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752 CHAPTER 12 Indefinite Integrals ✓ CHECKPOINT ANSWERS Expressions (a) and (b) can be evaluated with the Power Rule Expressions (c) and (d) not fit the format of the Power Rule (c) (2x ϩ 5)Ϫ2(6x 2) dx ϭ Ϫ(2x ϩ 5)Ϫ1 ϩ C True (x ϩ 9)6 ϩC (a) (x ϩ 9)5(3x dx) ϭ (x ϩ 9)16 (b) 3(x ϩ 9)15(x dx) ϭ ϩC 48 x 18x 81x (c) 3(x ϩ 9)2(x dx) ϭ 3(x ϩ 18x ϩ 81)x dx ϭ ϩ ϩ ϩC | EXERCISES | 12.2 In Problems and 2, find du u ϭ 2x ϩ u ϭ 3x Ϫ 4x In each of Problems and 4, one of parts (a) and (b) can be integrated with the Power Rule and the other cannot Integrate the part that can be done with the Power Rule, and explain why the Power Rule cannot be used to evaluate the other (a) (3x Ϫ 7)12 (12x dx) (b) (5x ϩ 11)7 (15x dx) (b) 14x 2(2x ϩ 9)6 dx (a) 5(6 ϩ 5x)10 dx Evaluate the integrals in Problems 5–34 Check your results by differentiation (x ϩ 3)3 2x dx (3x ϩ 1)4 9x dx (8x ϩ5)3 (32x 3) dx (5x ϩ 11) 15x dx (3x Ϫ x 3)2 (3 Ϫ 3x 2) dx 10 (4x Ϫ 3x)4 (8x Ϫ 3) dx 11 4x 3(7x ϩ 12)3 dx 12 9x 5(3x Ϫ 4)6 dx 13 15 7(4x Ϫ 1)6 dx 8x 5(4x ϩ 15)Ϫ3 dx 17 19 20 (x Ϫ 1)(x Ϫ 2x ϩ 5)4 dx (2x Ϫ x)(x Ϫ x 2)6 dx 2(x Ϫ 1)(x Ϫ 4x ϩ 3)Ϫ5 dx 3(x Ϫ 2x)(x Ϫ 6x ϩ 7)Ϫ2 dx 21 23 25 7x x ϩ dx (x ϩ 1)2 (3x dx) (3x Ϫ 1)2 12x dx 18 27 28 29 31 33 35 14 16 22 24 26 3(5 Ϫ x)Ϫ3 dx 5x 3(3x ϩ 7)Ϫ4 dx In Problems 37 and 38, (a) evaluate each integral and (b) graph the members of the solution family for C ؍؊5, C ؍0, and C ؍5 37 x(x Ϫ 1)3 dx 38 (3x Ϫ 11)1 dx Each of Problems 39 and 40 has the form f (x) dx (a) Evaluate each integral to obtain a family of functions (b) Find and graph the family member that passes through the point (0, 2) Call that function F(x) (c) Find any x-values where f(x) is not defined but F(x) is (d) At the x-values found in part (c), what kind of tangent line does F(x) have? dx x dx 40 3 39 3 (2x Ϫ 1) (x Ϫ 1)1 In each of Problems 41 and 42, a family of functions is given, together with the graphs of some functions in the family Write the indefinite integral that gives the family 41 F(x) ϭ (x Ϫ 1)4 ϩ C F(x) (x 1) C y 3x Ϫ x dx (x Ϫ 5)2 (2x dx) (2x ϩ 3)2 (8x dx) x Ϫ 3x (x Ϫ 1) dx x ϩ 2x (x ϩ 1) dx 3x dx 5x dx 30 (2x Ϫ 5) (x Ϫ 8)3 x3 Ϫ 3x Ϫ 2x dx 32 dx (x Ϫ 4x) (x Ϫ x 4)5 x Ϫ 4x x2 ϩ dx 34 dx 3 x Ϫ 6x ϩ x ϩ 3x ϩ 10 If f (x) dx ϭ (7x Ϫ 13)10 ϩ C, find f(x) x 42 F(x) ϭ 54(4x ϩ 9)Ϫ1 ϩ C F(x) 54(4x 9) C y x 36 If g(x) dx ϭ (5x ϩ 2)6 ϩ C, find g(x) Copyright 2016 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it SECTION 12.2 In parts (a)–(c) of Problems 43 and 44, three integrals are given Integrate those that can be done by the methods studied so far Additionally, as part (d), give your own example of an integral that looks as though it might use the Power Rule but that cannot be integrated by using methods studied so far 7x dx 7x dx 43 (a) 3 (b) 3 (x ϩ 4) (x ϩ 4)2 (c) x2 ϩ dx x (2x ϩ 1)7 dx 44 (a) 3x (c) (b) 10x(2x ϩ 1)7 dx 5x dx (2x ϩ 1)7 APPLICATIONS 45 Revenue Suppose that the marginal revenue for a product is given by Ϫ30 MR ϭ ϩ 30 (2x ϩ 1)2 where x is the number of units and revenue is in dollars Find the total revenue 46 Revenue The marginal revenue for a new calculator is given by MR ϭ 60,000 Ϫ 40,000 (10 ϩ x)2 where x represents hundreds of calculators and revenue is in dollars Find the total revenue function for these calculators 47 Physical productivity The total physical output of a number of machines or workers is called physical productivity and is a function of the number of machines or workers If P ϭ f (x) is the productivity, dP dx is the marginal physical productivity If the marginal physical productivity for bricklayers is dP dx ϭ 90(x ϩ 1)2, where P is the number of bricks laid per day and x is the number of bricklayers, find the physical productivity of bricklayers (Note: P ϭ when x ϭ 0.) 48 Production The rate of production of a new line of products is given by dx 400 ϭ 200 ϩ dt (t ϩ 40)2 where x is the number of items and t is the number of weeks the product has been in production (a) Assuming that x ϭ when t ϭ 0, find the total number of items produced as a function of time t (b) How many items were produced in the fifth week? 49 Data entry speed The rate of change in data entry speed of the average student is ds dx ϭ 5(x ϩ 1)Ϫ1 2, where x is the number of lessons the student has had and s is in entries per minute The Power Rule 753 (a) Find the data entry speed as a function of the number of lessons if the average student can complete 10 entries per minute with no lessons (x ϭ 0) (b) How many entries per minute can the average student complete after 24 lessons? 50 Productivity Because a new employee must learn an assigned task, production will increase with time Suppose that for the average new employee, the rate of performance is given by dN ϭ dt tϩ1 where N is the number of units completed t hours after beginning a new task If units are completed after 3 hours, how many units are completed after hours? 51 Film attendance An excellent film with a very small advertising budget must depend largely on wordof-mouth advertising In this case, the rate at which weekly attendance might grow can be given by dA Ϫ100 2000 ϭ ϩ dt (t ϩ 10)2 (t ϩ 10)3 where t is the time in weeks since release and A is attendance in millions (a) Find the function that describes weekly attendance at this film (b) Find the attendance at this film in the tenth week 52 Product quality and advertising An inferior product with a large advertising budget does well when it is introduced, but sales decline as people discontinue use of the product Suppose that the rate of weekly sales revenue is given by SЈ(t) ϭ 400 200 Ϫ (t ϩ 1) (t ϩ 1)2 where S is sales in thousands of dollars and t is time in weeks (a) Find the function that describes the weekly sales (b) Find the sales for the first week and the ninth week 53 Demographics Because of job outsourcing, a Pennsylvania town predicts that its public school population will decrease at the rate dN Ϫ300 ϭ dx xϩ9 where x is the number of years and N is the total school population If the present population (x ϭ 0) is 8000, what population size is expected in years? 54 Franchise growth A new fast-food firm predicts that the number of franchises for its products will grow at the rate dn ϭ9 tϩ1 dt where t is the number of years, Յ t Յ 10 If there is one franchise (n ϭ 1) at present (t ϭ 0), how many franchises are predicted for years from now? Copyright 2016 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it ... attendance An excellent film with a very small advertising budget must depend largely on wordof-mouth advertising In this case, the rate at which weekly attendance might grow can be given by... population (x ϭ 0) is 8000, what population size is expected in years? 54 Franchise growth A new fast-food firm predicts that the number of franchises for its products will grow at the rate dn ϭ9 tϩ1
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