Explanation Given Information: The provided integral is, ∫ − ∞ 0 x ( x 2 + 1 ) 2 d x Formula used: According to the power rule of integrals, ∫ x n d x = x n + 1 n + 1 + C If the limit defining the improper integral is a unique finite number, then integral converges else it diverges, ∫ − ∞ b f ( x ) d x = lim a → − ∞ ∫ a b f ( x ) d x Calculation: Consider the provided integral, ∫ − ∞ 0 x ( x 2 + 1 ) 2 d x Now, use the formula, ∫ − ∞ b f ( x ) d x = lim a → − ∞ ∫ a b f ( x ) d x To rewrite the integral by multiplying and dividing it by 2 as, ∫ − ∞ 0 x ( x 2 + 1 ) 2 d x = 1 2 lim a → − ∞ ∫ a 0 2 x ( x 2 + 1 ) 2 d x Now, let x 2 + 1 = t , then on obtaining differentials, 2 x d x = d t Thus, the integral becomes, ∫ − ∞ 0 x ( x 2 + 1 ) 2 d x = 1 2 lim a → − ∞ ∫ a 0 2 x ( x 2 + 1 ) 2 d x = 1 2 lim a → − ∞ ∫ a 0 d t t 2

Mathematical Applications for the Management, Life, and Social Sciences

11th Edition

ISBN: 9781305108042

Author: Ronald J. Harshbarger, James J. Reynolds

Publisher: Cengage Learning

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Chapter 13, Problem 34RE

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To calculate: The value of the improper integral ∫−∞0x(x2+1)2dx if it converges.

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Chapter 13, Problem 33RE

Chapter 13, Problem 35RE

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Chapter 13 Solutions

Mathematical Applications for the Management, Life, and Social Sciences

Ch. 13.1 - Prob. 1CP Ch. 13.1 - Prob. 2CP Ch. 13.1 - Prob. 1E Ch. 13.1 - Prob. 2E Ch. 13.1 - Prob. 3E Ch. 13.1 - Prob. 4E Ch. 13.1 - Prob. 5E Ch. 13.1 - Prob. 6E Ch. 13.1 - Prob. 7E Ch. 13.1 - Prob. 8E

Ch. 13.1 - Prob. 9E Ch. 13.1 - Prob. 10E Ch. 13.1 - Prob. 11E Ch. 13.1 - Prob. 12E Ch. 13.1 - Prob. 13E Ch. 13.1 - Prob. 14E Ch. 13.1 - In Problems 14-19, find the value of each sum. 15....Ch. 13.1 - Prob. 16E Ch. 13.1 - Prob. 17E Ch. 13.1 - Prob. 18E Ch. 13.1 - Prob. 19E Ch. 13.1 - Prob. 20E Ch. 13.1 - Prob. 21E Ch. 13.1 - Prob. 22E Ch. 13.1 - Prob. 23E Ch. 13.1 - Prob. 24E Ch. 13.1 - Prob. 25E Ch. 13.1 - Prob. 26E Ch. 13.1 - Prob. 27E Ch. 13.1 - Prob. 28E Ch. 13.1 - Prob. 29E Ch. 13.1 - Prob. 30E Ch. 13.1 - Prob. 31E Ch. 13.1 - Prob. 32E Ch. 13.1 - Prob. 33E Ch. 13.1 - Prob. 34E Ch. 13.1 - Prob. 35E Ch. 13.1 - 36. Speed trials The figure gives the time it...Ch. 13.1 - Prob. 37E Ch. 13.1 - 38. Drug levels in the blood The manufacturer of a...Ch. 13.1 - Prob. 39E Ch. 13.1 - Prob. 40E Ch. 13.2 - CHECKPOINT 1. True or false: (a) For any integral,...Ch. 13.2 - Prob. 2CP Ch. 13.2 - Evaluate the definite integrals in Problems...Ch. 13.2 - Prob. 2E Ch. 13.2 - Prob. 3E Ch. 13.2 - Prob. 4E Ch. 13.2 - Prob. 5E Ch. 13.2 - Prob. 6E Ch. 13.2 - Prob. 9E Ch. 13.2 - Prob. 10E Ch. 13.2 - Prob. 11E Ch. 13.2 - Prob. 12E Ch. 13.2 - Evaluate the definite integrals in Problems...Ch. 13.2 - Prob. 8E Ch. 13.2 - Prob. 23E Ch. 13.2 - Evaluate the definite integrals in Problems...Ch. 13.2 - Evaluate the definite integrals in Problems...Ch. 13.2 - Prob. 14E Ch. 13.2 - Prob. 15E Ch. 13.2 - Prob. 16E Ch. 13.2 - Prob. 17E Ch. 13.2 - Prob. 18E Ch. 13.2 - Prob. 19E Ch. 13.2 - Prob. 20E Ch. 13.2 - Evaluate the definite integrals in Problems...Ch. 13.2 - Prob. 22E Ch. 13.2 - Evaluate the definite integrals in Problems...Ch. 13.2 - Prob. 26E Ch. 13.2 - Prob. 27E Ch. 13.2 - Prob. 28E Ch. 13.2 - Prob. 29E Ch. 13.2 - Evaluate the definite integrals in Problems...Ch. 13.2 - Prob. 31E Ch. 13.2 - Prob. 32E Ch. 13.2 - Prob. 33E Ch. 13.2 - Prob. 34E Ch. 13.2 - Prob. 35E Ch. 13.2 - Prob. 36E Ch. 13.2 - In Problems 39-42, (a) write the integral that...Ch. 13.2 - Prob. 38E Ch. 13.2 - In Problems 39-42, (a) write the integral that...Ch. 13.2 - In Problems 39-42, (a) write the integral that...Ch. 13.2 - Prob. 41E Ch. 13.2 - 44. Find the area between the curve and the...Ch. 13.2 - Prob. 43E Ch. 13.2 - 46. Find the area between the curve and the...Ch. 13.2 - Prob. 45E Ch. 13.2 - Prob. 46E Ch. 13.2 - Prob. 47E Ch. 13.2 - Prob. 48E Ch. 13.2 - Prob. 49E Ch. 13.2 - Prob. 50E Ch. 13.2 - Prob. 51E Ch. 13.2 - Prob. 52E Ch. 13.2 - 55. Depreciation The rate of depreciation of a...Ch. 13.2 - Depreciation The rate of depreciation of a...Ch. 13.2 - 57. Sales and advertising A store finds that its...Ch. 13.2 - Prob. 56E Ch. 13.2 - 59. Total income The income from an oil change...Ch. 13.2 - 60. Total income Suppose that a vending machine...Ch. 13.2 - Prob. 59E Ch. 13.2 - Prob. 60E Ch. 13.2 - Prob. 61E Ch. 13.2 - Prob. 62E Ch. 13.2 - Prob. 63E Ch. 13.2 - Prob. 64E Ch. 13.2 - 67. Testing The time t (in minutes) needed to read...Ch. 13.2 - 68. Response time In a small city, the response...Ch. 13.2 - Customer service The duration t (in minutes) of...Ch. 13.2 - 70. Product life The useful life of a car battery...Ch. 13.3 - CHECKPOINT 1. True or false: (a) Over the interval...Ch. 13.3 - Prob. 2CP Ch. 13.3 - Prob. 3CP Ch. 13.3 - For each shaded region in Problems 1-6, (a) form...Ch. 13.3 - For each shaded region in Problems 1-6, (a) form...Ch. 13.3 - For each shaded region in Problems 1-6, (a) form...Ch. 13.3 - For each shaded region in Problems 1-6, (a) form...Ch. 13.3 - Prob. 5E Ch. 13.3 - For each shaded region in Problems 1-6, (a) form...Ch. 13.3 - For each shaded region in Problems 7-12, (a) find...Ch. 13.3 - For each shaded region in Problems 7-12, (a) find...Ch. 13.3 - Prob. 9E Ch. 13.3 - Prob. 10E Ch. 13.3 - For each shaded region in Problems 7-12, (a) find...Ch. 13.3 - Prob. 12E Ch. 13.3 - In Problems 13-26, equations are given whose...Ch. 13.3 - Prob. 14E Ch. 13.3 - In Problems 13-26, equations are given whose...Ch. 13.3 - Prob. 16E Ch. 13.3 - Prob. 17E Ch. 13.3 - Prob. 18E Ch. 13.3 - Prob. 19E Ch. 13.3 - Prob. 20E Ch. 13.3 - Prob. 21E Ch. 13.3 - Prob. 22E Ch. 13.3 - Prob. 23E Ch. 13.3 - In Problems 13-26, equations are given whose...Ch. 13.3 - Prob. 25E Ch. 13.3 - Prob. 26E Ch. 13.3 - Prob. 27E Ch. 13.3 - Prob. 28E Ch. 13.3 - Prob. 29E Ch. 13.3 - In Problems 27-32, find the average value of each...Ch. 13.3 - Prob. 31E Ch. 13.3 - Prob. 32E Ch. 13.3 - Prob. 33E Ch. 13.3 - Prob. 34E Ch. 13.3 - Prob. 35E Ch. 13.3 - 36. Sales and advertising The figure shows the...Ch. 13.3 - Prob. 37E Ch. 13.3 - 38. Inventory management The figure shows how an...Ch. 13.3 - 39. Sales and advertising The number of daily...Ch. 13.3 - 40. Demand The demand function for a certain...Ch. 13.3 - 41. Social Security beneficiaries With data from...Ch. 13.3 - Prob. 42E Ch. 13.3 - Prob. 43E Ch. 13.3 - Prob. 44E Ch. 13.3 - Prob. 45E Ch. 13.3 - 46. Income distribution In an effort to make the...Ch. 13.3 - Prob. 47E Ch. 13.3 - Prob. 48E Ch. 13.3 - Prob. 49E Ch. 13.3 - Prob. 50E Ch. 13.4 - CHECKPOINT 1. Suppose that a continuous income...Ch. 13.4 - Prob. 2CP Ch. 13.4 - CHECKPOINT 3. If demand is , supply is , and the...Ch. 13.4 - 1. Find the total income over the next 10 years...Ch. 13.4 - 2. Find the total income over the next 8 years...Ch. 13.4 - 3. Suppose that a steel company views the...Ch. 13.4 - 4. Suppose that the Quick-Fix Car Service...Ch. 13.4 - 5. A small brewery considers the output of its...Ch. 13.4 - 6. A company that services a number of vending...Ch. 13.4 - 7. A franchise models the profit from its store as...Ch. 13.4 - 8. The Medi Spa franchise has a continuous income...Ch. 13.4 - 9. A continuous income stream has an annual rate...Ch. 13.4 - 10. A continuous income stream has an annual rate...Ch. 13.4 - 11. The income from an established chain of...Ch. 13.4 - Prob. 12E Ch. 13.4 - 13. Suppose that a printing firm considers its...Ch. 13.4 - 14. Suppose that a vending machine company is...Ch. 13.4 - 15. A 58-year-old couple are considering opening a...Ch. 13.4 - Prob. 16E Ch. 13.4 - Prob. 17E Ch. 13.4 - Prob. 18E Ch. 13.4 - In Problems 17-26, p and C are in dollars and x is...Ch. 13.4 - Prob. 20E Ch. 13.4 - Prob. 21E Ch. 13.4 - Prob. 22E Ch. 13.4 - In Problems 17-26, p and C are in dollars and x is...Ch. 13.4 - Prob. 24E Ch. 13.4 - In Problems 17-26, p and C are in dollars and x is...Ch. 13.4 - Prob. 26E Ch. 13.4 - 27. Suppose that the supply function for a good...Ch. 13.4 - Prob. 28E Ch. 13.4 - 29. If the supply function for a commodity is ,...Ch. 13.4 - Prob. 30E Ch. 13.4 - Find the producers surplus at market equilibrium...Ch. 13.4 - 32. Find the producer's surplus at market...Ch. 13.4 - Prob. 33E Ch. 13.4 - Prob. 34E Ch. 13.4 - 35. The demand function for a certain product is ...Ch. 13.4 - Prob. 36E Ch. 13.5 - Prob. 1CP Ch. 13.5 - Prob. 2CP Ch. 13.5 - Prob. 3CP Ch. 13.5 - Prob. 4CP Ch. 13.5 - Prob. 5CP Ch. 13.5 - Prob. 6CP Ch. 13.5 - Prob. 1E Ch. 13.5 - Prob. 2E Ch. 13.5 - Prob. 3E Ch. 13.5 - Prob. 4E Ch. 13.5 - Prob. 5E Ch. 13.5 - Prob. 6E Ch. 13.5 - Prob. 7E Ch. 13.5 - Prob. 8E Ch. 13.5 - Prob. 9E Ch. 13.5 - Prob. 10E Ch. 13.5 - Prob. 11E Ch. 13.5 - Prob. 12E Ch. 13.5 - Prob. 13E Ch. 13.5 - Prob. 14E Ch. 13.5 - Prob. 15E Ch. 13.5 - Prob. 16E Ch. 13.5 - Prob. 17E Ch. 13.5 - Prob. 18E Ch. 13.5 - Evaluate the integrals in Problems 1-32. Identify...Ch. 13.5 - Prob. 20E Ch. 13.5 - Prob. 21E Ch. 13.5 - Prob. 22E Ch. 13.5 - Prob. 23E Ch. 13.5 - Prob. 24E Ch. 13.5 - Prob. 25E Ch. 13.5 - Prob. 26E Ch. 13.5 - Prob. 27E Ch. 13.5 - Prob. 28E Ch. 13.5 - Prob. 29E Ch. 13.5 - Prob. 30E Ch. 13.5 - Prob. 31E Ch. 13.5 - Prob. 32E Ch. 13.5 - Prob. 33E Ch. 13.5 - Prob. 34E Ch. 13.5 - Prob. 35E Ch. 13.5 - Prob. 36E Ch. 13.5 - Prob. 37E Ch. 13.5 - Prob. 38E Ch. 13.5 - Prob. 39E Ch. 13.5 - Prob. 40E Ch. 13.5 - 41. Income streams Suppose that when a new oil...Ch. 13.5 - Prob. 42E Ch. 13.6 - Prob. 1CP Ch. 13.6 - Prob. 2CP Ch. 13.6 - Prob. 1E Ch. 13.6 - Prob. 2E Ch. 13.6 - Prob. 3E Ch. 13.6 - Prob. 4E Ch. 13.6 - Prob. 5E Ch. 13.6 - Prob. 6E Ch. 13.6 - Prob. 7E Ch. 13.6 - Prob. 8E Ch. 13.6 - Prob. 9E Ch. 13.6 - Prob. 10E Ch. 13.6 - Prob. 11E Ch. 13.6 - In Problems 1-16, use integration by parts to...Ch. 13.6 - Prob. 13E Ch. 13.6 - Prob. 14E Ch. 13.6 - Prob. 15E Ch. 13.6 - Prob. 16E Ch. 13.6 - Prob. 17E Ch. 13.6 - Prob. 18E Ch. 13.6 - Prob. 19E Ch. 13.6 - Prob. 20E Ch. 13.6 - Prob. 21E Ch. 13.6 - Prob. 22E Ch. 13.6 - Prob. 23E Ch. 13.6 - Prob. 24E Ch. 13.6 - Prob. 25E Ch. 13.6 - Prob. 26E Ch. 13.6 - Prob. 27E Ch. 13.6 - Prob. 28E Ch. 13.6 - Prob. 29E Ch. 13.6 - Prob. 30E Ch. 13.6 - Prob. 31E Ch. 13.6 - Prob. 32E Ch. 13.6 - Prob. 33E Ch. 13.6 - Prob. 34E Ch. 13.6 - Prob. 35E Ch. 13.6 - Prob. 36E Ch. 13.6 - Prob. 37E Ch. 13.6 - Prob. 38E Ch. 13.7 - Prob. 1CP Ch. 13.7 - Prob. 2CP Ch. 13.7 - In Problems 1-20, evaluate the improper integrals...Ch. 13.7 - Prob. 2E Ch. 13.7 - In Problems 1-20, evaluate the improper integrals...Ch. 13.7 - Prob. 4E Ch. 13.7 - Prob. 5E Ch. 13.7 - Prob. 6E Ch. 13.7 - Prob. 7E Ch. 13.7 - Prob. 8E Ch. 13.7 - Prob. 9E Ch. 13.7 - Prob. 10E Ch. 13.7 - Prob. 11E Ch. 13.7 - Prob. 12E Ch. 13.7 - Prob. 13E Ch. 13.7 - Prob. 14E Ch. 13.7 - Prob. 15E Ch. 13.7 - Prob. 16E Ch. 13.7 - Prob. 17E Ch. 13.7 - Prob. 18E Ch. 13.7 - Prob. 19E Ch. 13.7 - Prob. 20E Ch. 13.7 - Prob. 21E Ch. 13.7 - 22. For what value of c does Ch. 13.7 - Prob. 23E Ch. 13.7 - Prob. 24E Ch. 13.7 - Prob. 25E Ch. 13.7 - Prob. 26E Ch. 13.7 - Prob. 27E Ch. 13.7 - Prob. 28E Ch. 13.7 - Prob. 29E Ch. 13.7 - Prob. 30E Ch. 13.7 - 31. Find the value of c so that is a probability...Ch. 13.7 - Prob. 32E Ch. 13.7 - Prob. 33E Ch. 13.7 - Prob. 34E Ch. 13.7 - 35. Find the area below the graph of and above...Ch. 13.7 - Prob. 36E Ch. 13.7 - Prob. 37E Ch. 13.7 - 38. Capital value Suppose that a donor wants to...Ch. 13.7 - 39. Capital value Suppose that a business provides...Ch. 13.7 - Prob. 40E Ch. 13.7 - Prob. 41E Ch. 13.7 - 42. Capital value Suppose that a business provides...Ch. 13.7 - 43. Repair time In a manufacturing process...Ch. 13.7 - 44. Customer service The duration t (in minutes)...Ch. 13.7 - Prob. 45E Ch. 13.7 - Prob. 47E Ch. 13.8 - CHECKPOINT 1. Suppose [1, 4] is divided into 6...Ch. 13.8 - Prob. 2CP Ch. 13.8 - Prob. 3CP Ch. 13.8 - Prob. 1E Ch. 13.8 - Prob. 2E Ch. 13.8 - Prob. 3E Ch. 13.8 - Prob. 4E Ch. 13.8 - Prob. 5E Ch. 13.8 - Prob. 6E Ch. 13.8 - Prob. 7E Ch. 13.8 - Prob. 8E Ch. 13.8 - Prob. 9E Ch. 13.8 - Prob. 10E Ch. 13.8 - Prob. 11E Ch. 13.8 - Prob. 12E Ch. 13.8 - Prob. 13E Ch. 13.8 - Prob. 14E Ch. 13.8 - Prob. 15E Ch. 13.8 - Prob. 16E Ch. 13.8 - Prob. 17E Ch. 13.8 - Prob. 18E Ch. 13.8 - Prob. 19E Ch. 13.8 - Prob. 20E Ch. 13.8 - Prob. 21E Ch. 13.8 - Prob. 22E Ch. 13.8 - In Problems 23-30, round all calculations to 2...Ch. 13.8 - In Problems 23-30, round all calculations to 2...Ch. 13.8 - Prob. 25E Ch. 13.8 - Prob. 26E Ch. 13.8 - In Problems 23-30, round all calculations to 2...Ch. 13.8 - In Problems 23-30, round all calculations to 2...Ch. 13.8 - Prob. 29E Ch. 13.8 - Prob. 30E Ch. 13.8 - Prob. 31E Ch. 13.8 - Prob. 32E Ch. 13.8 - Pollution monitoring Suppose that the presence of...Ch. 13.8 - Prob. 34E Ch. 13 - 1. Calculate . Ch. 13 - 2. Use formulas to simplify Ch. 13 - Prob. 3RE Ch. 13 - Prob. 4RE Ch. 13 - 5. Use a definite integral to find the area under...Ch. 13 - Prob. 6RE Ch. 13 - Prob. 7RE Ch. 13 - Prob. 8RE Ch. 13 - Prob. 9RE Ch. 13 - Prob. 10RE Ch. 13 - Prob. 11RE Ch. 13 - Prob. 12RE Ch. 13 - Prob. 13RE Ch. 13 - Prob. 14RE Ch. 13 - Prob. 15RE Ch. 13 - Prob. 16RE Ch. 13 - Prob. 17RE Ch. 13 - Prob. 18RE Ch. 13 - Prob. 19RE Ch. 13 - Prob. 20RE Ch. 13 - Prob. 21RE Ch. 13 - Prob. 22RE Ch. 13 - Prob. 23RE Ch. 13 - Prob. 24RE Ch. 13 - Prob. 25RE Ch. 13 - Prob. 26RE Ch. 13 - Prob. 27RE Ch. 13 - Prob. 28RE Ch. 13 - Prob. 29RE Ch. 13 - In Problems 27-30, use integration by parts to...Ch. 13 - Prob. 31RE Ch. 13 - Prob. 32RE Ch. 13 - Prob. 33RE Ch. 13 - Prob. 34RE Ch. 13 - Prob. 35RE Ch. 13 - Prob. 36RE Ch. 13 - 38. Use the table that follows to...Ch. 13 - 39. Suppose that a definite integral is to be...Ch. 13 - 40. Maintenance Maintenance costs for buildings...Ch. 13 - Prob. 40RE Ch. 13 - Prob. 41RE Ch. 13 - Prob. 42RE Ch. 13 - Prob. 43RE Ch. 13 - Prob. 44RE Ch. 13 - Prob. 45RE Ch. 13 - Prob. 46RE Ch. 13 - 46. Income streams Suppose that a machine’s...Ch. 13 - Prob. 48RE Ch. 13 - Prob. 49RE Ch. 13 - Prob. 50RE Ch. 13 - Prob. 51RE Ch. 13 - 54. Capital value Find the capital value of a...Ch. 13 - 55. Total income Suppose that a continuous income...Ch. 13 - Prob. 55RE Ch. 13 - 1. Use left-hand endpoints and subdivisions to...Ch. 13 - 2. Consider from with n equal subdivisions. (a)...Ch. 13 - Prob. 3T Ch. 13 - Prob. 4T Ch. 13 - Prob. 5T Ch. 13 - Prob. 6T Ch. 13 - Prob. 7T Ch. 13 - Prob. 8T Ch. 13 - 9. Suppose the supply function for a product is ...Ch. 13 - 10. Suppose a continuous income stream has an...Ch. 13 - Prob. 11T Ch. 13 - Prob. 12T Ch. 13 - Prob. 13T Ch. 13 - Prob. 14T Ch. 13 - Prob. 15T Ch. 13 - 16. The environmental effects of a chemical spill...

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The value of the improper integral ∫ − ∞ 0 x ( x 2 + 1 ) 2 d x if it converges.

Solution Summary: The author calculates the value of the provided integral if it converges. If the limit defining the improper integral is a unique finite number, it diverges.

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To calculate: The value of the improper integral ∫−∞0x(x2+1)2dx if it converges.

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Chapter 13 Solutions