3. 求下列不定积分 (1)
?(1?3xx2)dx=x?x3?C
2x13?x?C (2) ?(2?x)dx=
ln232xxx?12??C=x2?2x2?C (3) ?dx=?(x?x)dx=113x?1??122(4)
121?21?121??1231?2x(x?3)dx=?x?3x)dx=x2?2x2?C
53212531?x2?1x21??dxdx(5) ?==1?dx=x?arctanx?C ??1?x2??1?x2?1?x2?1?x2?1x211?x?1?dxdx(6) ?===?1dx???1?x2??1?x2?2ln1?x?x?C 1?x2(7) (8)
1?cosx12xsindxdx(x?sinx)?C ==?2?22?cot2xdx=??csc2x?1?dx=?cotx?x?C
335??441???2444(9) ??1?2?xxdx=??1?x?xdx=?(x?x)dx=x?4x4?C
7?x?71e2x?1dx=?(ex?1)dx=ex?x?C (10) ?xe?1(11)
cos2x?cosx?sinxdx=?(cosx?sinx)dx=sinx?cosx?C 111??1??arctanx?C dx==?dx22??x2(1?x2)??x?x1?x?11??1dx=?dx=tanx?cotx?C ??cos2xsin2x??cos2xsin2x??或
(12)
(13)
112?2cscdx?4dx?2xd(2x)??2cot2x?C ?cos2xsin2x?sin22x1?x21?x4(14)
?dx=?11?x2dx=arcsinx?C
(15)
11??x?cosx?sinx?C =1?sinx?cosxdx????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)
52527dx1d(1?2x)1==??(1?2x)22?(1?2x)22(1?2x)?C
?3??d??2x????3??31???2x???2(3)
?dx2?3x2=
?=
13arcsinx?C
23(4)
dx?1?cosx=?dx2sin2x2??csc2xx?x?d??=?cot?C
22?2?(5)
3xa?dx=
13x13xad(3x)a?C =?33lna2x?1d(x2?x?3)2dx=?2(6) ?2=lnx?x?3?C
x?x?3x?x?31?2x?x?x?2x(e?e)dx?e?e?C =?21(8) ?(sin5x?sin5a)dx=?cos5x?x?sin5a?C
5(7) (9)
?1d(1?x2)=?1?x2?C dx=??21?x21?x2x1423311333(10) ?x?1?xdx=?(1?x)3d(1?x)=(1?x)3?C
43x1dx(11) ?=4?x44??x2?1?x2???d??=arctan?C ??22?2??x???4?2??1???2???1(12)
?dx1dx=2?x(1?x)1?x??2=2arctanx?C
(13)
?xe?x21?x21?x22dx=??ed(?x)=?e?C
22(14)
?sinxcos3xdx=??cos?32xd(cosx)=2cosx?C=
?122?C cosx(15)
3?1414?C 4d(cotx)=???cotx???cotx=dx??sin2x?4cotx3(16)
arctanx12arctanxd(arctanx)dx(arctanx)?C ==??1?x221exdexxdx=?2xdx=?arctane?C(填空题5) (17) ?x=?xx2e?ee?11?(e)11?x11?x?1?x?1?1?x?lndx(18) ?=lnd?ln?=?ln??C ?1?x21?x21?x?1?x?4?1?x?(19)
2111?11?1x?1??lnx?1?lnx?3?Cdx===?dxln?C ???(x?1)(x?3)4??x?1x?3?44x?3(20)
1x11??1arctanx?arctan?C dx==?dx???(x2?1)(x2?2)??x2?1x2?2?22(21)
?sin3xsinxdx=??411111??cos4x?cos2x?dx=?1?sin2x?sin4x?C =sin4x?sin2x?C??2482?42?21311?1?cos2x?2??1?2cos2x?cos2xdxx?sin2x?sin4x?C (22) ?sinxdx=??==dx??484322??231522245(1?sinx)dsinx(1?2sinx?sinx)dsinxcosxdxsinx?sinx?sinx?C ===???351223(24) ?tanxdx=?tanx(secx?1)dx=?tanxdtanx??tanxdx=tanx?lncosx?C
2(23)
e?1?(25) ?2dx=??exd??=?ex?C
x?x?(26) (27)
1x1112132????????lnxdlnxlnx?dxlnx?C ==??x3?esinxcosxdx=?esinxdsinx=esinx?C
(28)
11?11?1?1112?3x?dx===?dx?ln2?3x?ln2?3x?Cln?C ?????4?9x24??2?3x2?3x?4?33122?3x?(29)
?(arcsinx)dx21?x2=
1d(arcsinx)?=?(arcsinx)2arcsinx?C
(30)
dxd(x?1)=?x2?2x?2?(x?1)2?1=arctan(x?1)?C
?2ex?(1?ex)21?2ex?e2xx?1?dxdx=?dx=??x?2arctane?C (31) ?=x22x2x?1?(e)?1?e1?e???x2?7(x2?2x?3)?2x?102x?10?24??dx=?dx(32) ?2==1??dx 1?dx??????x?2x?3x2?2x?3x?1x?3(x?1)(x?3)????=x?2ln|x?1|?4ln|x?3|?C 5. 求下列不定积分
(1)
3467310322369??????u?u?u?C ?(1?u)?u?(?3u)du(?3u?6u?3u)dux?1?xdx==???dx??3u2du471023令31?x?u363=?(1?x)3?(1?x)3?(1?x)3?C
4710(2)
4710?令2?x?u28(2?u2)2x2?(?2u)du=?(?2u4?8u2?8)du=?u5?u?8u3?C dx??????dx??2udu53u2?x=?232?8x?3x215??2?x?C
(3)
?2du1?ex?11?u?1?1?C =??dx????????C=ln?du=ln2?2uduxxu?1u?1u?1u?1dx???1?e?11?eu2?11令1?ex?u(4)
arctanxarctanx=dx2?x(1?x)?1?(x)2d(x)=2?arctanxd(arctanx)=(arctanx)2?C
令x?uarctanuarctanx22?2arctanud(arctanu)?2udu或?=?(arctanu)?C(arctanx)?C dx???????2?dx?2uduu(1?u)x(1?x)(5)
?(1?x)2dx32dx?cosudu??????令x?atanu令x?sinucosudu(1?sinu)232=
duxtanu?C==?C ?cos2u21?x=
2(6)
?(x?dx2?a2)32dx?asecudu???????2asec2udu(a2tan2u?a2)311xcosudusinu?C==?C 2?2222aaax?a(7)
2tanu?secudu令3x?2secu111dx23lnsecu?tanu?Cln3x?9x?4?C secudu===??????????222334secu?439x?4dx?3tanusecudu(8)
?令x?asecua2sec2u?a2x2?a2dx?????????atanu?secudu=a?tan2udu=a??sec2u?1?du=a(tanu?u)?C dx?atanusecuduxasecu?x2?a2?aa?=a?arccos?C?=x2?a2?a?arccos?C
??xax??(9)
?cos5xsinxdx=?1?sinx?2?2591?222?sinx?2sinx?sinx?d(sinx) sinxd(sinx)=????=
1123247242?2?sinx?sin2x?sin2x?C=sinx?sinx?sin3x?sin5x??C 3711711?3?312lnx1?lnx?11??2dx=?d(1?lnx)=??1?lnx?(10) ??d(1?lnx)=(1?lnx)?2(1?lnx)2?C
31?lnxx1?lnx1?lnx??=
21?lnx?lnx?2??C 3令1?lnx?t232t2?1lnx2t2?1?2(t?1)dt?t?2t?C?1?lnx?lnx?2??C dx????????或??2tedt??t2?1t2?133dx?2tedtx1?lnxte(11)
?edxx2?ex1du?e2????????x2dx?1udx令u?ex2212du1?11???1?==2??du2ln1?u?lnu????C 2??u?u2u??u???1?uuu??1?ex2?1?u1?1????C=2?ln1?e?x2?e?x2??C =2?=ln??C?2ln???xx??????22uu??ee??dx13222??cscx?cscxdx?cotx?1d(cotx)?cotx?cotx?C (12) ?===?sin4x?3(13)
?1?u2du??13??12?66?????66u?1?du===6u?u?ln1?u?C6x?x?ln1?x?????C ????32dx?6u5du1?u1?u???2??2?x?xdx令x?u6(14)
?tan3u211?1???1??????secudu==???dcosu???C ???243??2dx?sec2udu?secucosucosucosu3cosu????1?xx3dx令x?tanu31222=(1?x)?1?x?C 36. 求下列不定积分 (1)
?arctanxdx=x?arctanx??nx1dxx?arctanx?ln(1?x2)?C =21?x2111?n?1xn?1?n?1n?1nlnxdx=x?lnx??xdx=??(2) ?xlnxdx (n??1)=x?lnx??C ??n?1?n?1n?1?n?1?????
医用高等数学第三章习题部分答案
