7–58. Improper integrals Evaluate the following integrals or state that they diverge.

47. ∫ (from 0 to 10) 1/∜(10 - x) dx

7–58. Improper integrals Evaluate the following integrals or state that they diverge.

50. ∫ (from 0 to 9) 1/(x - 1)¹ᐟ³ dx

7–58. Improper integrals Evaluate the following integrals or state that they diverge.

53. ∫ (from 0 to 1) ln x dx

7–58. Improper integrals Evaluate the following integrals or state that they diverge.

56. ∫ (from 0 to 1) 1/(x + √x) dx

59. Perpetual Annuity

Imagine that today you deposit $B in a savings account that earns interest at a rate of *p*% per year compounded continuously (see Section 7.2). The goal is to draw an income of $I per year from the account forever. The amount of money that must be deposited is:

B = I × ∫(from 0 to ∞) e^(-rt) dt

where r = p/100.

Suppose you find an account that earns 12% interest annually, and you wish to have an income from the account of $5000 per year. How much must you deposit today?

62. Electronic Chips Suppose the probability that a particular computer chip fails after a hours of operation is 0.00005 ∫(from a to ∞) e^(-0.00005t) dt.

a. Find the probability that the computer chip fails after 15,000 hr of operation.

63. Average Lifetime The average time until a computer chip fails (see Exercise 62) is 0.00005 ∫(from 0 to ∞) t e^(-0.00005t) dt. Find this value.

65-76. Volumes Find the volume of the described solid of revolution or state that it does not exist.

66. The region bounded by f(x) = (x^2 + 1)^(-1/2) and the x-axis on the interval [2, ∞) is revolved about the x-axis.

65-76. Volumes Find the volume of the described solid of revolution or state that it does not exist.

69. The region bounded by f(x) = 1/√(x ln x) and the x-axis on the interval [e, ∞) is revolved about the x-axis.

65-76. Volumes Find the volume of the described solid of revolution or state that it does not exist.

72. The region bounded by f(x) = (x + 1)^(-3/2) and the x-axis on the interval (-1, 1] is revolved about the line y = -1.

65-76. Volumes Find the volume of the described solid of revolution or state that it does not exist.

75. The region bounded by f(x) = (4 - x)^(-1/3) and the x-axis on the interval [0, 4) is revolved about the y-axis.

77–86. Comparison Test Determine whether the following integrals converge or diverge.

78. ∫(from 0 to ∞) dx / (eˣ + x + 1)

77–86. Comparison Test Determine whether the following integrals converge or diverge.

81. ∫(from 1 to ∞) (sin²x) / x² dx

77–86. Comparison Test Determine whether the following integrals converge or diverge.

84. ∫(from 1 to ∞) (2 + cos x) / x² dx

87. Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

d. If ∫(from 1 to ∞) x^(-p) dx exists, then ∫(from 1 to ∞) x^(-q) dx exists (where q > p).

88. Incorrect Calculation

b. Evaluate ∫(from -1 to 1) dx/x or show that the integral does not exist.

What are the two general ways in which an improper integral may occur?

7–58. Improper integrals Evaluate the following integrals or state that they diverge.

28. ∫ (from 1 to ∞) tan⁻¹(s)/(s² + 1) ds

7–58. Improper integrals Evaluate the following integrals or state that they diverge.

31. ∫ (from 1 to ∞) 1/[v(v + 1)] dv

108. Draining a tank Water is drained from a 3000-gal tank at a rate that starts at 100 gal/hr and decreases continuously by 5%/hr. If the drain is left open indefinitely, how much water drains from the tank? Can a full tank be emptied at this rate? 

101. Many methods needed Show that the integral from ∫(from 0 to ∞)(sqrt(x) * ln x) / (1 + x)^2 dx equals pi, following these steps

d. Evaluate the remaining integral using the change of variables z = sqrt(x)

102–106. Laplace transforms A powerful tool in solving problems in engineering and physics is the Laplace transform. Given a function f(t), the Laplace transform is a new function F(s) defined by F(s) = ∫[0 to ∞] e^(-st) f(t) dt, where we assume s is a positive real number. For example, to find the Laplace transform of f(t) = e^(-t), the following improper integral is evaluated using integration by parts:

F(s) = ∫[0 to ∞] e^(-st) e^(-t) dt = ∫[0 to ∞] e^(-(s+1)t) dt = 1/(s+1).

Verify the following Laplace transforms, where a is a real number.

104. f(t) = t → F(s) = 1/s²

94. The family f(x) = 1/xᵖ revisited Consider the family of functions f(x) = 1/xᵖ, where p is a real number.

For what values of p does the integral ∫(1 to ∞) 1/xᵖ dx exist?

What is its value when it exists?

91. [Use of Tech] Regions bounded by exponentials Let a > 0 and let R be the region bounded by the graph of y = e^(-a·x) and the x-axis

on the interval [b, ∞).

a. Find A(a,b), the area of R as a function of a and b.

91. [Use of Tech] Regions bounded by exponentials Let a > 0 and let R be the region bounded by the graph of y = e^(-a·x) and the x-axis

on the interval [b, ∞).

c. Find the minimum value b* such that when b > b*, there exists some a > 0 where A(a,b) = 2.

Gaussians An important function in statistics is the Gaussian (or normal distribution, or bell-shaped curve), f(x) = e^(-ax²).

c. Complete the square to evaluate ∫ from -∞ to ∞ of e^(-(ax² + bx + c)) dx, where a > 0, b, and c are real numbers.

109. Escape velocity and black holes The work required to launch an object from the surface of Earth to outer space is given by W = ∫ from R to ∞ of F(x) dx, where R = 6370 km is the approximate radius of Earth, F(x) = (GMm)/x² is the gravitational force between Earth and the object, G is the gravitational constant, M is the mass of Earth, m is the mass of the object, and GM = 4 × 10¹⁴ m³/s².

c. The French scientist Laplace anticipated the existence of black holes in the 18th century with the following argument: If a body has an escape velocity that equals or exceeds the speed of light, c = 300,000 km/s, then light cannot escape the body and it cannot be seen. Show that such a body has a radius R ≤ 2GM/c². For Earth to be a black hole, what would its radius need to be?

102–106. Laplace transforms A powerful tool in solving problems in engineering and physics is the Laplace transform. Given a function f(t), the Laplace transform is a new function F(s) defined by F(s) = ∫[0 to ∞] e^(-st) f(t) dt, where we assume s is a positive real number. For example, to find the Laplace transform of f(t) = e^(-t), the following improper integral is evaluated using integration by parts:

F(s) = ∫[0 to ∞] e^(-st) e^(-t) dt = ∫[0 to ∞] e^(-(s+1)t) dt = 1/(s+1).

Verify the following Laplace transforms, where a is a real number.

106. f(t) = cos(at) → F(s) = s/(s² + a²)

95–98. {Use of Tech} Numerical methods Use numerical methods or a calculator to approximate the following integrals as closely as possible. The exact value of each integral is given.

98. ∫(from 0 to 1) (ln x)/(1+x) dx = -π²/12

95–98. {Use of Tech} Numerical methods Use numerical methods or a calculator to approximate the following integrals as closely as possible. The exact value of each integral is given.

96. ∫(from 0 to ∞) (sin²x)/x² dx = π/2