＝
14sinu?C?x?44x2?C
四、分部积分法（有时还用了换元积分法） 【例1】 求下列不定积分 （1）?xedx
（3）??3x?x?1?cosxdx
2?x（2）?xe2?2xdx
解 （1）?xedx???xde（2）?xe＝?12122?2x?x?x??xe?x?2?edx??(x?1)e?2x?x?x?C
dx??12?xde2?2x??212xe1?12?e
?2xdx
2xe2?2x??12xe?2xdx??1212xe?2x??xde2e?2x?2x＝?xe2?2x?xe?2x??e?2xdx??121?2x?x??2????C ?（3）??3x?x?1?cosxdx?22??3x22?x?1?dsinx
＝?3x?x?1?sinx??sinxd?3x?x?1? ＝?3x?x?1?sinx?2??6x?1?sinxdx ??6x?1?dcosx
＝?3x?x?1?sinx?22＝?3x?x?1?sinx??6x?1?cosx?6?cosxdx ＝?3x?x?1?sinx??6x?1?cosx?6sinx?C
2【例2】 求下列不定积分 （1）?xlnxdx(n??1) （3）?arctanxdx
解 （1）?xlnxdx(n??1)?＝＝
1n?1xn?1nn（2）?arcsinxdx
1n?1?lnxd(xn?1)
lnx?1n?1?xn?11?dx x1n?1xn?1xn?1lnx?1?n?1?2xn?1?C
＝
1??lnx????C n?1?n?1?
（2）解一
?arcsinxdx?xarcsinx??xdarcsinx
＝xarcsinx??xdx1?x2?xarcsinx?1?2d(1?x)1?x22 ＝xarcsinx?1?x2?C
解二 令arcsinx?t，则x?sint
?arcsinxdx??tdsint?tsint??sintdt
＝tsint+cost+C=xarcsinx+（3）?arctanxdx?xarctanx?＝xarctanx?1-x+C
2?xdarctanx
12?1?x12x2dx?xarctanx??d(1?x)1?x22
＝xarctanx?ln(1?x)?C
2
【例3】 求下列不定积分 （1）?lnxdx
3323（2）?arcsinxdx
1322解 （1）?lnxdx?xlnx?3??xlnx??dx?xlnx?3?lnxdx
x＝xlnx?3xlnx?3?x?2lnx??dx
x321＝xlnx?3xlnx?6?lnxdx
＝xlnx?3xlnx?6xlnx?6?x?dx
x32321＝x?lnx?3lnx?6lnx?6??C
32
（2）解一
??arcsinx?22dx?x?arcsinx??2?xd?arcsinx?
2＝x?arcsinx??2?2xarcsinx1?x2dx
＝x?arcsinx??2?arcsinxd1?x ＝x?arcsinx??2?1?xarcsinx?222??21?xdarcsinx?
?
＝x?arcsinx??2?1?xarcsinx??dx?
22??＝x?arcsinx??21?x2arcsinx?2x?C 解二 令arcsinx?t，则x?sint
2??arcsinx?22dx??tdsint?tsint?2?tsintdt
222＝tsint?2?tdcost?tsint?2tcost?2?costdt ＝tsint?2tcost?2sint?C
＝x(arcsinx)+21-xarcsinx-2x+C 【例4】 求下列不定积分
（1）?esinbxdx （a?0,b?0） 解 （1）?esinbxdx?＝
1aeaxaxax222（2）??1aeaxx?adx a?0 1a221a?sinbxde1aeaxsinbx?ba2?edsinbx
axaxsinbx?ba?e2axcosbxdx?axsinbx??cosbxde
＝
1aeaxsinbx?baeaxcosbx?ba22?eaxsinbxdx 2?b?1axbaxax1?esinbxdx?esinbx?ecosbx?C? ?2??2aaa???eaxsinbxdx?e2ax2a?b2?asinbx?bcosbx??C
22（2）?2x?adx?xx?a?x2222?xd2x?a dxx?a2222＝xx?a?222?dx?xx?a?2x?a2?x?adx?a222? 2?x?adx?xx?a?alnx?22?x?a22??C?
?x?adx?22x2x?a?22a22lnx??x?a22??C
arctaneexx【例5】 求下列不定积分 （1）?dx1
xex3（2）?dx
解 （1）?dx1xex3???1?1?exd??x?x???t1令1x?t??tedt
?t＝?tde＝
1xe??t?te?1x?t??edt?te?t?e?t?C
1x?e?C
（2）令e?t，则
arctaneexxx?dx??arctantt21?1?dt???arctantd????arctant?t?t??11?dt 2t1?t＝?arctant????dt 2?t?t1?t?＝?arctant?lnt?t11212ln?1?t21?1t???C
2x＝?e?xarctane?x?xln?1?e??C
2五、其他
【例1】 设f?x?的一个原函数F?x??ln解 I??x?x?1，求I?2??xf??x?dx
FxC?xdx?f????xf??x??x?1?ln2??df?x?x??x?F? x?2xx?12lnx???2?x?x?1?C
2?【例2】 设F??x??f?x?，当x?0时，f?x?F?x??xex22?1?x?，又F?0??1，
F?x??0，求f?x? ?x?0?.
2解 2?f?x?F?x?dx?2?F?x?dF?x??F而
x?x??C1
ex?xex2?1?x?dx?????x?1??1??ex?1?x?2dx??1?x???C2
dex?1?x?dx 2?e1?x??ex2?1?x?exdx??ex2?1?x?dx?ex1?x?F2?x??1?x?C，?F?0??1，?C?0，又F?x??0
因此 F?x??ex1?x?e21?x1e2xx
12e21?x?x则 f?x??F??x??21?x1?xxsinx?xe22?1?x?32x
【例3】 设f?sinx??2，求I??x1?xf?x?dx
arcsinuu解一 令u?sinx，则sinx?2u，x?arcsinu，f?u??
则 I??arcsin1?xxdx???arcsin1?xxd?1?x???2?arcsinxd1?x
??21?xarcsinx?2?1?x?11?xdx ??21?xarcsin
解二 令x?sint，则sint2 x?2x?C
x1?x?sintcost，dx?2costsintdt
则 I??t??2sintcostdt??2?tdcost costsint ??2tcost?2?costdt??2tcost?2sint?C ??21?xarcsin
【例4】 设In?x?2x?C
?dx?x2?a2?n
?n?2正,整数，a?，求证 ?0??x?In=??2n?3?In?1? n?1222?2?n?1?a??x?a???1证In=1a2?x?a??x??x?a?2222n2dx?1a2In?1?12a2?xd?x?a22n??x2?a2?
=1a2In?1?12?n?1?a2???1? xd?n?1?x2?a2?????=1a2In?1??x???In?1? n?1222?2?n?1?a??x?a???11??=?x2?n?1?a2?n?1??2n?3?In?1???x2?a2?? ?
其他参看PPT讲义和题型小节
