If and f 共x兲 艋 t共x兲 x 0 t共x兲 dx diverges, then x 0 f 共x兲 dx also
14. If and f 共x兲 艋 t共x兲 x 0 t共x兲 dx diverges, then x 0 f 共x兲 dx also
7. If is f continuous, then x ⫺⬁ 共x兲 dx 苷 lim f t l ⬁ x t ⫺t 共x兲 dx f .
diverges.
EXERCISES Note: Additional practice in techniques of integration is provided
5. y 0 sin cos 2 d
6. y
in Exercises 7.5. 兾2 3 1
y 2 ⫺ 4y ⫺ 12 dy 1– 40 Evaluate the integral.
y 8. y y
sin 共ln t兲
7. y dt
dx
dx
1. ye ⫺0.6y
sarctan x 1 ⫹ sin
3. y 0 d
4 dt
y 1 共2t ⫹ 1兲 3 9. y 1 x ln x dx
10. y 0
1⫹x 2 dx
CHAPTER 7 REVIEW
2 sx 2 ⫺1
11. y dx
1 sin x
12. y 2 dx
dx
⬁ tan ⫺1 x
y ⫺⬁ 50. 1 4x 2 ⫹ 4x ⫹ 5 y x 2 dx
⫺1 1⫹x
y ⫹2
13. e s x
dx
14. y dx
x⫹ 2 ; 51–52 Evaluate the indefinite integral. Illustrate and check that
your answer is reasonable by graphing both the function and its
15. y dx
y tan 2
x⫺ 1 sec 6
16. d
antiderivative (take C苷 0
⫹ 2x
2 51. 2 x x 3 ⫹ 8x ⫺ 3 y ln 共x ⫹ 2x ⫹ 2 兲 dx 52. y 2 ⫹1 dx
17. y x sec x tan x dx
18. y 3 2 dx
sx
x ⫹ 3x
20. tan 5 sec 3 2 y 3 9x 2 ⫹ 6x ⫹ 5 d y
x⫹ 1 19. dx
; 53. Graph the function f 共x兲 苷 cos x sin x and use the graph to guess the value of the integral 2 x 0 f 共x兲 dx . Then evaluate the
21. y st
dx
sx 2 22. y
integral to confirm your guess.
te dt
⫺ 4x
x 5 x e ⫺2x dx by hand? (Don’t
CAS 54. (a) How would you evaluate
dx actually carry out the integration.)
23. y 2 24. e x cos x dx
(b) How would you evaluate x x 5 e ⫺2x dx using tables?
x sx ⫹1
(Don’t actually do it.)
(c) Use a CAS to evaluate x x 5 e ⫺2x dx .
3x 3 ⫺x 2 ⫹ 6x ⫺ 4
25. y 2 2 dx
26. x sin x cos x dx
共x ⫹1 兲共x ⫹2 兲
(d) Graph the integrand and the indefinite integral on the
3 x same screen.
27. y 0 cos 3 ⫹1 x sin 2x dx 28. y s 3 dx
s x ⫺1 55–58 Use the Table of Integrals on the Reference Pages to evaluate the integral.
29. y ⫺1 x sec x dx
30. y
1 5 dx
e x s1 ⫺ e ⫺2x
55. y 5 s4x 2 ⫺ 4x ⫺ 3 dx 56. y csc t dt
ln 10 e x se x ⫺1
31. y 0 dx
兾4 x sin x
y 0 cos 3 x
e x ⫹8
32. dx
57. cos x s4 ⫹ sin 2 y x dx
cot x
58. y dx
34. 共arcsin x兲 y 2 2 dx
59. Verify Formula 33 in the Table of Integrals (a) by differentia-
36. sx ⫹ x y
35. y 3 兾2 dx
1 1 ⫺ tan
tion and (b) by using a trigonometric substitution.
d
1 ⫹ tan
60. Verify Formula 62 in the Table of Integrals. x 2 Is it possible to find a number such that n ⬁ x n dx is
61. y 0 y x
37. 共cos x ⫹ sin x兲 2 cos 2x dx
38. dx
共x ⫹ 2兲 3 convergent?
62. For what values of is ⬁ 1 ax 兾2 xe 兾3 stan a x 0 e cos x dx convergent? Evaluate
2x
39. y 0 2 dx
40. y d
共1 ⫹ 2x兲
the integral for those values of . a 63–64 Use (a) the Trapezoidal Rule, (b) the Midpoint Rule,
sin 2
41–50 Evaluate the integral or show that it is divergent. and (c) Simpson’s Rule with n苷 10 to approximate the given integral. Round your answers to six decimal places.
41. y 1 3 dx
1 ⬁ ln x
42. y 1 4 dx
4 1 共2x ⫹ 1兲 4
63. y 2 dx
y 1 sx cos x dx
ln x
dx
43. y 2 44.
y 2 sy ⫺ 2 dy
x ln x
65. Estimate the errors involved in Exercise 63, parts (a) and (b). 4 ln x
45. y 0 dx
46. y How large should be in each case to guarantee an error of 0 dx n
sx
2 ⫺ 3x
less than 0.00001?
48. 66. Use Simpson’s Rule with n苷 6 y to estimate the area under 0
sx y ⫺1
the curve y苷e x 兾x from x苷 1 to x苷 4 .
CHAPTER 7 TECHNIQUES OF INTEGRATION
67. The speedometer reading ( ) on a car was observed at v 71. Use the Comparison Theorem to determine whether the 1-minute intervals and recorded in the chart. Use Simpson’s
integral
Rule to estimate the distance traveled by the car.
1 x 5 ⫹2 dx t (min)
v (mi
t (min)
v (mi
is convergent or divergent.
1 42 7 57 72. Find the area of the region bounded by the hyperbola 2
2 45 8 y ⫺x 57 2 苷 1 and the line y苷 3 .
3 49 9 55 73. Find the area bounded by the curves y苷 cos x and y苷 cos 2 x
4 52 10 56 between and . x苷 0 x苷 5 54
74. Find the area of the region bounded by the curves
y苷 1 ( 2⫹ sx ) , y苷 1 ( 2⫺ sx ) , and . x苷 1
68. A population of honeybees increased at a rate of r bees per week, where the graph of is as shown. Use Simpson’s Rule r
75. The region under the curve y苷 cos 2 x ,0艋x艋 , is with six subintervals to estimate the increase in the bee popu-
rotated about the -axis. Find the volume of the resulting solid. x lation during the first 24 weeks.
76. The region in Exercise 75 is rotated about the -axis. Find the y r
volume of the resulting solid.
77. If f⬘ is continuous on
and lim xl⬁ f , show that
y 0 f⬘
78. We can extend our definition of average value of a continuous 4000
function to an infinite interval by defining the average value
of on the interval f to be
4 8 12 16 20 24 t
t lim ⬁ l t⫺a y a f
(weeks)
(a) Find the average value of
y苷 tan x
(b) If f and x a f is divergent, show that the
on the interval .
CAS 69. (a) If f , use a graph to find an upper bound
for . f average value of on the interval f is lim xl⬁ f , if
(b) Use Simpson’s Rule with n苷 10 to approximate
x ⬁ (c) If x a f is convergent, what is the average value of f
this limit exists.
0 f and use part (a) to estimate the error.
(c) How large should be to guarantee that the size of the n
on the interval
error in using S n is less than 0.00001 ? (d) Find the average value of y苷 sin x on the interval . 70. Suppose you are asked to estimate the volume of a football.
79. Use the substitution u苷 1 兾x to show that You measure and find that a football is 28 cm long. You use a
piece of string and measure the circumference at its widest ⬁ ln x point to be 53 cm. The circumference 7 cm from each end is
y 0 1⫹x 2 dx 苷 0
45 cm. Use Simpson’s Rule to make your estimate. 80. The magnitude of the repulsive force between two point charges with the same sign, one of size 1 and the other of size q , is
F苷
4 0 r 2 where is the distance between the charges and r 0 is a con-
stant. The potential V at a point due to the charge is P q defined to be the work expended in bringing a unit charge to P from infinity along the straight line that joins and . Find q P
28 cm
a formula for . V