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

CHAPTER 5 Review 423
;
;
27. y
x 1 2
sx 2 1 4x dx
28. y
csc 2 x
1 1 cot x dx
29. y sin t cos t dt 30. y sin x cosscos xd dx
31. y esx
sx dx
32. y
33. y tan x lnscos xd dx 34. y
sinsln xd
dx
x
x
dx
s1 2 x 4
x
35. y
3
1 1 x dx 36. y sinhs1 1 4xd dx
4
sec tan
37. y
1 1 sec d 38. y y4
s1 1 tan td 3 sec 2 t dt
0
39. y 3
| x 2 2 4 | dx 40. y 4
0 0
| sx 2 1 | dx
41–42 Evaluate the indefinite integral. Illustrate and check
that your answer is reasonable by graphing both the function
and its antiderivative (take C − 0).
41. y
cos x
s1 1 sin x dx
42. y
x 3
sx 2 1 1 dx
43. Use a graph to give a rough estimate of the area of the
region that lies under the curve y − xsx , 0 < x < 4.
Then find the exact area.
; 44. Graph the function f sxd − cos 2 x sin x and use the graph
to guess the value of the integral y 2
0
f sxd dx. Then evaluate
the integral to confirm your guess.
45–50 Find the derivative of the function.
t 2
45. Fsxd − y x
0 1 1 t dt 46. Fsxd − y 1
st 1 sin t dt
3 x
47. tsxd − y x4
cosst 2 d dt 48. tsxd − y sin x 1 2 t 2
0
1 1 1 t dt 4
49. y − y x e t
dt
50. y − y 3x11
sinst 4 d dt
sx t
2x
51–52 Use Property 8 of integrals to estimate the value of the
integral.
51. y 3
sx 2 1 3 dx 52. y 5 1
1 3 x 1 1 dx
53–56 Use the properties of integrals to verify the inequality.
53. y 1
x 2 cos x dx < 1
0 3
54. y y2
y4
sin x
x
dx < s2
2
55. y 1
e x cos x dx < e 2 1 56. y 1
x sin 21 x dx < y4
0
0
57. Use the Midpoint Rule with n − 6 to approximate
y 3 0 sinsx 3 d dx.
58. A particle moves along a line with velocity function
vstd − t 2 2 t, where v is measured in meters per second.
Find (a) the displacement and (b) the distance traveled by
the particle during the time interval f0, 5g.
59. Let rstd be the rate at which the world’s oil is consumed,
where t is measured in years starting at t − 0 on January 1,
2000, and rstd is measured in barrels per year. What does
y 8 0
rstd dt represent?
60. A radar gun was used to record the speed of a runner at the
times given in the table. Use the Midpoint Rule to estimate
the distance the runner covered during those 5 seconds.
t (s) v smysd t (s) v smysd
0 0 3.0 10.51
0.5 4.67 3.5 10.67
1.0 7.34 4.0 10.76
1.5 8.86 4.5 10.81
2.0 9.73 5.0 10.81
2.5 10.22
61. A population of honeybees increased at a rate of rstd bees
per week, where the graph of r is as shown. Use the Midpoint
Rule with six subintervals to estimate the increase in
the bee population during the first 24 weeks.
62. Let
r
12000
8000
4000
0 4 8 12 16 20 24
f sxd −H 2x 2 1 if 23 < x < 0
2s1 2 x 2 if 0 < x < 1
t
(weeks)
Evaluate y 1 23
f sxd dx by interpreting the integral as a difference
of areas.
63. If f is continuous and y 2 0
f sxd dx − 6, evaluate
y y2
f s2 sin d cos d.
0
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