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