（4）?axexdx
解：?aedx??(ae)dx?xxx(ae)xlnae?C?aexxlna?1?C
（5）?解：?dx4?9xdx22
?14?9x?4?4dx1?(3x2)2?12?43?11?(3x2)2d(3x2)?16arctan3x2?C
（6）?解：?dxcos(2x?dx2
)?121cos(2x?2cos(2x?2?4)??4d(2x?)?4)?12tan(2x??4)?C
（7）?解：?sec2xtanx2dx
secxtanxdx??tan1xdtanx?lntanx?C
（8）?解：?x1?2xx1?2x22dx
dx?1?411?2x2d(1?2x)?2121(1?2x)2?C2
4．用第一类换元法求下列不定积分： （1）?解：?x23dx
2?xx2dx??312?x23x?312?x233d(2?x)??31?(2?x33)?12d(2?x)
31??(2?x)2?C??dx32?x3?C
（2）?解：?e?edx?x
?e?ex?x?eedx2xx?1??e12x?1de?arctane?Cxx
26
（3）?解：?14dxx?2x?5dx22
?x?2x?5?x2dx2?2x?1?412??(x?1)2?Cdx2?4?14?dx1?(x?12)2
??2?11?(x?12)2d(x?1)?arctanx?1
（4）?解：?1x?x1x?xarctan22dx
dx?1x?1x1?xdx?2?11?(x)2dx?2arcsinx?C
（5）?解：?1?x2dx
1x1x121xarctan1?x21xdx???arctandarctan??(arctan)?C2
（10）?tan3xdx
解：?tan3xdx=?tanx(sec2x?1)dx??tanxsec2xdx??tanxdx
??tanxdtanx??tanxdx?12tan2x?lncosx?C
（11）?sin3xdx
解：?sin3xdx??sinxsin2xdx???(1?cos2x)dcosx
???dcosx??cos2xdcosx??cosx?13cos3x?C
5．用第二类换元法求下列不定积分： （1）?x解：设t所以?x?3?xdx ，则x?t23?x?3，dx?2tdt
33?xdx?42?(t2?3)?t?2tdt??(t2?3)?t?2tdt5?2?(t?3t)dt=?25t5?2t3?C?25(3?x)2?2(3?x)2?C
（3）?dxx?3x
27
解：设t?6x，则x?t6，dx?6t5dt
?dxx?23?x?t6tdt35?t2?6?1t?1t3t?1dt?6?32t?1?1t?13dt?6?((t?1)(t?t?1)t?12?1t?1)dt
?6?(t?t?1)dt?6?dt?2t?3t?6t?6lnt?1?C6
?2x?33x?66x?6lnx?1?C
（5）?dx1?e2x
12解：设t?1?e2x，则x?tln(t2?1)，dx?tt2?1dt
所以?dx1?e12122x??t?1dt?t2?t12?1dt?12?(2x2x1t?1?1t?1)dt
e(1?e2x2x2?(lnt?1?lnt?1)?C?122122ln1?e1?e?1?1?C?12ln?C
?1)?lne2x?ln(1?e2x?1)??C?x?ln(1?e2x?1)?C
（8）?x34?xdx
解：设x?2sint，则dx?2costdt
所以?x34?x2dx??8sin3t?4?4sin2t?2costdt
?32?sint?costdt??32?(cost?cost)dcost??32(3224cost33?cost55)?C
??323(4?x22)?3325(4?x22)?C??5433(4?x)2?155(4?x)2?C
（10）?dxx2(1?x)23
解：设x?sint，则dx?costdt 所以?dxx2(1?x)23??sincostdt2t(1?sint)23??sincostdt2tcost3??sin2t?cost22sintcost2dt
28
??(sec2t?csct)dt?tant?cost?C?xdx(1?x)2322x1?x2?1?xx2?C
（12）?
2解：设x?tant，则dx?sectdt
所以?xdx(1?x)232??tan2tsectdt22(1?tant)3??tan2tsectdt32sect
x1?x2??(sect?cost)dt?lnsect?tant?sint?C?ln1?x2?x??C
6．用分部积分法求下列不定积分： （1）?(ex?lnx)dx
解：?(ex?lnx)dx??exdx??lnxdx?ex?xlnx??xdlnx
?e?xlnx?x?dx?e?xlnx?x?Cx
（2） ?x2e?2xdx 解：?x2e?2xdx????????121212xexexe222?2x1?2xexde?2x2?2x??12122xe2?2x?1121212?e?2xdx2
????1212dx????1214xe?2x??xde2xee2?2x
12xe?2x?2xxexe?2x?ee?2xdx???2x?2?1?e4
?2xd(?2x)
?2x?2x?2x?C???2x(x?x?12)?C（3）?(2x?3x2)arctanxdx 解：?(2x?3x2)arctanxdx
??arctanxd(x?x)?(x?x)arctanx?232323?(x?x)darctanx23
?(x?x)arctanx???x?x1?x2223dx?(x?x)arctanx?23x(x?1)?x?1?x?11?x22dx
29
?(x?x)arctanx?23?xxdx?2?dx?12?1?x12x2dx??211?x2dx
?(x?x)arctanx?232?x??1?x12d(1?x)?arctanx
?(x?x?1)arctanx?23x22?x?ln(1?x)?C
2（4） ?(解：?(lnxx1x1x1xlnxx2)dx2
22)dx??(lnx)x2dx???(lnx)d1xdx??1x1x1x??1x(lnx)?1xx2?xd(ln
1x)2
??????(lnx)?22?x1x1x12lnx?(lnx)?2?lnx?d1x(lnx)?2222(lnx)?2(lnx)?2sinxe?x2lnx?2?lnx?21x?dlnx??1x1lnx?2?1x2?dx
?C??[(lnx)?2lnx?2]?C
（5） ?解：?sinxex
?x?esinxdx???e?xdcosx??e?xcosx??cosxde
sinx??x
??e??e?xcosx??e?xcosxdx??e?xcosx???e?e?xdsinx?x?xcosx?e??xsinx??sin12exde?x?x?xcosx?e?e?xsinxdx
所以?sinxex?e?xsinxdx??(cosx?sinx)?C
（6） ?xsec2xdx
解：?xsec2xdx??xdtanx?xtanx??tanxdx
?xtanx??cossinxxdx?xtanx??cos1xdcosx
?xtanx?lncosx?C
（7） ?xtan2xdx
解： ?xtan2xdx??x(sec2x?1)dx??xsec2xdx??xdx??xdtanx??xtanx?12x2
?tanxdx?12x?xtanx?lncosx?212x?C2
（8） ?ln(1?x2)dx
30