functions and their applications indefinite integration ...€¦ · indefinite integration and its...

© 2010 Chung Tai Educational Press. All rights reserved.

Chapter 1

Binomial Expansion

Chapter 2

Limits and Derivatives

Chapter 3

Differentiation

Chapter 4

Applications of Differentiation

Chapter 5

Exponential and Logarithmic Functions

Chapter 6

Derivatives of Exponential and Logarithmic Functions and their Applications

Chapter 7

Indefinite Integration and its Applications

Chapter 8

Definite Integration

Chapter 9

Applications of Definite Integration

Chapter 10

Further Probability

Chapter 11

Discrete Random Variables

Chapter 12

Discrete Probability Distributions

Chapter 13

Continuous Random Variables and Normal Distribution

Chapter 14

Parameter Estimation

7.1


Find the following indefinite integrals. (1 − 7)

1. 4

dx 2. dxx23

3. dxx5

2 4. dxx 3

5. − dxx1

6. dxex

2

1

7. dxx7


8. −+ dxxx )946( 2 9. + dxx

x )2

3(2

10. − dxxx )2(11 4

3

11. +−− dxxxx )82( 342

7.2


12. +− dxxx )54)(1( 13. −+ dx

xxx 22

14. ++ dx

xx

1

13

15. dxx 3ln

1

16. dxe x6

1ln

17. −+ dxx

xx )4

2(

18. +− dxxx

x )1

)(13( 19. −+ dx

xxx

2

432

20. − dxex 12 21. + dxe x 34

22. (a) Show that Ca

baxdxbax

++=+

21, where a and b are constants, 0≠a and

abx −≠ .

(b) Hence find the following indefinite integrals.

(i) −dx

x 1

1

(ii) +dx

x 52

3

23. (a) Show that Cnp

pxdxpxn

n ++

−=−+

)1(

)1()1(

1

, where p and n are constants, 0≠p and 1 −≠n .


(i) dxx −3 16

(ii) dxx − 5)41(

7.3


24. (a) Show that Ckedxke

e xx

x++=

+ )ln( , where k is a positive constant.


(i) +dx

eex

x

42

(ii) −+dx

e x31

1

25. (a) Find )6()9( 2

3

+− xxdxd

.

(b) Hence find − dxxx 9 .

26. (a) Find xxe

dxd

.

(b) Hence find dxxex .

27. (a) Find )22ln( 2 ++ xxdxd

.


(i) +++ dxxx

x22

12

(ii) ++dx

xxx

222

2


28. − dxx 7)4( 29. − dxx 6)1(

7.4


30. + dxx 5)12( 31. −dx

x 4)23(

1

32. + dxx 14 33. −dx

x83

1

34. +dx

x 25

1


35. + dxxx 42 )1(2 36. + dxxx 89 32

37. +−− dxxxx 32 )72)(1( 38. −++ dxxx

x32 )15(

52

39. +

dxx

x

12 2 40. −

dxx

x4

3

6

4

41. +−− dxxx

xx13

)2(23

42. − dxxx 8)5(

43. + dxxx 13 44. − xdxx

1

2

45. +− dxe x 45 46. −− dxex xx 42

)2(

47. dxxe x

3

12

6 48.

+ dxx

xln3

7.5



49. − dxxx 62 23

50. −

dxx

x

252

3

51. Use the substitution 1−= xu to find − )1( xxdx

.

52. Use the substitution 1+= xu to find +− dx

xx

1

1.

53. Given that 0>x , use the substitution

xu 1= to find

+122 xx

dx.

54. (a) If

6)6(

1

−+≡

− xB

xA

xx, find the values of constants A and B.

(b) Hence find − )6(xxdx

.

55. (a) If 323 )4()4(4)4(

2

++

++

+≡

++

xC

xB

xA

xx

, find the values of constants A, B and C.

(b) Hence find ++ dx

xx

3)4(

2.

56. (a) Find −

−

−dx

ee

x

x

1.

(b) Hence find −−dx

e x1

1.

(c) Using the results of (a) and (b), find −

−

−+ dx

ee

x

x

1

1.

7.6


57. In each of the following, )(' tS is the rate of change of )(tS with respect to t. Find )(tS .

(a) 1)2( ;14)(' =+= SttS

(b) 3

4)0( ;)('

3 == SetS t

(c) 4)0( ;12

1

1

1)(' =

++

+= S

tttS

58. It is given that 3=dxdy

. When 8 ,2 == yx . Find y in terms of x.

59. It is given that 12 ++= xkxdxdy

, where k is a constant. When 7 ,12

2

==dx

ydx and 3=y . Find y in terms of x.

60. The slope at any point (x, y) of a curve is 32 +x . If (1, 5) is a point on the curve, find the equation of the

curve.

61. The slope at any point (x, y) of a curve is 23 2 +x . If the curve passes through (0, 4), find the equation of

the curve.

62. The slope at any point (x, y) of a curve is

323 xex . If the y-intercept of the curve is −2, find the equation of

the curve.

63. At any point on a certain curve, 122

2

+= xdx

yd. Find the equation of the curve if it passes through )

6

29 ,1(

and the slope is 5 at that point.

7.7


64. It is given that 212 22

2

+= xdx

yd. When 7 ,1 ==

dxdyx and 6=y . Find y in terms of x.

65. At any point on a certain curve, xdx

yd =2

2

. Find the equation of the curve if it passes through (1, 2)

and (4, 4).

66. The growth rate of the population of a city is given by )0(12)(' 015.0 ≥= tetP t , where t is the time measured

in years from the beginning of 2000, )(tP (in thousands) is the population at time t. It is known that the

population of the city was 900 thousand at the beginning of 2004.

(a) Find )(tP .

(b) Find the population of the city at the beginning of 2020. (Give your answer correct to 4 significant figures.)

67. The rate of change of the number of flats sold in a private housing estate can be modelled by

)0()(

90023.03.0

≥+

= − teedt

dNtt ,

where t is the number of days elapsed since the start of the selling of the flats, N is the number of flats sold at time t. It is known that 100=N when 0=t .

(a) (i) Prove that 26.0

6.0

)1(

900

+= t

t

ee

dtdN

.

(ii) Using the substitution 16.0 += teu , or otherwise, express N in terms of t.

(b) Can the number of flats sold be 900? Explain briefly.

68. The rate of change of the daily number of people infected with common cold in a town can be modelled by

)70(3

)25(32 3

12

<<−= tttdtdN

,

where t is the time measured in days with 1=t corresponds to last Monday, N is the daily number of

infected people.

(a) When did the daily number of people infected with common cold become the greatest?

(b) If the daily number of infected people was 150 on last Monday, find the daily number of infected people on the day obtained in (a). (Give your answer correct to the nearest integer.)

7.8


69. The slope at any point (x, y) of the curve C is given by xdxdy 48 −= . The line 32 += xy is a tangent to the

curve at the point P.

x

y

y = 2x + 3

O

C

P

(a) Find the coordinates of P.

(b) Find the equation of C.

70. The slope at any point (x, y) of the curve C is given by )5)(2(6 +−−= xxdxdy . The y-intercept of C is 10.

(a) Find the equation of C.

(b) (i) Prove that the slope of C cannot exceed

2147 .

(ii) Find the point of C with the greatest slope.

71. The rate of change of the temperature of a city yesterday can be modelled by

)100(1 ≤≤+−=θ − thedtd kt ,

where t is the time in hours measured from 9:00 a.m., θ (in °C) is the temperature at time t. At 9:00 a.m., the temperature was 7.4°C, h and k are positive constants.

(a) (i) Express )1ln( +θdtd

as a linear function of t.

(ii) If the slope and the intercept on the vertical axis of the graph of the linear function in (a)(i) are −0.5 and 2 respectively, find the values of h and k. (Give your answers correct to 4 significant figures if necessary.)

Take 4.7=h and 5.0=k .

(b) Express θ in terms of t.

(c) Find the greatest temperature. (Give your answer correct to 1 decimal place.)

7.9


72. The rate of change of the number of visitors in a library during a day can be modelled by

)120(10016

)8(1602

≤≤+−−= t

ttt

dtdN

,

where t is the time elapsed in hours since 8:00 a.m., N is the number of visitors in the library at time t. When the library is just open (i.e. t = 0), there are 88 visitors.

(a) (i) Let 100162 +−= ttu . Find

dtdu

.

(ii) Hence express N in terms of t.

(b) There is a period of time where the number of visitors in the library exceeds 160. How long does the period last for? (Give your answer correct to the nearest 0.1 hour.)

(c) Can the number of visitors in the library reach 180? Explain briefly.

7.10

2009 Chung Tai Educational Press. All rights reserved.

Exercise 7A (page 7.1)

1. Cx 41

2. Cx 3

3. Cx

42

1

4. Cx 34

43

5. Cx ln

6. Cex 21

7. Cx

7ln

7

8. Cxxx 922 23

9. Cxx 23

10. Cxx 24

11

114

11. Cx

xx 831 23

12. Cxxx 521

34 23

13. Cxxx 21

23

25

432

52

14. Cxxx 23

21

31

15. Cx

3ln

ln

16. Cx ln61

17. Cxxx

ln421

2ln2 2

18. Cxxxx 23

21

ln3

19. Cxxx ln223

41 2

20. Cex 12

21. Ce x 34

41

22. (b) (i) Cx 12

(ii) Cx 523

23. (b) (i) Cx

8)16( 3

4

(ii) Cx 24

)14( 6

24. (b) (i) Cex )2ln(21

(ii) Cex )3ln(

25. (a) 925 xx

(b) Cxx )6()9(52 2

3

26. (a) xx exe

(b) Cexe xx

27. (a) 22

222

xx

x

(b) (i) Cxx )22ln(21 2

(ii) Cxxx )22ln( 2

Exercise 7B (page 7.3)

28. Cx 8)4(81

29. Cx 7)1(71

30. Cx 6)12(121

31. Cx

3)23(9

1

32. Cx 23

)14(61

33. Cx

483

34. Cx 25ln51

35. Cx 52 )1(51

36. Cx 23

3 )8(2

37. Cxx 42 )72 (81

38. Cxx

22 )15(2

1

7.11


39. Cx ++2

12 2

40. Cx +−− 46ln

41. Cxx ++− 13ln31 23

42. Cxx +−+− 910 )5(95)5(

101

43. Cxx ++−+ 23

25

)13(272)13(

452

44. Cxxx +−−−+−− 21

23

25

)1(2)1(34)1(

52

45. Ce x +− +− 45

51

46. Ce xx +− 42

21

47. Ce x +− 21

3

48. Cx ++ 2)ln3(21

49. Cxx +−+− 23

225

2 )6(4)6(52

50. Cxx +−+− 21

223

2 )25(25)25(31

51. Cx +− 1ln2

52. Cxxx ++++−+ )1ln(4)1(6)1( 2

53. Cx

x ++− 12

54. (a) 61 ,

61

=−= BA

(b) Cx

x +−

6ln61

55. (a) 2 ,1 ,0 −=== CBA

(b) Cxx

++

++

− 2)4(1

41

56. (a) Ce x +− − 1ln

(b) Cex x +−+ − 1ln

(c) Cex x +−+ − 1ln2

Exercise 7C (page 7.6)

57. (a) 92)( 2 −+= tttS

(b) 131)( 3

+= tetS

(c) 412ln211ln)( ++++= tttS

58. 23 += xy

59. 21

2

23 +++= xxxy

60. 132 ++= xxy

61. 423 ++= xxy

62. 33

−= xey

63. 1323

23+++= xxxy

64. 324 +++= xxxy

65. 45

1724594

154 2

5

+−= xxy

66. (a) 06.0015.0 800900800)( eetP t −+= (b) 1 130 thousand

67. (a) (ii) 8501

500 16.0 +

+−= te

N

(b) No

68. (a) Last Friday (b) 427

69. (a) )6 ,23(

(b) 2382 2

−+−= xxy

70. (a) 106092 23 ++−−= xxxy

(b) (ii) )2

187 ,23( −−

71. (a) (i) hktdtd ln)1ln( +−=+θ

(ii) 5.0 ,389.7 == kh

(b) 2.228.14 5.0 +−−=θ − tet

(c) C2.16 °

72. (a) (i) 162 −t (ii) 100ln8088)10016(ln80

2 +++−−= ttN

(b) 4.3 hours (c) No

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