© 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
© 2009 Chung Tai Educational Press. All rights reserved.
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
Find the following indefinite integrals. (8 − 21)
8. −+ dxxx )946( 2 9. + dxx
x )2
3(2
10. − dxxx )2(11 4
3
11. +−− dxxxx )82( 342
7.2
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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 .
(b) Hence find the following indefinite integrals.
(i) dxx −3 16
(ii) dxx − 5)41(
7.3
© 2009 Chung Tai Educational Press. All rights reserved.
24. (a) Show that Ckedxke
e xx
x++=
+ )ln( , where k is a positive constant.
(b) Hence find the following indefinite integrals.
(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
.
(b) Hence find the following indefinite integrals.
(i) +++ dxxx
x22
12
(ii) ++dx
xxx
222
2
Find the following indefinite integrals. (28 − 34)
28. − dxx 7)4( 29. − dxx 6)1(
7.4
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30. + dxx 5)12( 31. −dx
x 4)23(
1
32. + dxx 14 33. −dx
x83
1
34. +dx
x 25
1
Find the following indefinite integrals. (35 − 48)
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
© 2009 Chung Tai Educational Press. All rights reserved.
Find the following indefinite integrals. (49 − 50)
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
© 2009 Chung Tai Educational Press. All rights reserved.
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
© 2009 Chung Tai Educational Press. All rights reserved.
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
© 2009 Chung Tai Educational Press. All rights reserved.
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
© 2009 Chung Tai Educational Press. All rights reserved.
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
© 2009 Chung Tai Educational Press. All rights reserved.
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