高数a第四章不定积分习题答案

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五、(不定积分)联系题参考答案 习题4－1(A) 1.

1252 32

ln 3 C

x2 C (4)10x(1)x C

C (3)

x35553

4322122

(5)x2 x2 C (6) v2 v v2 C

35523

3xex

C (9) 2ex 3lnx C (7) 2arcsinx 3arctanx C (8)

1 ln3

(10) 3cosx cotx c 2.x3 C,x3 2 4.y

525

x c，y x2 1 22

5.y lnx 1 6.(1) 27m

1

7.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

x32x

tanx secx C (8)(x sinx) C

(6) (7)e CC

2(9)2(tanx x) C (10) 2.s s0

1

1

tanx C 2

(1 cos t)

x 5cost 103.

y 2sint

13

3(x 5)

4.F(x) x 1

1

(x3 1) 3

x 1x 1 x 1

习题4－2(A) 1. 111211

(1) (2) (3) (4) 2 (5) (6) (7) (8) 1 71035312

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3.

11C

(1) (3 2x)3 C (2) ln 2x C

(3)3213 C

(5) C (6)arctanx C

62(7)

4111 C (8) e 5t C (9)x sin4x C

5281 111x 2

(10) cot(2x ) C (11)cos2x cos4x C (12) ln C

24483x 1

4.

11(1)ln(x2 2) C (2)arctan(x2) C

(3) C 223

12x292232

C (6)(1 x) C (4) ln(x 9) C

(5)arcsinx

23229

11212

(7)(lnx)2 C (8) ex C (9) e2x C (10) ln cosx C 224

1

(11)sinx sin3x C (12)arctanex C

3

5.

1 C (2)arccos C

(3)(1)ln C

xa2x1

C

(5)(4)arcsin2a9 C

x

C

31C

(8) C

ln(1 C

(7)(1 2x)26(10) C

习题4－2(B)

1.

112

F(ax2 b) C (2) F() C

(3)F( C (1)2axa

1

(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.

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1111

(1)arctan(2sinx) C (2) cos5x cos7x C (3)sec3x secx C 2573

2

1x3

(4)cox C (5)tan C (6)tanx secx C (7)(sinx cosx)3 C

x22

11

C (8)(lntanx)2 C (9)lnlnlnx C (10)

4xlnx3.

(1)C (2)arcsin

x 2 C

(3) C

2(4)1) C

(5)arcsinx1

(6)(arcsinx lnx C 24.

11

(1)f(x) 1 (2)f(x) x x2

22

习题4－3(A)

c

x 1

C ex

1 1 11

(4) xcos(3x ) sin(3x ) C (5) xcos2x sin2x C

349448

1

(6)xarccosxC (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 24222

1

(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(xC (3) 4113

(5)(x2 3x )e2x C (6)x (1 e x)ln(1 ex) C 22

2xxx21

(7)arctanx (x arctanx) C (8) e 2x(cos 4sin) C

172222

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11

(9)-e(x4 x2 1) C (10)(xsec2x tanx) C

22

-x2

xx3x21

(11)arctanx ln(1 x2) C (12) cos(lnx) sin(lnx) C

2366

(13)32) C

(14) C

1n 2

secn 2xtanx n 2 n 1n 1习题4－4(A)

2. n

1x21x6(x 4)4

1.(1)ln2 C (2) ln6 C (3)ln C 3

2x 124x 4(x 3)x3x2

x 8lnx 4lnx 1 3lnx 1 C

(4)

32

x

C (6)ln tan C

(7) C

22

132

(8)ln(4tanx 9) C

(9)(x 1)3 3ln C 82

lnx C 2.

x

1x3111e 1

C (3)3arctan3 C a 0

(1) lnx C (2)2

3aa2(1 x)1 x2e 1

(4)lnx 2ln(1 C (5)xln(1 x2) 2x arctanx C

11xx

(6)tanxsecx lnsecx tanx C (7)2(sin cos) C

2222

x4arctanx

(8) C

(9) C

4x(10) C

(11)2

11 C C (12) 2x

2e 1x1

(13)lnsinx cosx C (14) lncosx sinx C

22

sinx1x22x C (16)ln C (15) (arcsinx)

441 sinx

习题4－4(B)

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1. (1)lnx

3 C

1) 1) C x 112111

C (4)lntanx tanx C (3)ln2 3x

992 3x22

x

(5)2sinxesinx 2esinx C (6)xtan C

(7)(4 2xC

2

(8) (9)

C

1

(x lnx 1 C

2

(10) 2.

C

x

3 11

(1)4 C

(2) 33a

3

(3)

2

33xxx32x

csc cot C (4)(1 e)(e ) C

25522

(5) esinx(x secx) C (6)x ln(1 ex)

1

C 1 ex

1x3xxex

) C (8)x ln(1 ex) C

(7)(arctan 2

5439 xe 1

(9)x(arcsinx)2 x 2x C

11

(10) (2 x2x x(x2 6) C

39x1 C (11)earctanxC (12)

lnx23.

xcosx 2sinx

C

x

x 1 C

4.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

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x1 x22

二、(1) 2cos C

(2)C (3) e(x 1) C

22

3

122

f 1(x) (4) x e C (5) (1 x) C (6)xf 1(x) F C 3

x

3

1lnx122

三、

(1) (1 x)C (2) ln C

31 xx

1

(3) (1 )ln(1 x) C

(4)1)x

C

(5)2C

1

(6) e xarccotex x ln(1 e2x) C

211

(7) xarctanx ln(1 x2) (arctanx)2 C (8)e2xtanx C

22

1x2 1

C (9)arctan

42

(10)当a 0,b 0时，

11

tanx C cotx C a 0,b 0 . 当时，b2a2

1atanx

) C 当ab 0时，arctan(

abb

1x1x1x1x

(11) xcsc2 cot C (12) lntan tan2 C

82424282

(13)x2cosx 4xsinx 6cosx C (14)2lnx x C

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 x2

x(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.

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x3

x 1 3 1

F1(x) x(其中 1, 2为待定常数，由F1(x)在x 1 …… 此处隐藏：5599字，全部文档内容请下载后查看。喜欢就下载吧 ……