1211xln(x?1)??(x?1?)dx 22x?1121211 =xln(x?1)?x?x?lnx?1?C
2422 =例3.43．
?t?xxln(x?1)dx
2ln(t?1)dt3 ?3221232t3)dt dt=t3ln(t?1)??(t2?t?1? =tln(t?1)??33t?133t?1232t312 =tln(t?1)?(?t?t?ln(t?1))?C
33323232122x?ln(x?1)?C =x2ln(x?1)?x2?x?39333解：原式??tln(t?1)2tdt?例3.44．
??2x?1?ln?2xdx
22222解：原式?lnxd(x?x)?(x?x)lnx?2(x?x)?1lnxdx xx2 ?(x?x)lnx?2?(x?1)lnxdx?(x?x)lnx?2?lnxd(?x)
2x2x2122 ?(x?x)lnx?2(?x)lnx?2?(?x)dx
22x12222 ?(x?x)lnx?(x?2x)lnx?x?2x?c
22222例3.45．xarctan2xdx
?112x22dx 解：原式??arctan2xdx?xarctan2x??221?4x21214x2?1?1dx ?xarctan2x??2241?4x111?x2arctan2x?x?arctan2x?c 248例3.46．(x?1)arcsinxdx 解：原式?x?sint??(sint?1)tcostdt?1tsin2tdt??tcostdt 2???1tdcos2t??tdsint ?4tcos2t1????cos2tdt?tsint??sintdt
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第三章 不定积分
11??tcos2t?sin2t?tsint?cost?C
4811??arcsinx?(1?2x2)?x1?x2?xarcsinx?1?x2?C
444．四类杂例
（１）含绝对值的不定积分 例3.47．?xdx ?x2?c1,x?0??2解：原式?F(x)??，F(x)可导必连续：c1?c2， 2??x?c,x?02??2?x2??c1,x?0??2故原式?F(x)?? 。 2??x?c,x?01??2例3.48．?x2?2x?3dx ?x2?2x?3,x??1?22解： f(x)?x?2x?3?(x?3)(x?1)???x?2x?3,?1?x?3， ?x2?2x?3,x?3??132x?x?3x?c1,x??1?3??1f(x)dx???x3?x2?3x?c2,?1?x?3 ， ?3?132x?x?3x?c3,x?3?3?原式?F(x)??1?1??1?3?c??1?3?c21??33由F(x)可导知，成立?，
2727???9?9?c??9?9?c32?3?3- 89 -
10?c???c1??23解得：? ，
1044?c?18?c?18??c??c1321?33??132x?x?3x?c1,x??1?3?10?132?c1,?1?x?3 。 所以，F(x)???x?x?3x?3?344?132x?x?3x??c1,x?3?33?（2）分段函数积分
x?1?x,?例3.49．f(x)??2x?1,1?x?2，求?f(x)dx 。
?x?1,x?2?解：F(x)???x2x?1?2?c1,??f(x)dx??x2?x?c2,1?x?2，
?2?x?x?c3,x?2??2?1??c1?2?c2由F(x)可导知，成立?2
??6?c2?4?c3解得：c2??31?c1，c3?2?c2??c1 22?x2x?1?2?c1,?3?2所以，?f(x)dx??x?x??c1,1?x?2 。
2??x21?x??c1,x?2?22?（3）递推关系
2n例3.50．In?sinxdx
?n?1解：In??sinxdcosx
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第三章 不定积分
In??sinn?1xcosx???n?1?cos2xsinn?2xdx
In??sinn?1xcosx??n?1??sinn?2xdx??n?1??sinnxdx
nIn??sinn?1xcosx??n?1?In?2
?n?1?I ?n?1,2,......? 1In??sinn?1xcosx?n?2nn2n例3.55．In?tanxdx ?2n2n?2x)dx?tan2(n?1)dx 解：In?(tanx?tan??In??tan2n?2xdtanx??tan2(n?1)dx In?1tan2n?1x?In?1 ?n?1,2,......? 2n?13例3.56．I?secxdx ?解：I?secxdtanx?secxtanx???2tanxsecxdx ＝secxtanx?I?secxdx， I??111?sinxsecxtanx?ln?C 241?sinx（4）一些特殊的变换 例3.57．1?x6(1?x2)dx 1， x解：令t?t6?1??2?dt???dt 原式??21??t1?t????1?2??t?15131????t?t?t?arctant?C ????t4?t2?1?dt2?531?t????111111???arctan?C 5x53x3xx- 91 -
t6例3.58．
?1?xdx 1?x1?x1?t21?x222解：令t?，解得：t?，t?xt?1?x，x?，则
1?x1?t21?xdx??4tdt 22(1?t)t?tanu?4t2原式???dt22(1?t)?4tan2u2??secudu 4secu??4?sin2udu??2?(1?cos2u)du??2u?sin2u?C。
（5）一些特殊积分
2x2例3.59．e(tanx?1)dx
?2x22x2x2x解：原式＝esecxdx?2etanxdx?edtanx?2etanxdx
???? ＝e2xtanx?2?e2xtanxdx?2?e2xtanxdx?e2xtanx?C
1x?1例3.60．?(1?x?)exdx
x1x?1解：原式＝?edx??x(1?2)exdx
x1111x?x?x?x?1xxx ＝?edx??xed(x?)??edx??xdex
xx?1x ＝e?x?1xdx?xex22x?1x??ex?1xdx?xex?1x?C
例3.61．(x?1)edx 解：原式＝xde
单元练习题3
?2?x22??edx?xe??edx??edx?xe?C
x22x22x22x22x221．dcos2x? 。
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