2(x2?1)?2dx2?xln(1?x)??dx?xln(1?x)?2dx?2??1?x2 1?x2?xln(1?x2)?2x?2arctanx?C.2★（3）
?arctanxdx
思路：同上题。
dx1d(1?x2)?xarctanx??解：?arctanxdx?xarctanx??x 1?x221?x21?xarctanx?ln(1?x2)?C
2x?2xsindx ★★(4)e?2思路：严格按照“反、对、幂、三、指”顺序凑微分即可。 解：?e??2xxx11x11xsindx??sind(?e?2x)??e?2xsin??e?2xcosdx
222222221x1x1??e?2xsin??cosd(?e?2x)224221x11x1x??e?2xsin?(?e?2xcos??e?2xsindx)2242242
1x1x1x??e?2xsin?e?2xcos??e?2xsindx2282162x2e?2xxx?2x??esindx??(4sin?cos)?C.21722★★(5)
2x?arctanxdx
思路：严格按照“反、对、幂、三、指”顺序凑微分即可。
x313131dx 解：?xarctanxdx??arctanxd()?xarctanx??x3331?x22131x131x3?x?x?xarctanx?(x?)dx dx ?xarctanx??22?331?x331?x1311x1312112xarctanx??xdx??dx?xarctanx?x?d(1?x)22?3331?x3661?x
13121?xarctanx?x?ln(1?x2)?C.366?★(6)
xxcosdx ?2思路：严格按照“反、对、幂、三、指”顺序凑微分即可。
21
解：xcosdx?2xdsin ?★★(7)
?x2?xxxxxx?2xsin?2?sindx?2xsin?4?sind 222222xx2xsin?4cos?C.
222?xtanxdx
思路：严格按照“反、对、幂、三、指”顺序凑微分即可。
2解：xtanxdx??222x(secx?1)dx?(xsecx?x)dx?xsecxdx??xdx ???11??xd(tanx)??xdx?xtanx??tanxdx?x2?xtanx?lncosx?x2?C.
22★★(8)
?ln22xdx
思路：严格按照“反、对、幂、三、指”顺序凑微分即可。
解：lnxdx?xlnx?x?2lnx?dx?xlnx?2lnxdx?xlnx?2xlnx?2x?dx
?2?1x2?2?1x?xln2x?2xlnx?2?dx?xln2x?2xlnx?2x?C.
★★(9)
?xln(x?1)dx
思路：严格按照“反、对、幂、三、指”顺序凑微分即可。
x2121x2?xln(x?1)??dx 解：?xln(x?1)dx??ln(x?1)d222x?1111121x2?1?1)dx dx?x2ln(x?1)??(x?1? ?xln(x?1)??22x?122x?1?12111xln(x?1)?x2?x?ln(x?1)?C 2422ln2x★★(10)?x2dx
思路：严格按照“反、对、幂、三、指”顺序凑微分即可。
ln2x1121112lnx2解：?2dx??lnxd(?)??lnx??2lnx?dx??lnx?2?2dx
xxxxxxx11121122??ln2x?2?lnxd(?)??ln2x?lnx?2?2dx??ln2x?lnx??C
xxxxxxxx12 ??(lnx?lnx?2)?C
x★★(11)
?coslnxdx
思路：严格按照“反、对、幂、三、指”顺序凑微分即可。
22
解：?coslnxdx?xcoslnx?xsinlnx?dx?xcoslnx?sinlnxdx
??1x?1?xcoslnx?xsinlnx??xcoslnx?dx?xcoslnx?xsinlnx??coslnxdxx
x??coslnxdx?(coslnx?sinlnx)?C.2★★(12)
lnx?x2dx
思路：详见第(10) 小题解答中间，解答略。
★★(13)
?xnnlnxdx(n??1)
思路：严格按照“反、对、幂、三、指”顺序凑微分即可。
xn?11n?11n?11?xlnx??x?dx 解：?xlnxdx??lnxdn?1n?1n?1x?1n?11n1n?1?1?xlnx??xdx?x?lnx???C. n?1n?1n?1(n?1)??★★(14)
?xe2?xdx
思路：严格按照“反、对、幂、三、指”顺序凑微分即可。
2?x2?x?x2?x?x?x解：xedx??xe?e2xdx??xe?2xe?2edx
?????x2e?x?2xe?x?2e?x?C??e?x(x2?2x?2)?C
★★(15)
32x(lnx)dx ?思路：严格按照“反、对、幂、三、指”顺序凑微分即可。 解：x(lnx)dx?(lnx)d(x)??32?21441411x(lnx)2??x4?2lnx?dx 44x14111x(lnx)2??x3lnxdx?x4(lnx)2??lnxdx442481111111?x4(lnx)2?x4lnx??x4?dx?x4(lnx)2?x4lnx??x3dx 488x48811111?x4(lnx)2?x4lnx?x4?C?x4(2ln2x?lnx?)?C.483284?lnlnx?xdx
lnlnxdx写成lnlnxd(lnx)，将lnx看作一个整体变量积分即可。 思路： 将积分表达式
xlnlnx111dx??lnlnxd(lnx)?lnxlnlnx??lnx??dx?lnxlnlnx??dx 解：?xlnxxx★★(16)
23
?lnxlnlnx?lnx?C?lnx(lnlnx?1)?C.
★★★ (17)
?xsinxcosxdx
思路：严格按照“反、对、幂、三、指”顺序凑微分即可。
11111xsin2xdx?xd(?cos2x)??xcos2x?cos2xdx ??22?244?1111??xcos2x??cos2xd2x??xcos2x?sin2x?C.
484822xdx ★★(18)xcos?21?cosx2x思路：先将cos降幂得，然后分项积分；第二个积分严格按照“反、对、幂、三、指”顺
22解：xsinxcosxdx?序凑微分即可。
解：xcos?22x1111dx??(x2?x2cosx)dx??x2dx??x2cosxdx 222221312111x??xdsinx?x3?x2sinx??2xsinxdx62622
13121312?x?xsinx??xdcosx?x?xsinx?xcosx??cosxdx6262??1312x?xsinx?xcosx?sinx?C 62★★(19)
?(x22?1)sin2xdx
思路：分项后对第一个积分分部积分。 解：(x?1)sin2xdx??1122xsin2xdx?sin2xdx?xd(?cos2x)?cos2x ???2211111??x2cos2x??2xcos2xdx?cos2x??x2cos2x??xdsin2x2222211111?cos2x??x2cos2x?xsin2x??sin2xdx?cos2x22222
12111??xcos2x?xsin2x?cos2x?cos2x?C224211313x??x2cos2x?xsin2x?cos2x?C??(xsin2x?)cos2x?sin2x?C.224222★★★(20)
xe?dx
3思路：首先换元，后分部积分。 解：令t?3x，则x?t3,dx?3t2dt,
24
??exdx??et3t2dt?3?ett2dt?3?t2det?3t2et?3?2tetdt?3t2et?3?2tdet?3t2et?6ett?6?etdt?3t2et?6ett?6et?C ?33x2e★★★(21)
33x?6e3x3x?6e3x?C?3ex(3x2?23x?2)?C.3?(arcsinx)dx
222思路：严格按照“反、对、幂、三、指”顺序凑微分即可。 解：(arcsinx)dx?x(arcsinx)?x???2arcsinx1?x2dx
?x(arcsinx)2??arcsinx1?x2d(1?x2)?x(arcsinx)2?2?arcsinxd(1?x2)
?x(arcsinx)2?21?x2arcsinx?2?1?x2?11?x2dx?x(arcsinx)2?21?x2arcsinx?2?dx?x(arcsinx)2?21?x2arcsinx?2x?C.★★★(22)
?exsin2xdx
思路：严格按照“反、对、幂、三、指”顺序凑微分即可。 解：方法一：
x22xx2xesinxdx?sinxde?esinx?e???2sinxcosxdx
?exsin2x??exsin2xdx??esin2xdx??sin2xde?esin2x??e2cos2xdx?esin2x?2?cos2xde?exsin2x?2excos2x?4?exsin2xdxex(sin2x?2cos2x)??esin2xdx??C
5exx2??esinxdx?(5sin2x?sin2x?2cos2x)?C5xxxxxxx
方法二： esinxdx?e?x2?x1?cos2x1111dx??exdx??excos2xdx?ex??excos2xdx 22222??excos2xdx??cos2xdex?excos2x??ex2sin2xdx?excos2x?2?sin2xdex
?excos2x?2exsin2x?4?excos2xdxex(cos2x?2sin2x)??ecos2xdx??C
5ex1x1x2??esinxdx??esin2x?excos2x?C2510x 25