(完整版)高等数学课后习题答案

531122222 ?1??1?x???1?x??C; ?2?21?t?2ln1?1?t?C;

53??2?112xln ?3??21??ln?C; ?4?2ax11??1x1? ?5?1?e2xa2?b2x?aa?bx?a22?C;

11?e2x?11?ln?C; ?6?arccos?C; 2x1?e2x?1 ?7?11?lnx?1?C. 3ex?2?3ex?4??C; (8) ?8?21?lnx?ln271?lnx?13 (1)3?4 略

?3; (2)

7?27??; (5) 2; (3)2?; (4) ln9216?3?1; (6)

?72?. 1445 ?1?xarccosx?1?x2?C; ?2?lnx?ln(lnx)?1??C;?3?? ?4?2xex?1?4212x?2x?2??C; x?e?ex?1?arctanex?1?C;

?ex?5??1?xarcsinx?x?C; ?6??5?(cos2x?2sin2x)??C;

10x2x7xtanx?lncosx??C; 8?????sin(lnx)?cos(lnx)??C.

221??135???1??; (6) . 6 (1) ??1?;(2) ?ex?2?;(3) 2?1??;(4) 0;(5)

2?2?12852?e?习题3.6

(1)

?2; (2) 1; (3) ?; (4) 发散.

习题3.7

25. 32 18. 13 . 34 12. 1

45. 816 ln2?.

21287 .

35 8

5003. 39 (1) 256; ?2?643; ?3?363?18ln2?3.

??103?. 211(1)

?2; (2)2?.

812 ?.

513ln2?3.

??14 2?2a.

复习题3

52723x13949131223333?x?tanx?C;?2?ax?ax?ax?x?C;?3??x?11?3?C; 1 ?1?3ln33573?4?1x?342x?1ln?C;?5?lnx2?x?3?arctan?C;?6?x?ln1?ex?C; 5x?21111??211?x?238arctanx??C;7(3x?1)(x?2)?C;9arcsinx?x1?x?C; ????????22?1?x?5xx(10) ?10?2xsin?4cos?C;?11??xe?2x?C; ?12?tanxlnsinx?x?C.

22251??12 (1) ?ln26;(2)0; (3) 4?2arctan2; (4) 2?; (5) ?; (6) ?;

28422?2?12?5e;2?. 6?2e;73?; (8) (9) (10) ??e33 (1)

?2?1;(2) 1.

14 (1) 1; (2) .

25 e.

6 最小值为7 690.

533, 最小值为0.

8 y?ln2x?x2.

19 e??2.

e110 ??33?2.

31511 ?.

2??习题 4.1

1（1）√;（2）×;（3）×;（4）√. 2（1）n!;（2）(?1)n?1n2xn11n?2. ;（4）;（3）;（5）(?1)n?1;（6）2n!n!nln(n?1)2n?1n?133（1）收敛 ;（2）发散;（3）收敛

124;（4）发散;（5）发散;（6）发散;（7）发散;（8）115;（9）发散. 354 收敛 .

45 m.

收敛

习题 4.2

1（1）收敛;（2）收敛;（3）收敛;（4）发散.

2（1）收敛;（2）发散;（3）发散;（4）发散;（5）发散;（6）收敛;（7）收敛;（8）收敛;（9）收敛.

3（1）绝对收敛;（2）绝对收敛;（3）条件收敛;（4）发散;（5）条件收敛;（6）绝对收敛;（7）发散;（8）绝对收敛;（9）绝对收敛.

习题 4.3 1（1）(-1,1);（2）(-∞,+∞);（3）[-2,2);（4）[-1,1];（5）(-2,2);（6）(-∞,+∞);（7）{0};（8）[-1,1];（9）[-

42,). 332 (1)

111?x,ln, x???1,1?; (3)(1?x)ln(1?x)?x,x???1,1?. (2)x??1,1;??(1?x)221?x习题 4.4

12n1(1)?x,n?0n!?1?(?1)n?4n2nx????,???;?2??(?1)x,x???1,1?;?3??x,

2(2n)!n?0n?0?n2n??(?1)n(?1)n?1n2n?1x,x????,???;?5??x????,???;?4??2n?1x,x???2,2?; n(2n?1)!n?12n?1n2

1??(?1)n?11?n?n6(?1)(n?1)x,x?7?2x,x??1,1;?????????n??,?; ?5n?0?2?22?n?0??nn(?1)nx2n?1,x????,???. ?8??n?0(2n?1)(2n?1)!??(?1)n(n?1)1??1n(x?1)n. 2 ?1???n?1?n?1?(x?4),x???6,?2?;（2）?2??n?233?n?0n?0?2?3（1）0.156;（2）1.099;（3）3.003;（4）0.946.

习题 4.5 1（1）相等;（2）0 , 0 , 2 , (?1)2（1）f(x)?n?122n;（3）? , 2(?1)?1 , 0. nn???4Asin(2n?1)x, ????x???,x?k?,k?Z?;

?n?12n?1??3?2?sin(2n?1)x??, ????x???,x?k?,k?Z?; （2）f(x)?2?n?12n?1（3）f(x)????2cos(2n?1)xnsinnx?????(?1), ????x???,x?k?,k?Z?;

4n?1??(2n?1)2n???（4）f(x)??2?cos(2n?1)x, ????x????; ?2?n?1(2n?1)cos(2n?1)x, ????x????; ??n?1(2n?1)24?4?（5）f(x)??22??（6）f(x)??3n??3??(?1)sinnx, ????x???,x?k?,k?Z?; ????n?1?nn?（7）f(x)?183?8(?1)nnsinnx, ????x???,x?k?,k?Z?; ?2n?11?9n??(?1)n1??sin(2n?1)x, ????x???,x?(k?)?,k?Z?. （8）f(x)?2???n?12n?12??习题 4.6

n??2(?1)nn?x]sin, ????x???,x?2k,k?Z?; 1（1）f(x)??[222?n?n?2n?14sin(?1)n?1nxsin, ????x???,x?2k?,k?Z?; （2）f(x)?8??n2n?1?

1?cos2(2n?1)?xnsin2n?x?(?1)], (3) f(x)???[224n?1(2n?1)?n?2k?1??,k?Z?; ????x???,x?2??n（4）f(x)?131(?1)??[2?16?n?1n??sinnn?2]cos2nx,

(2k?1)???,k?Z?; ????x???,x?4??E2?2E?14n?t?cost, ????x????; 2 f(x)??sin?2?23?n?14n?13E(?1)nn?sinx, ??2?x?2?; 3 f(x)??1???n?1n24?4 f(x)?1????sin(2n?1)x???x??,x?0 , （1）,（2）; x?x??2n?123n?1??4?3(?1)n1?(?1)n1?]sin2n?x5 f(x)??[, (0?x?); 33n?n?2n?1111f(x)??12?2(?1)n?11cos2nx, (0?x?). ?2n2n?1?习题 4.7

1 f(x)?ee?4?n????n?0????sinn?n?4ei2x . n复习题4

1 （1）×;（2）√;（3）√;（4）√;（5）×. 2 （1）A;（2）C;（3）B;（4）B;（5）C.

3 （1）收敛;（2）收敛;（3）绝对收敛;（4）发散;（5）当0?a?1时，发散；当a?1时收敛;（6）收敛;（7）收敛;（8）收敛;（9）发散;（10）发散;（11）收敛;（12）发散. 4 （1）

211?x1, x???1,1?; ln?arctanx?x , x???1,1?;（2）3(1?x)41?x2??1(lna)nn5 （1）?x, x????,???;（2）?nxn, x???2,2?;

n!n?1n2n?0