James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

Appendix I Answers to Odd-Numbered Exercises A99
Exercises 7.5 • page 507
1. 2lns1 tan21S 2 sin xd 1 C 3. 32 3 ln 2 2 28 9
1
5.
t 2
7. e
2s2 s2D1C
y4 2 e 2y4
9. 4 5 ln 2 1 1 5 ln 3 (or 1 5 ln 48) 11. 1 2 sec21 x 1 sx 2 2 1
1 C
2x 2
13. 2 1 5 cos5 t 1 2 7 cos 7 t 2 1 9 cos9 t 1 C
15. x sec x 2 ln | sec x 1 tan x | 1 C
17. 1 4 2 19. e e x 1 C 21. sx 1 1d arctan sx 2 sx 1 C
23. 4097
45 25. 4 2 ln 4 27. x 2 ln s1 1 e x d 1 C
29. x lnsx 1 sx 2 2 1d 2 sx 2 2 1 1 C
31. sin 21 x 2 s1 2 x 2 1 C
21S 33. 2 sin D x 1 1 1 x 1 1 s3 2 2x 2 x
2 2
2 1 C
35. 0 37. 1 4 39. ln | sec 2 1 | 2 ln | sec | 1 C
41. tan 2 1 2 2 2 ln | sec | 1 C 43. 2 3 tan21 sx 3y2 d 1 C
45. 2 1 3 sx 3 1 1de 2x 3 1 C
47. ln | x 2 1 | 2 3sx 2 1d21 2 3 2 sx 2 1d22 2 1 3 sx 2 1d23 1 C
49. ln Z
s4x 1 1 2 1
s4x
Z 1 C 51. 2ln Z 2 1 1 1 1
s4x 1 1 1 1 2x
1
53.
m x 2 cosh mx 2 2 m x sinh mx 1 2 2 3
cosh mx 1 C
m
55. 2 ln sx 2 2 lns1 1 sx d 1 C
57. 3 7 sx 1 cd 7y3 2 3 4 csx 1 cd4y3 1 C
59.
1
32
Z ln x 2 2
x 1 2
Z tan21S 2 1 x 1 C
16 2D
Z 1 C
61. csc 2 cot 1 C or tansy2d 1 C
63. 2(x 2 2sx 1 2)e sx 1 C
65. 2tan 21 scos 2 xd 1 C 67. 2 3 fsx 1 1d3y2 2 x 3y2 g 1 C
69. s2 2 2ys3 1 ln(2 1 s3 ) 2 ln(1 1 s2 )
71. e x 2 lns1 1 e x d 1 C
73. 2s1 2 x 2 1 1 2 sarcsin xd2 1 C 75. ln | ln x 2 1 | 1 C
77. 2sx 2 2ds1 1 e x 1 2 ln s1 1 e x 1 1
s1 1 e x 2 1 1 C
79. 1 3 x sin3 x 1 1 3 cos x 2 1 9 cos3 x 1 C
81. 2s1 1 sin x 1 C 83. xe x 2 1 C
Exercises 7.6 • page 512
1. 2 5
21 3. s13 2 3 4 lns4 1 s13 d 2 1 2 1 3 4 ln 3
5.
8 arctan 4 2 1 4 lns1 1 1 16 2 d 7. 1 6 Z ln sin x 2 3
sin x 1 3
Z 1 C
9. 2 s9x 2 1 4
1 3 ln(3x 1 s9x 2 1 4 ) 1 C
x
11. 5y16 13. 2sx arctan sx 2 lns1 1 xd 1 C
15. 2ln | sinhs1/yd | 1 C
17.
2y 2 1
8
s6 1 4y 2 4y 2 1 7 2y 2 1
sin21S
8 s7
2 1 12 s6 1 4y 2 4y 2 d 3y2 1 C
D
19. 1 9 sin3 x f3 lnssin xd 2 1g 1 C
21.
1
2s3 ln Z
e x 1 s3
e x 2 s3
Z 1 C
23. 1 4 tan x sec3 x 1 3 8 tan x sec x 1 3 8 ln | sec x 1 tan x | 1 C
25. 1 2 sln xds4 1 sln xd2 1 2 lnfln x 1 s4 1 sln xd 2 g 1 C
27. 2 1 2 x 22 cos 21 sx 22 d 1 1 2s1 2 x 24 1 C
29. se 2x 2 1 2 cos 21 se 2x d 1 C
31. 1 5 ln | x 5 1 sx 10 2 2 | 1 C 33. 3 8 2
37. 1 3 tan x sec2 x 1 2 3 tan x 1 C
39. 1 4 xsx 2 1 2dsx 2 1 4 2 2 lnssx 2 1 4 1 xd 1 C
41. 1 4 cos3 x sin x 1 3 8 x 1 3 8 sin x cos x 1 C
43. 2ln| cos x | 2 1 2 tan2 x 1 1 4 tan4 x 1 C
45. (a) 2ln Z
1 1 s1 2 x 2
x
Z 1 C;
both have domain s21, 0d ø s0, 1d
Exercises 7.7 • page 524
1. (a) L 2 − 6, R 2 − 12, M 2 < 9.6
(b) L 2 is an underestimate, R 2 and M 2 are overestimates.
(c) T 2 − 9 , I (d) L n , T n , I , M n , R n
3. (a) T 4 < 0.895759 (underestimate)
(b) M 4 < 0.908907 (overestimate); T 4 , I , M 4
5. (a) M 10 < 0.806598, E M < 20.001879
(b) S 10 < 0.804779, E S < 20.000060
7. (a) 1.506361 (b) 1.518362 (c) 1.511519
9. (a) 2.660833 (b) 2.664377 (c) 2.663244
11. (a) 27.276910 (b) 24.818251 (c) 25.605350
13. (a) 22.364034 (b) 22.310690 (c) 22.346520
15. (a) 0.243747 (b) 0.243748 (c) 0.243751
17. (a) 8.814278 (b) 8.799212 (c) 8.804229
19. (a) T 8 < 0.902333, M 8 < 0.905620
(b) | | ET < 0.0078, | | EM < 0.0039
(c) n − 71 for T n, n − 50 for M n
21. (a) T 10 < 1.983524, E T < 0.016476;
M 10 < 2.008248, E M < 20.008248;
S 10 < 2.000110, E S < 20.000110
(b) | | ET < 0.025839, | | EM < 0.012919, | | ES < 0.000170
(c) n − 509 for T n, n − 360 for M n, n − 22 for S n
23. (a) 2.8 (b) 7.954926518 (c) 0.2894
(d) 7.954926521 (e) The actual error is much smaller.
(f) 10.9 (g) 7.953789422 (h) 0.0593
(i) The actual error is smaller. (j) n > 50
25.
n L n R n T n M n
5 0.742943 1.286599 1.014771 0.992621
10 0.867782 1.139610 1.003696 0.998152
20 0.932967 1.068881 1.000924 0.999538
n E L E R E T E M
5 0.257057 20.286599 20.014771 0.007379
10 0.132218 20.139610 20.003696 0.001848
20 0.067033 20.068881 20.000924 0.000462
Observations are the same as after Example 1.
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