专插本高等数学例题和习题ch3不定积分(5)

第三章 不定积分

t2dt （19）原式??arctant?2tdt??arctantdt?tarctant??1?t2t?x22 ?t2arctant?t?arctant?C?xarctanx?x?arctanx?C （20）原式

arcsinx?t?222t?costdt?tdsint?tsint?2tdt ???tsin ?t2sint?2?tdcost?t2sint?2tcost?2?costdt

?t2sint?2tcost?2sint?C?xarcsin2x?21?x2arcsinx?2x?C （21）解：原式?x?t23t?sint?2tdt?2t??sintdt

??2?t3dcost??2t3cost?6?t2?costdt??2t3cost?6?t2dsint ??2t3cost?6t2sint?12?tsintdt ??2t3cost?6t2sint?12?tdcost

??2t3cost?6t2sint?12tcost?12sint?C

??2xcosx?6xsinx?12xcosx?12sinx?C

32（22）原式Iarctanx?tt?tanx?tantetsintttdt??sect?cost?cost?edt??esintdt

??sintdet?etsint??etcostdt?etsint??costdet ?etsint?etcost??etsintdt?etsint?etcost?I

I?1t111esint?etcost?C?earctanx?sin(tanx)?earctanx?cos(tanx)?C 2222（23）解：I??sin(lnx)dx?xsin(lnx)??cos?lnx?dx

?xsin(lnx)?xcos?lnx???sinxlnxdx?xsin?lnx??xcos?lnx??C?I

1(xsin(lnx)?xcos(lnx))?C 21?cosx11dx?e2x??e2xcosxdx,令I??e2xcos2xdx （24）原式??e2x?24211I=?cos2xde2x?e2xcos2x??e2xsin2xdx 2211?e2xcos2x??sin2xde2x22I?

98 / 26

第三章 不定积分

?12x1ecos2x?sin2x?e2x?C??e2xcos2xdx2211?e2xcos2x?e2xsin2x?C?I 221I?e2xcos2x?e2xsin2x?C

4111原式?e2x?e2xcos2x?e2xsin2x?c

488?xxx?x?xx?x

exdx （25）原式??earctanedx???arctanede??earctane??e2x1?earctanex1?e2x?e2xarctanex1???dx???x?ln(1?e2x)?c x2xx?2e1?ee（26）原式??xdtanx?xtanx??tanxdx?xtanx?ln|cosx|?c

11ex1x（27）原式??e( ?)dx?dx?ed2??x?1(x?1)1?xx?1xexexexex??dx???dx??c 1?x1?x1?xx?11sinxdcosxdx?dx??（28）原式?? 424?sin2xcos?(1sinxcos4xx?cosx)cosxu?cosxdu(1?u2?u2)du11???????du?du 2424422???(1?u)u1?uuu1?uu????1111?3111?udu?(?)du?u??ln||?c 422?uu1?u3u21?u11111?cosx???ln||?c 33cosxcosx21?cosx???4（29）原式?4x?tx?t1t34tdt?4?t2(1?t)3?(1?t)3dt

?4?(?1141?)dt???2?C 232(1?t)(1?t)1?t(1?t)?42??C 2441?x(1?x)u?tanxtanx?sec2xtanxududx?dtanx?（30）原式???tan3x?1?1?u3 tan3x?11?u?1??u?1??u2?u?11u?111??du?du?du 33?u2?u?13?u?1?u?1?u2?u?1???? 99 / 26

第三章 不定积分

13u??122du?1ln|1?u| ??23?31?3u????2?4?112?ln(u2?u?1)??arctan623u?12?2?1ln|1?u|?C

3311?2tanx?1?1?ln(tan2x?tanx?1)?arctan???3ln|1?tanx|?C 633??sint?2dcost2?costdt???2csc?sin2tcost?sin2t?tdt

11?cost|?2cot(t)?C ??ln|21?costu?x2111u22du （32）原式??ln(4?x)dx??ln(4?u)du?uln(4?u)??2224?u1?uln(4?u)?u?4ln(4?u)?C 21?x2ln(4?x2)?x2?4ln(4?x2)?C 211112)dx?arctanxd?arctanx （33）原式=?arctanx(2?2?x2x1?xarctanx111??dx?arctan2x =?2xx1?x2（31）原式?x?sintarctanx1?x2?x212 =???dx?arctanx 2x2x1?x?? =? =?arctanx1x1??dx??dx?arctan2x 2xx21?x??arctanx11?ln|x|?ln(1?x2)?arctan2x?C x22（34）原式=?e2x(tan2x?1?2tanx)dx??e2xdtanx?2?e2xtanxdx =e2xtanx?2?tanxe2xdx?2?e2xtanxdx?C=e2xtanx?C （35）原式=? =?111lnx11lnxd()???dx 222?2?221?x1?x(1?x)x1lnx11x?(?)dx

21?x22?x1?x2 100 / 26

第三章 不定积分

=?1lnx11?ln|x|?ln(1?x2)?C 221?x24（36）原式=?11?cos4xdsinx???2dcos2x1?(cos2x)2??ln(cos2x?1?cos4x)?c

??x?1?(1?x)??2,x??1?1（37）令f(x)?|1?x|?|1?x|??1?x?(1?x)?2x,?1?x?

?1?x?(x?1)?2,x?1? F(x)????2x?c1,x??1?f(x)dx??x2?c2,?1?x?1

?2x?c,x?13?由连续特性知：2?c1?1?c2,1?c2?2?c3，?c2?1?c1,c3??1?c2?c1

x??1??2x?c1,?故F(x)??x2?1?c1,?1?x?1

?2x?c,x?11?3??x,x?1（38）解：f?x??max?x,x?? ?2

??x,x?123?14x?c1,x?1??4原式=?

11?x3??c,x?11?12?3xx??sin?cos??22?x?（39）原式=?e?dx

x??2cos2???2??0.5?ex(tanxxx?1)2dx?0.5?ex(sec2?2tan)dx 2222x?xxxx? ??exd?tan???extandx?extan??extandx??extandx?C

2?2222??extanx?C 2（40）原式=?1??1xxxdx??dxe?lnxe?ln1?xe?C ???xx??xxxe?1?xe??xe1?xe??x?1?ex 101 / 26

第三章 不定积分

（41）令u?x?3x?3，u2?，xu2?u2?x?3

x?1x?1u2?3?8u 所以x?2,dx?du 22u?1?u?1?1??8uu2?11?原式=??u??du??8du??8du 222??22u??u?1?u?1??u?1? ?4ln1?u?C?4ln1?ux?1?x?3?C

x?1?x?3

本章测试答案

21. 3 2.cosx?C 3. x3ex?C

xcos2x4. ?2?C??cot2x?C

sinxx2x2x215．原式=?lnxd(?x)?(?x)lnx??(?x)?dx

222xx2x2?(?x)lnx??x?C 24lnu22udu?4?lnudu?4ulnu?4u?C?4xlnx?4x?C 6．原式u?x?ux?2sint1?2costdt 7．原式??24sint?2cost?1?cot(t)?C 414?x2???C

4x2sinxcosxsinxdx?2dx??2lncosx?C 2?cosxcosxx9. tanx??C

28．原式=?10.?2sinx?C

5211.(xlnx)2?C

5x2?412.ln(x?4?x)??C

x2 102 / 26

第三章 不定积分

1213.?x?ln|2sinx?cosx|?C

55sinxxcosx?sinx)??, 14.证明：f(x)?(xx2 ?x3f?(x)dx??x3df(x)?x3f(x)?3?x2f(x)dx

＝x(xcosx?sinx)?3?xdsinx?3?sinxdx?x2cosx?4xsinx?6cosx?C 15.?xf??(2x?1)dx?x2x＝211??xf(2x?1)d(2x?1)?xdf?(2x?1) ??221x1f?(2x?1)??f?(2x?1)dx?f?(2x?1)??f?(2x?1)d(2x?1)

2241f?(2x?1)?f(2x?1)?C

4＝

16.原

xln(x?x2?1)??xx2?1dx?xln(x?x2?1)?x2?1?C

103 / 26

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