医用高等数学习题解答（第1,2,3,6章） - 21 -
1?x2?1x211?x?1?dxdx?1dx(6) ?===????1?x2?2ln1?x?x?C ?1?x21?x2(7)
1?cosx12x==sindxdx(x?sinx)?C ?2?22?1?2?1dx=?cotx?x?C =cotxdx???sin2x???(8)
??441???2444(9) ??1?2?xxdx=?1?xxdx=?(x?x)dx=x?4x4?C
7x????33571e2x?1dx=?(ex?1)dx=ex?x?C (10) ?xe?1(11)
cos2x?cosx?sinxdx=?(cosx?sinx)dx=sinx?cosx?C 1?11?1?dxdx==??arctanx?C 22??x2(1?x2)??x?x1?x?1?1?1?dx=tanx?cotx?C =dx??cos2xsin2x??cos2xsin2x??(12)
(13)
(14)
?1?x21?x4dx=?11?x2dx=arcsinx?C
(15)
11??1?sinx?cosxdx=x?cosx?sinx?C ????22?4. 求下列不定积分
2(1) ?(2?x)dx=??(2?x)d(2?x)=?(2?x)2?C
7(2)
525271d(1?2x)dx1?==?(1?2x)22?(1?2x)22(1?2x)?C
(3)
?dx2?3x2=
??3??d??2x????3??31???2x???2=
13arcsinx?C
23?x?d??dxdxx2(4) ?=?=???=?cot?C
xx1?cosx2sin22sin222医用高等数学习题解答（第1,2,3,6章） - 22 -
(5)
3xa?dx=
13x13x=ad(3x)a?C
3?3lnad(x2?x?3)2x?1(6) ?2=lnx2?x?3?C dx=?2x?x?3x?x?31?2x?x?x?2x=?e?e?C (e?e)dx?21(8) ?(sin5x?sin5a)dx=?cos5x?x?sin5a?C
5(7) (9)
?1d(1?x2)dx=??=?1?x2?C
21?x21?x2x141133323(10) ?x?1?xdx=?(1?x)3d(1?x)=(1?x)3?C
4331x(11) ?=dx4?x44??x2?1?x2??d??=arctan????C 22?2?42?x??????1???2???1(12)
?1dxdx=2?x(1?x)1?x?x2??322=2arctanx?C
(13)
?xe?1?x21?x22dx=??ed(?x)=?e?C
22dx=??cosxd(cosx)=2cos??12(14)
sinxcos3xx?C=
2?C cosx(15)
3?1144d(cotx)=?4?C dx????=?cotxcotx?sin2x?4cotx?3(16)
arctanx12==dx(arctanx)?C arctanxd(arctanx)?1?x2?2dexex1x(17) ?x=arctane?C(填空题5) dx=?2xdx=?x2?x1?(e)e?1e?e11?x?1?x?1?1?x?11?xlnd?lnlndx(18) ?=?=?ln??C 2?1?x?1?x?4?1?x?1?x21?x(19)
21?11?1x?111?dxln?C dx??lnx?1?lnx?3?C===???(x?1)(x?3)4??x?1x?3?44x?31?11x?1?dxdxarctanx?arctan?C ==???(x2?1)(x2?2)??x2?1x2?2?22(20)
(21)
?sin3xsinxdx=??11111??cos4x?cos2x?dx=?1??sin4x?sin2x??C=sin2x?sin4x?C
2?42248?医用高等数学习题解答（第1,2,3,6章）
2 - 23 -
1311?1?cos2x?2(22) ?sinxdx=??==??1?2cos2x?cos2xdxx?sin2x?sin4x?C dx??484322??4231522245===sinx?sinx?sinx?C cosxdx(1?sinx)dsinx(1?2sinx?sinx)dsinx???3512(24) ?tan3xdx=?tanx(sec2x?1)dx=?tanxdtanx??tanxdx=tanx?lncosx?C
2(23)
e?1?(25) ?2dx=??exd??=?ex?C
x?x?(26) (27)
1x1112132????==lnx?dxlnx?C ????lnxdlnx??x3sinxsinxsinx==e?C ecosxdxedsinx??(28)
1?11?1?1112?3x1??dx?ln2?3x?ln2?3x?Cln?C ===dx?????4?9x24??2?3x2?3x?4?33122?3x?(29)
?(arcsinx)dx21?x2=
d(arcsinx)1=??(arcsinx)2arcsinx?C
(30)
d(x?1)dx=?x2?2x?2?(x?1)2?1=arctan(x?1)?C
?(1?ex)21?2ex?e2x2ex?x??dxdxx?2arctane?C (31) ?===1?dx2x2xx2????1?e1?e?1?(e)?d(x2?2x?3)d(x?1)(x2?2x?3)?2x?10x2?7(2x?2)?12x??12dxdx=?x?dx(32) ?2==?x2?2x?3?22?(x?1)2 ?x2?2x?3x2?2x?3x?2x?3=x?lnx?2x?3?12?2x?112?(x?1)?C ln?C=x?lnx2?2x?3?3lnx?32?22?(x?1)5. 求下列不定积分
363(1) ?x?1?xdx???????(1?u)?udu=??(u?2u?u)du=?u3?u3?u3?C
dx??du471023令1?x?u2131343734710363=?(1?x)3?(1?x)3?(1?x)3?C
4710(2)
4710?令2?x?u31?11x2(2?u)28322522222dx???????du=?(?4u?4u?u)du=?8u?u?u?C
dx??du35u2?x=?22u(60?20u?3u2)?C=?32?8x?3x21515??2?x?C
医用高等数学习题解答（第1,2,3,6章）
令1?ex?u - 24 -
(3)
?1?ex?11?u?12du?1?C dx????????C=ln=???du=ln2?2uduxxu?1u?1?u?1u?1?dx?21?e?11?eu?11(4)
arctanxarctanx2(arctanx)?C ===dx2d(x)2arctanxd(arctanx)?x(1?x)?1?(x)2?(5)
?(1?x)2dx32dx?cosudu??????令x?atanu令x?sinucosudu(1?sinu)232=
xdu?C ==tanu?C?cos2u21?x=
2(6)
?(x?dx2?a)232dx?asecudu???????2asec2udu(atanu?a)2223x11?C ==cosudusinu?C222a2?a2ax?a(7)
2tanu?secudu令3x?2secu11dx123lnsecu?tanu?Cln3x?9x?4?C ===secudu??????????222339x?4dx?3tanusecudu4secu?43令x?asecua2sec2u?a2x2?a2dx?????????atanu?secudu=a?tan2udu=a?sec2u?1du=a(tanu?u)?C dx?atanusecuduxasecu(8)
????x2?a2?aa??arccos?C?=x2?a2?a?arccos?C =a??axx??(9)
25?=1?sinx?sinxd(sinx)=??cosxsinxdx?sin???212x?2sin52x?sin92x??d(sinx)
?=
4223247211?2?sinx?sin2x?sin2x?C=sinx?sinx?sin3x?sin5x??C
7113711?3?311?lnx?1lnx21??2d(1?lnx)=??1?lnx?dx=?(10) ??d(1?lnx)=(1?lnx)?2(1?lnx)2?C
31?lnxx1?lnx1?lnx??=
21?lnx?lnx?2??C 3(11)
?edxx2?ex1du?e2????????x2令u?ex22dx?1udx11?1?12du?1?2ln1?u?lnu?2??du==????C ???u?u2u??1?uuu2??u?x??1?u1?1?e21????C=2?ln1?e?x2?e?x2??C =2?=2ln?ln??C???xx????22??uu??ee??dx132?cotx?cotx?C (12) ?==?cotx?1d(cotx)4?sinx3??(13)
?1?u2du??13??12?666u?1?du6u?u?ln1?u?C6x?x?ln1?x6===??????????C ????32dx?6u5du1?u221?u??????x?xdx令x?u6(14)
?tan3u11?1???12???dcosu???C secudu==????????24?3??2dx?sec2udu?secu?cosucosu??cosu3cosu?1?xx3dx令x?tanu医用高等数学习题解答（第1,2,3,6章）
31222=(1?x)?1?x?C 3 - 25 -
6. 求下列不定积分 (1)
?arctanxdx=x?arctanx??nx1=dxx?arctanx?ln(1?x2)?C 21?x21?n?1xn?1?11n?1n?1n??(2) ?xlnxdx (n??1)=x?lnx??C lnxdx=x?lnx??xdx=???n?1n?1n?1?n?1?????xn?1?1?=?lnx???C n?1?n?1?(3)
?xln2xdx=
332?322422?3222???2??2?xlnx?lnxdx==lnxdxxlnx?2xlnxdx??? ?????????????3??3?33232?248?23228?32232?23228?32?=xlnx??xlnx??xdx?=xlnx??xlnx?x??C=x?lnx?lnx???C
?33?39?9?339??(4) (5)
uuuuuu???C=2e2ue?e===edx?????2eudu2ude2ue?edu????x令x?udx?2udu??x?x?1?C
?22=xlnx?1?x??lnx?1?xdx??????x?1??x?1?x2?1?x2x?xdx?dx=xlnx?1?x2?? ?21?x???=xlnx?1?x2?1?x2?C (6) (7) =????xcosxdx=?xdsinx=xsinx??sinxdx=xsinx?cosx?C
2?2xx?edx=?12?2x12?2x12?2x12?2x?2x?2x?2x?2x?xe?2xedx?xe?xde?xe?xe?edx ===xde????2222??????1?2?2x1?2x1?2x?21???2x?xe?xe?e?C?=?e?x?x???C 2?222???233===secxdxsecxd(tanx)secxtanx?tanxsecxdxsecxtanx?(secx?secx)dx ????(8)
=secxtanx?sec3xdx?lnsecx?tanx，?sec3xdx=
??(9)
?sin(lnx)dx=xsin(lnx)??cos(lnx)dx=xsin(lnx)??xcos(lnx)??sin(lnx)dx?
1??sin(lnx)dx=x?sin(lnx)?cos(lnx)??C
21?secxtanx?lnsecx?tanx??C 2(10)
1axb?1axbax1axbax?axesinbx?ecosbx?esinbxdxesinbx?ecosbxdx==esinbxdx?? ???aa?aaaa?eax?asinbx?bcosbx??C ??esinbxdx=22a?bax