第八讲 不定积分的分部积分法
一、单项选择题（每小题4分，共24分） 1．设e?x是f?x?的一个原函数,则
?xf?x?dx?( )
A． e?x?x?1??c B．
?e?x?x?1??c
C．e?x?1?x??c D．
e?x?x?1??c
解：?F?x??e?x,f?x??F??x???e?x
?原式=?xdF?x? ?xF?x???F?x?dx
?xe?x??e?xdx??x?1?e?x?c
选A
2．若f?x?的一个原函数为ln2x,则
?xf??x?dx?( )
A．lnx?ln2x?c B．2lnx?ln2x?cC．2lnx?ln2x?c D．lnx?ln2x?c 解： F?x??ln2x,
f?x??F??x??2xlnx
?xf??x?dx??xdf?x?
?xf?x???f?x?dx
?2lnx?ln2x?c
选C
3．设f??lnx???1?x?lnx,则 f?x?=( )
1
A．xex?x22?c B．?x?1?ex?x22?c
C．xex?x2．?x?1?ex?x22?c D2?c
解：(1)?f??lnx???1?elnx?lnx
?f??x???1?ex?x
(2)f?x???x?1?ex?dx
?x2xx2xx2??xde?2?xe?e?c
选B
4．?xf???x?dx= ( ) A． xf??x???f?x?dx
B． xf?x??f?x??c C． xf??x??f?x??c D．f?x??xf??x??c 解： 原式=?xdf??x?
?xf??x???f??x?dx
?xf??x???df?x?
?xf??x??f?x??c 选C
5．
?xcos2xdx? ( )
A． xtanx?lncosx?c B． xtanx?lncosx?c C． xtanx?lnsinx?c D． xtanx?lnsinx?c 解： 原式=?xdtanx
2
?xtanx??sinxcosxdx =xtanx??dcosxcosx
=xtanx?lncosx?c 选B 6．?1x2? ( )
1?x2?dx? A． 1x?arctanx?c B．
1x?arctanx?c C． ?1x?arctanx?c
D．?1x?arctanx?c
2解： 原式??x?1?x2x2?1?x2?dx
??1x2dx??11?x2dx
??1x?arctanx?c
选C
二、填空题（每小题4分，共24分）7．?lnxdx= 解： 原式?xlnx??xdlnx
?xlnx??x?1xdx ?xlnx?x?c
8．?exdx?
解： 原式x?t?et?2tdt
?2?tdet?2tet?2et?c
回代2xex?2ex?c
3
9．?1?x?1??x?2?dx= 解： 原式
拆项??x?2???x?1??x?1??x?2?dx
??1dxx?1dx??x?2
?lnx?1?lnx?2?c
?lnx?1x?2?c
10．
若f??ex??1?x?x?0?,则
f?x?= 解：(1)?f??ex??1?lnex ?f??x??1?lnx
(2)f?x??x??xlnx?x??c?xlnx?c 11．?xsin2xdx? 解： 原式=∫xd??cotx?
??xcotx?∫
cosxsinxdx
??xcotx?lnsinx?c
12．?xf?x2?f??x2?dx? 解： 原式=1??x22f2??f??x2?dx
凑微分12?f?x2?df?x2??1?2?24?f?x???c三、计算题（每小题8分，共64分） 13．?lnsinxcos2xdx
.解：原式=?lnsinxdtanx =tanx?ln?sinx???tanx?cosxsinxdx
=tanx?ln?sinx???dx
4
=tanx?ln?sinx??x?c
1?x214．??11?x2arctanxdx
解：原式=?x21?x2arctanxdx
=xarctan??x1?x2dx??arctanxdarctanx
=xarctanx?12ln?1?x2??12?arctanx?2?c
15．?xcosxsinxdx 解：原式=?x?12sin2xdx
=?14?xdcos2x
=?1?4?xcos2x??cos2xdx?? =
?114xcos2x?8sin2x?e
16．?x3e?x2dx
x2e?x2dx2.解：原式=?2
x2?t?te?tdt?11t2??2?tde?
??1?te?t?e?t2??dt??
??1?t?t2??te?e???c 回代?1??x2?1?e?x2?2???c
17．?x2cosxdx 解： 原式??x2dsinx
?x2sinx??sinxdx2
?x2sinx?2?xsinxdx
5
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