五、(不定积分)练习题参考答案 习题4－1(A) 1.
1252?32h2ln?3?C ?x2?C (4)10x(1)x?C (2)?C (3)x3595534322122(5)2x?x2?x2?C (6) v2?v?v2?C
355233xex?C (9) 2ex?3lnx?C (7) 2arcsinx?3arctanx?C (8)
1?ln3 (10) ?3cosx?cotx?c 2.x3?C,x3?2 4.y?525x?c，y?x2?1 225.y?lnx?1 6.(1) 27m (2)3360S
17.D(p)?1000()p
3习题4－1(B) 1.
1(1)arcsinx?C (2)x2?arctanx?C (3) x?arctanx?C (4) ??3arctanx?C
x34(x2?7)xtanx?secx?C (8)(x?sinx)?C (5) (6) (7)e?2x?C?C274x(9)2(tanx?x)?C (10) 2.s?s0?11tanx?C 2?(1?cos?t)
?x??5cost?103.?
y?2sint??13?3(x?5)?4.F(x)??x?1?1?(x3?1)?3x?1x?1 x??1
习题4－2(A) 1. 111211(1) (2) (3)? (4)?2 (5)? (6)? (7) (8) ?1 710353123.
1122?5x?C (1) ?(3?2x)3?C (2)?ln1?2x?C (3)?325(4)11313arcsinx?C (5) ?5?3x2?C (6)arctanx?C
36253(7)
2ln41112x?1?C (8)?e?5t?C (9)x?sin4x?C
5282x?11?111x?2(10)?cot(2x?)?C (11)cos2x?cos4x?C (12) ln?C
24483x?14.
111(1)ln(x2?2)?C (2)arctan(x2)?C (3) ?1?x4?C 2223121x29222329?4x?C (6)(1?x)?C (4)?ln(x?9)?C (5)arcsinx?23422911212(7)(lnx)2?C (8) ex?C (9) e2x?C (10)?ln1?cosx?C 2241(11)sinx?sin3x?C (12)arctanex?C
35.
11?1?x21?x2?C (2)arccos?C (3)?(1)ln?C
xxxa2xx21a?x2?C (5)(4)arcsin?2a29x2?91?C (6)2axxx?a22?C
3x111?e?11?2x?C (8)ln?C (9)2x?ln(1?2x)?C (7)(1?2x)2?x621?e?1(10)2(x?1?arctanx?1)?C
习题4－2(B)
1.
112F(ax2?b)?C (2) ?F()?C (3)F(ax)?C (1)2axa1(4)F(lnx)?C (5)F(eax)?C (6)F(sinx)?C (7)?F(cosx)?C
a(8)F(tanx)?C (9)-F(arccosx)?C (10)F(arctanx)?C 2.
1111(1)arctan(2sinx)?C (2)?cos5x?cos7x?C (3)sec3x?secx?C 257321x3(4)cox?C (5)tan?C (6)tanx?secx?C (7)(sinx?cosx)3?C
x2211?C (8)(lntanx)2?C (9)lnlnlnx?C (10)?4xlnx3.
(1)2arctanx?C (2)arcsinx?23?2x?3?C (3) 23?2x?3ln?C 23?2x?3(4)2x?44x?4ln(4x?1)?C (5)arcsinx?1(6)(arcsinx?lnx?1?x2)?C 24.
11(1)f(x)?2x?1 (2)f(x)?x?x2?
22习题4－3(A)
x1?1?x2?c
x?1?C ex1?1?11(4)?xcos(3x?)?sin(3x?)?C (5)?xcos2x?sin2x?C
3494481(6)xarccosx?1?x2?C (7)xarctanx?ln(1?x2)?C
2(1)xsinx?cosx?C (2)x(lnx?1)?C(3)?121211ex(8)xln(x?1)?x?x?ln(x?1)?C (9)(sinx?cosx)?C 242221(10) xtanx?lncosx?x2?C
2习题4－3(B) 1.
?lnx1?(1)(x2?2)sinx?2xcosx?C (2)xn?1???C 2??n?1(n?1)?x2?2(lnx)2?2lnx?1??C (4)xln(x?1?x2)?1?x2?C (3) ??4113(5)(x2?3x?)e2x?C (6)x?(1?e?x)ln(1?ex)?C 222xxx21(7)arctanx?(x?arctanx)?C (8)?e?2x(cos?4sin)?C
17222211(9)-e(x4?x2?1)?C (10)(xsec2x?tanx)?C
22-x2xx3x21(11)arctanx??ln(1?x2)?C (12)?cos(lnx)?sin(lnx)?C?
2366(13)3e3x(3x2?23x?2)?C (14)?2x?1cosx?1?2sinx?1?C
1n?2secn?2xtanx??n?2 n?1n?1习题4－4(A)
2.?n?1x21x6(x?4)41.(1)ln2?C (2) ln6?C (3)ln?C 32x?124x?4(x?3)x3x2??x?8lnx?4lnx?1?3lnx?1?C (4)
32(5)123arctansinx2tanxx?C (6)ln1?tan?C (7)ln?C
2231?sinx2132(8)ln(4tanx?9)?C (9)(x?1)3?33x?1?3ln1?3x?1?C 82 (10)x2?1?lnx?x2?1?C 2.
x1x3111e?1??C (3)3arctan3?C a?0 (1) lnx?C (2)23aa2(1?x)1?x2e?1(4)lnx?2ln(1?3x)?C (5)xln(1?x2)?2x?arctanx?C
11xx(6)tanxsecx?lnsecx?tanx?C (7)2(sin?cos)?C 222244xx4x?1arctanx(8)?ln4?C (9)??ln?C
24x?2xx?1(10) (arctanx)?C (11)6ln26611x?C ?C (12)?2x2e?1x?1x1(13)lnsinx?cosx?C (14)?lncosx?sinx?C
22sinx1xx2221?xarcsinx??C (16)ln?C (15) (arcsinx)?4241?sinx习题4－4(B)
1. (1)lnx?31??C (2)arctan(2x?1)?arctan(2x?1)??C ??x?1212111?C (4)lntanx?tanx?C (3)ln2?3x?992?3x22x(5)2sinxesinx?2esinx?C (6)xtan?C (7)(4?2x)cosx?4xsinx?C
2(8) ?(9)
33x?1?C
2x?11(x?1)(x?1)2?4?4x2?2x?3?lnx?1?x2?2x?3?C 2(10) 2.
2x2?2x?1?C
x3?1?1?x21??x3x?(1)4????22??C (2)??223?3a??a?x?a?x??x???1?x2 ???x?3(3)
233xxx32x2lncsc?cot?C (4)(1?e)(e?)?C
25522(5) esinx(x?secx)?C (6)x?ln(1?ex)?1?C 1?ex1x3xxex)?C (8)x?ln(1?ex)?C (7)(arctan?25439?xe?1(9)x(arcsinx)2?21?x2arcsinx?2x?C
11(10)?(2?x2)1?x2arccosx?x(x2?6)?C
39x1x?1?C (11)earctanx?C (12)
2lnx21?x3.
xcosx?2sinx?C
x?x?1?C4.f(x)??x?e?C???x?00?x???0?x?1?lnx?1?C,f(lnx)??
x?C1?x????总复习题四
一、(1)C (2)D (3)C (4)B (5)A (6)B