y-parts formula on fe dy, and explain why it does not e * dx Exercises 6.3 Choose 0 for the constant. (3) Skills Warm-up Exercises (2), we have W In Problems 1-3, find the derivative of f(x) and the indefinite 22. [ in ( =) ax e -* ) + c integral of 8 (x). (If necessary, review Sections 3.2 and 5.1.) 21. In 2x dx Add an arbitrary constant here. 1 . f ( x ) = 5x;8 ( x ) = 13 2. f ( x ) = x2;8(x ) = ex x + C 3 . f ( x ) = x3:8(x) = 5x 4. f ( x ) = e;8(x) = x 23 . 7 + 1 dx 24 . + 5 5 . f ( x ) = ett ; 8 (x ) = 6. f(x) = Vx; 8(x) =e-2x 25. In * dx 7 . f ( x ) = :8 (x ) = efx 26. + 1 re -* - 2xe *+ 2xe * - 2e * + 2e-x 8. f ( x ) = e-2x; 8(x) = Vx A In Problems 9-12, integrate by parts. Assume that x > 0 when- 27 . Vx In x dx 28 . ~dx ever the natural logarithm function is involved. In Problems 29-34, the integral can be found in more than one . / xett dx 10 . xett dx way. First use integration by parts, then use a method that does not involve integration by parts. Which method do you prefer? and interpret the result geometrically. 11. x2 In x dx 12. x3 In x dx 29. ( 2* + 5 ) x dx 30 . ( 4x - 3 ) x dx return to the definite integral. Follow- choose 13. If you want to use integration by parts to find S (x + 1) s (x + 2) dx, which is the better choice for u: u = (x + 1)5 31. (7x - 1) x2 dx 2. ( 9x + 8 ) x dx v = dx or u = x + 2? Explain your choice and then integrate. 14. If you want to use integration by parts to find . ( x + 4 ) (x + 1 ) 2 dx 34. ( 3x - 2 ) ( x - 1 ) 2 dx (5x - 7) (x - 1)#dx, which is the better choice for u: u = 5x - 7 or u = (x - 1)4? Explain your choice and = X In Problems 35-38, illustrate each integral graphically and then integrate. describe what the integral represents in terms of areas. B Problems 15-28 are mixed-some require integration by parts, 35. Problem 19 36. Problem 20 and others can be solved with techniques considered earlier. 37. Problem 21 38. Problem 22 Integrate as indicated, assuming x > 0 whenever the natural logarithm function is involved. C Problems 39-66 are mixed-some may require use of the integration- C by-parts formula along with techniques we have considered earlier; ning of this section. Now we have 15. xe * dx 080.52 - 8 10 16. / (x - 1)e * dx others may require repeated use of the integration-by-parts formula. Assume that g (x) > 0 whenever In g (x) is involved. 17. xet dx 18 . xedx 39 . we'dx 40 . re' dx ( 1 In 1 - 1) 19 . / ( x - 3 ) e dx 20 . ( x + 1) edx 41. xeat dx , a * 0 42 . In ( ax ) dx , a > 0 1 )416 CHAPTER 6 Additional Integration Topics Use an app 43. In x tion ( to the * 2 " dx operation. 73. Profit. In 45 . In ( x + 4 ) dx 46 . In ( 4 - x ) dx and a desc 74. Production graph and 47 . xet - 2 dx 48. / xext dx mon zon-ve 75. Continuo compoun come stre 49. ( x In ( 1 + x 2 ) dox 50. / x In ( 1 + x) dx 76. Continu 51. ( e'In ( 1 + ex ) dx 52. In (1 + Vx) dx Vx 4.15%, C ous inco 53. ( In x ) 2 dx 54 . / x ( In x ) 2 dx 77. Income tration f 55. ( In x ) 3 dx 56. / x( In x) 3 dx 78. Income 57 . ( in ( x 2 ) ax 58. / In ( x4 ) dx tration 60. In (xet) dx 79. Income both a 61. ( ( In x) 4 rin say not 62. ( In x) 5 80. Incom x dx both a NOT ob bonion AsiaWay 63. Sx2In(ex) dx 81. Sales 64. fx'In(er) dx puter 65. fx3In(x2) dx 66. Sx2 In(x3) dx In Problems 67-70, use absolute value on a graphing calculator comp to find the area between the curve and the x axis over the given is the interval. Find answers to two decimal places. plans 67. y = xet; - 25x52 sales 2.00