理式萬能代換簡(jiǎn)單無理函數(shù)三角代換根式代換高等數(shù)學(xué)Mathematic is the Queen of Scienc

13、e 高等數(shù)學(xué)Mathematic is the Queen of Science 2. 2. 需求留意的問題需求留意的問題(1) 普通方法不一定是最簡(jiǎn)便的方法 ,(2) 初等函數(shù)的原函數(shù)不一定是初等函數(shù) ,要留意綜合運(yùn)用各種根本積分法, 簡(jiǎn)便計(jì)算 . 因此不一定都能積出.例如例如 , ,d2xex,dsinxxx,dsin2xx,dln1xx,1d4 xx,d13xx, ) 10(dsin122kxxk高等數(shù)學(xué)Mathematic is the Queen of Science 高等數(shù)學(xué)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高等數(shù)學(xué)Mathematic is the Queen of Science 高等數(shù)學(xué)Mathematic is the Queen of Science 例例2.2. 求不定積分.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高等數(shù)學(xué)Mathematic is the Queen of Science 高等數(shù)學(xué)Mathematic is the Queen of Science 例例3 3I12xxdx解法解法1 12tttdI當(dāng)1x時(shí)，令,20secttxsec tansec tanttdtIttct cx1arccos當(dāng)1x時(shí)，令, 1ttx12tttdct1arccoscx1arccos高等數(shù)學(xué)Mathematic is the Queen of Science

16、高等數(shù)學(xué)Mathematic is the Queen of Science 解法解法2 令,1tx 21 tdtIcx1arccos,12dttdxct arccos, 10 t21 tdtIcx1arccos,12dttdx1arcsinct , 01t高等數(shù)學(xué)Mathematic is the Queen of Science 高等數(shù)學(xué)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高等數(shù)學(xué)Mathematic is the Queen of Science 高等數(shù)學(xué)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高等數(shù)學(xué)Mathematic is the Queen of Science 高等數(shù)學(xué)Mathematic is the Queen of Science 例例6 6： 設(shè),2ln) 1(222xxxf且xxfln)(求(

18、 ).x dx解：解：1111ln22xx) 1(2xf1)(1)(ln)(xxxfxlnxxx1)(1)(2( )11xx xdx)(xdx)121 (Cxx1ln2高等數(shù)學(xué)Mathematic is the Queen of Science 高等數(shù)學(xué)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高等數(shù)學(xué)Mathematic is the Queen of S

19、cience 高等數(shù)學(xué)Mathematic is the Queen of Science dxxx53cossin1提示：xx53cossin1xxxx5322cossincossinxxxx335cossin1cossin11cossin22xxxx5cossinxx3cossin2xxcossin13xx5cossinxx3cossin2x2sin6xx3sincos例例8 8高等數(shù)學(xué)Mathematic is the Queen of Science 高等數(shù)學(xué)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高等數(shù)學(xué)Mathematic is the Queen of Science 高等數(shù)學(xué)Mathematic is the Queen of Science 例例10 10 求求dxxxxx23cossincosxesin提示: 原式xdxexsinsinxesinxdxxsectandxexsinxxesinxexsinsec dxexecxsxsincos 高等數(shù)學(xué)Mathematic is the Queen of Sc

21、ience 高等數(shù)學(xué)Mathematic is the Queen of Science 作業(yè)作業(yè)P221 1, 4, 6 , 9 ,14, 17 , 21 ,24, 26 , 30 , 38 , 39, 40 高等數(shù)學(xué)Mathematic is the Queen of Science 高等數(shù)學(xué)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高等數(shù)學(xué)Mathematic is the Queen of Science 高等數(shù)學(xué)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)(闡明闡明: 此法特別適用于此法特別適用于如下類型的積分: 高等數(shù)學(xué)Mathematic is the Queen of Science 高等數(shù)學(xué)Mathematic is the Queen of Science 例例8. 8. 求求.942xxd解法解法1. 查積分表查積分表令,2xu 那么原式P349 公式 37Cuu33ln3122Cxx239