x1?x22二、(1)?2cos?C (2)2arctanx?1?C (3)?e(x?1)?C
223122f?1(x)?(4) x?e?C (5)?(1?x)?C (6)xf?1(x)?F????C 3x31lnx122三、(1) (1?x)?1?x2?C (2)?ln1??C
31?xx1 (3) (1?)ln(1?x)?C (4)(2x?1?1)ex2x?1?C
(5)2xex?1?4ex?1?4arctanex?1?C
1 (6)?e?xarccotex?x?ln(1?e2x)?C
211(7) xarctanx?ln(1?x2)?(arctanx)2?C (8)e2xtanx?C
221x2?1?C (9)arctan42(10)当a?0,b?0时，
11tanx?C?cotx?C a?0,b?0 . 当时，b2a21atanx)?C 当ab?0时，arctan(abb1x1x1x1x(11)?xcsc2?cot?C (12) lntan?tan2?C
82424282(13)x2cosx?4xsinx?6cosx?C (14)2lnx?1?x?C (15)cx1?x2
六、(不定积分)练习题选解
1. 习题4－1(B)
1?2x21?x2?3x2131(4)?2dx?dx?(?)dx???3arctanx?C ?x2(1?x2)?x21?x2x(1?x2)x2. 习题4－1(B)
4.求f(x)?max{1,x2}的一个适合条件F(0)?1的原函数F(x)
?x2?解：f(x)??1?x2?x?1?1?x?1，f(x)为(??,??)上连续的分段函数，设F1(x)x?1为f(x)在(??,??)上的一个原函数，因F1(x)可导，所以F1(x)必连续.
?x3x?1?3??1?? F1(x)??x(其中?1,?2为待定常数，由F1(x)在x??1及x?1处?1??x1?x3???2x?1??3的连续性来决定)
12lim?F1(x)???1,lim?F1(x)??1 ??1?? x??1x??13312limF(x)???,limF(x)?1??? 1212?x?1?x?133??x32x?1?3?3?C??f(x)dx?F1(x)?C??x?C?1?x?1
?x32???Cx?1??33所求F(x)?F1(x)?C0，由F(0)?1?C0?1
?13x?1?3(x?1)??1?x?1 于是F(x)??x?1?1?(x3?5)x?1?312x212x22ed2x?e?C 4?411?sinx1?sinxdx??dx?dx 4. 习题4－2(B) 2(6) ?22?1?sinx1?sinxcosx1sinx??(2?)dx?tanx?secx?C 2cosxcosx2x3. 习题4－2(A) 4.（9）?e2?lnxdx??e2xelnxdx??e2xxdx?225. 习题4－2(B) 2.(10)?6. 习题4－2(B) 3.(2)?1?lnxd(xlnx)1dx????C 22?(xlnx)(xlnx)xlnxdxdx1dx????
22x(4?x)x?224?(x?2)1?()2?arcsinx?2?C 27. 习题4－2(B) 3.(6)?dxx?1?x2
令x?sint 原式?? ?
1in? ?(arcsx2xln??2x1?C )cost1(sint?cost)(cost?sint)dt??dt
sint?cost2sint?cost1?d(sitn?cot?s)1t??(t?lnsi?tn??2?sitn?ctos2??cto?s C)18. 习题4－3(A) (10)?xtan2xdx??x(sec2x?1)dx??xdtanx?x2
21?xtanx?lncosx?x2?C
2ln(1?ex)dx???ln(1?ex)de?x 9. 习题4－3(B) 1.(6)?xee ??ln(1?x)e??xex?x??eex?1dx?x?(1?e?xx)ln(?1e ?) C
xsinxxdcosx11dx??xd?cos3x2?cos2x cos3x1 ?(xse2cx?taxn?)C
2111. 习题4－3(B)1.(12)?cos(lnx)dx?xcos(lnx)??xsin(lnx)dx
x
10. 习题4－3(B) 1.(10)??xcos(lxn??)sinx(dlnx?xcos(lxn?)xlndx)? ??cos(xx?2sinx(?lnx)?cosx(?ln)1cxos (dlxn)xsi?n(ln C?x)12. 习题4－3(B)2.(2) 设?n??secnxdx试证 ?n?1n?2secn?2xtanx??n?2 n?1n?1(n?2,3,?)
证：?n??secnxdx??secn?2xdtanx
c?2xtaxn?? ?senc?2xtaxn?n?( ?sen2txan?(n?22)xdxse cn?2?2)2x(s?ec1xdx)s ec ?senc?2xtaxn?n?(?n2??)(n?2
)??n?1n?2secn?2xtanx??n?2 n?1n?1dxx5dx111613. 习题4－4(A)1.(2) ? ??(?)dx66666??x(x?4)x(x?4)24xx?41x6?ln6?C 24x?414. 习题4－4(A)1.(8)?tanxtanxdx?dtanx 222?4sinx?9cosc4tanx?91dta2nx12?ln(4taxn? ??224tanx?98?C9)
15. 习题4－4(A)1.(10)?x?1x?1x1dx??dx??dx??dx
222x?1x?1x?1x?1?x2?1?lnx?x2?1?C
x111x84dx?dx16. 习题4－4(A)2.(8)?8 484?x?3x?24x?3x?2113x4?2x414144dx??(?)dx4 ??dx??8444?44x?3x?244x?2x?144x4x?1 ??ln4?C
4x?217. 习题4－4(A)2.(11)? 令x?t6
xdx 3x(x?x)36t2?6t51x 原式??632dt?6?td?6ln6?C
t(t?t)t(t?1)x?118. 习题4－4(A)2.(12)?dxdx?
e2x?e?2x?2?(ex?e?x)2e2x11 ??2xdx???C
(e?1)22ex2?119. 习题4－4(A)2.(14)?dxcosx??dx
1?tanxcosx?sinx1(cosx?sinx)?(cosx?sinx)dx
2?cosx?sinxx11???d(cosx?sinx) 22cosx?sinxx1??lncosx?sinx?C 22?
20. 习题4－4(A)2.(15)?1?x2arcsinxdx 令x?sint
1?cos2tt21dt???tsin2tdt 原式??cost?tcostdt??t?244t211??tsintcost?sin2t?C 4241xx222?(arcsinx)?1?xarcsinx??C 424x2?1x2?121. 习题4－4(B)1.(2)?4dx??2dx
x?1(x?1)2?2x2x2?1 ??2dx 2(x?1?2x)(x?1?2x) ?1111dx?dx ??222x?1?2x2x?1?2x1111dx?dx
2?x2?1?2x2?x2?1?2x ?xx2sincosx?sinxx?sinxx22dx 22. 习题4－4(B)1.(6)?dx??dx??xdtan??xx1?cosx22cos22cos222xxxxtandx?xta?n C ?xtan??tandx? ?2222 23. 习题4－4(B)1.(8)?dx3(x?1)2(x?4)4??dx
x?123()(x?1)2x?1