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1 Page Calculus 2 Review for Test 3 math2414reviewtest3 1. Evaluate: a. b. c. d. e. 2. 2x - 3 dx. 9 + x2 ln(9 + x2) + C 1 x arctan + C 3 3 x ln(9 + x2) - arctan + C 3 ln 3 + x + C None of these 2x - 3 dx. 9 - x2 ln 9 - x2 + C x -2 9 - x2 - 3 arcsin + C 3 x arcsin + C 3 ln 3 - x + C None of these 1 dx. 1 - sin x x + csc x + C tan x + sec x + C ln 1 - sin x + C x + C x + cos x None of these Evaluate: a. b. c. d. e. 3. a. b. c. d. e. Evaluate: Page 2 math2414reviewtest3 4. Evaluate the indefinite integral: a. b. c. d. e. 5. Evaluate the indefinite integral: a. b. 3 (x - 3)1/3(x 4 3 (x - 3)2/3 + 4 3(x - 3)1/3 + 3 (x - 3)4/3 + 4 None of these 3(x + 6)1/3 + 3 (x + 6)2/3 4 3 (x + 6)4/3 4 3 (x + 6)1/3(x 4 None of these C 18(x + 6)1/6 + C 18(x + 6)1/3 + C + 18) + C x dx. (x + 6)2/3 x dx. (x - 3)2/3 - 9) + C 9(x - 3)1/6 + C C 9(x - 3)1/3 + C c. d. e. 6. Evaluate the definite integral: 8 x dx. 6 100 - x2 a. b. c. d. e. -2 6 - 2 2 2 2 - 6 2 None of these Page 3 math2414reviewtest3 7. Evaluate the definite integral: 12 x dx. 9 225 - x2 a. b. c. d. e. 8. 3 2 3 - 3 3 - 2 3 -3 None of these (tan x) ln(cos x) dx. Evaluate the integral: a. b. c. d. e. 9. Evaluate: a. b. c. d. e. 10. ln(cos x) + C 1 [ln(cos x)]2 + C 2 1 sec2 x tan x + C None of these x2 cos x dx. x3 sin x + C 3 x2 sin x + 2x cos x - 2 sin x + C x4 cos + C 4 x2 sin x + x2 cos x + C None of these Evaluate: xe2x dx. 2x(x - 2) + C a. 2e b. 2 2ex + C x c. e2x (2x - 1) + C 4 d. 1 x2e2x + C 2 e. None of these Page 4 math2414reviewtest3 11. Evaluate: a. arcsin t dt. b. c. d. e. 12. a. b. c. d. e. 13. 1 + C 1 - t2 -arccos t + C t arcsin t + 1 - t2 + C t arcsin t - ln 1 - t2 + C None of these ln 3t dt. t ln 3t - t + C 1 +C t 1 + C 3t 3t ln 3t - 3t + C None of these (ln x)3 dx. Evaluate: Evaluate the indefinite integral: a. b. c. d. e. 14. Evaluate: a. b. c. d. e. cos x sin x + cos x + sin x None of x(ln x(ln 1 (ln 3 3 (ln x None x)3 - 3(ln x)2 + C x)3 - 3x(ln x)2 + 6x(ln x) - 6x + C x)4 + C x)2 + C of these x sin x dx. x sin x + C x cos x + C x sin x + C x cos x + C these Page 5 math2414reviewtest3 15. a. b. c. d. e. 16. Evaluate the integral: a. b. c. d. e. 17. Evaluate the integral: a. b. c. d. e. 3 +C x ln x4 + C 3 (ln x)2 + C 2 3x(ln x - 1) + C None of these cos(ln x) dx. Evaluate: x cos 2x 1 1 sin 2x + x 4 2 1 1 cos 2x - x 4 2 1 1 cos 2x + x 4 2 1 1 sin 2x - x 4 2 None of these dx. cos 2x + C sin 2x + C sin 2x + C cos 2x + C ln x3 dx. x [cos(ln x) + sin(ln x)] + C 2 0 x ln x [cos x + sin x] + C 2 sin(ln x) - + C x None of these Page 6 math2414reviewtest3 18. Find the indefinite integral using integration by parts: a. b. x3ex + C 3 x+1 x3e + C 3(x + 1) ex(x2 - 2x + 2) + C xex(x + 2) + C None of these indefinite integral using integration by parts: x dx x3/2(ln x)2 + C ln x7/2 + C 2 x3/2[3 ln x - 2] + C 9 2 + ln x + C 2 x None of these x2exdx. c. d. e. 19. Find the x ln a. b. c. d. e. 20. x Find the indefinite integral using integration by parts: x2e- dx. a. xe-x(2 - x) + C b. x 3e-x + C 3 c. -xe-x(x + 2) + C d. -e-x(x2 + 2x + 2 + C e. None of these Page 7 math2414reviewtest3 21. Evaluate using integration by parts: 3 x ln x dx. 4/3 a. 3x (4 ln x - 3) + C 4 b. 3x4/3 (ln x + 3) + C 16 c. 3x4/3 (4 ln x - 3) + C 16 d. 3x4/3 (4 ln x - 3) + C 8 e. None of these Evaluate: ln 2x dx. a. x(ln 2x - 1) + C b. 2x(2 ln 2x - 1) + C c. 1 +C x d. 1 1 1 - + C 2 x e. None of these x4 ln x dx. 4 4 - x3 + C x 3 1 x5[5 ln x - 1] + C 25 x3 + ln 4x4 + C x3[1 + 4 ln x] + C None of these 22. 23. a. b. Evaluate: c. d. e. Page 8 math2414reviewtest3 24. Evaluate using integration by parts: a. x3 (ln x3 - 1) + C 9 x3 (ln x3 - 3) + C 9 x3 (3 ln x + 1) + C 3 x3 (3 ln x + 1) + C 9 None of these sin2 x cos3 x dx. 1 1 sin3 x - sin5 x + C 3 5 1 sin3 x cos4 x + C 12 1 1 cos4 x - cos6 x + C 4 6 1 1 1 cos2 x sin3 x - sin5 x + C 2 3 5 None of these x2 ln x dx. b. c. d. e. 25. a. b. c. d. e. 26. Evaluate: Evaluate: a. b. c. d. e. sin3 x cos2 x dx. cos3 x cos5 x - + C 3 3 1 sin4 x cos3 x + C 12 1 1 4 x - sin6 x + C sin 4 6 1 1 cos5 x - cos3 x + C 5 3 None of these Page 9 math2414reviewtest3 27. /4 Evaluate: sin2 2 d . /8 a. 4 - 2 12 b. 4 - 2 24 c. -2 16 d. +2 16 e. None of these /4 Evaluate: tan3 x dx. 0 a. 1 - 2 b. 1 2 c. 1 - ln 2 2 d. 1 + ln 2 2 e. None of these sec5 x tan3 x dx. 1 sec7 x - sec5 x + C 5 28. 29. Evaluate: a. b. c. d. e. 1 7 1 sec6 x tan4 x + C 24 1 sec4 x tan x + C 4 1 1 sec8 x - sec6 x + C 8 6 None of these Page 10 math2414reviewtest3 30. a. b. c. d. e. 31. Find the indefinite integral: a. b. c. d. e. 32. Find the indefinite integral: a. b. c. d. e. 33. 6 sin x - csc x + C -6 csc x + tan x + C sin x(6x - cot x) + C 6 cos x + tan x + C None of these 30 6 tan5 x - 10 tan3 x + 30 sec x + 6 tan5 x - 10 tan3 x + 30 tan x + 6 tan5 x - 10 tan3 x + 30 tan x 6 tan5 x + C None of these tan6 x dx. C C 30x + C 6 - cos x + tan x + C 6 cos x + sec x + C 6 csc x + tan x + C 6 - csc x - sec x + C None of these (cos x)(6 + csc2 x) dx. x x csc4 cot dx. 2 2 x x 1 csc5 cot2 + C 10 2 2 1 x - csc4 + C 2 2 1 x 1 x 1 x cot6 + cot4 + cot2 + C 6 2 2 2 2 2 1 x csc4 + C 2 2 None of these (sin x)(-6 + sec2 x) dx. Evaluate: Find the indefinite integral: a. b. c. d. e. Page 11 math2414reviewtest3 34. Evaluate the definite integral: a. b. c. d. e. 35. Evaluate the definite integral: a. b. c. d. e. 36. Evaluate the integral: a. b. c. d. e. 1 5 1 15 0 1 3 None of these cos 2x cos 4x dx. 3 5 1 3 7 15 0 None of these 1 0 1 + sin2 ( x) cos3 ( x) dx. 5 1 0 7 + sin2 ( x) cos3 ( x) dx. 15 1 sin 2x sin 4x + C 8 1 (3 sin 2x + sin 6x) + C 12 1 sin 6x + C 8 1 (3 cos 2x + cos 6x) + C 12 None of these Page 12 math2414reviewtest3 37. Evaluate the indefinite integral: a. b. c. d. e. 38. cos2x - tan x 2 sin x - sec x sin x - tan x tan3x sin x - 3 None of these cos3x - sin2x dx. cos2x +x+C +C +x+C +C Evaluate: a. b. c. d. e. 39. 2 dx. 3 + 2x - x2 arcsin (x - 1) + C 4 3 + 2x - x2 + C x-1 2 arcsin + C 2 ln 3 + 2x - x2 + C None of these 1 dx. 9 + x2 ln sec x + tan x + C ln x2 + 9 + x + C 1 x arctan + C 3 3 1 ln(9 + x2) + C 2x None of these Evaluate: a. b. c. d. e. Page 13 math2414reviewtest3 40. 2x2 dx. x2 + 9 ln(x2 + 9) + C x 2x - 6 arctan + C 3 x arctan + C 3 2x - 2 ln(x2 + 9) + C None of these Evaluate: a. b. c. d. e. 41. a. Evaluate: b. c. d. e. 42. 1 dx. (16 - x2)3/2 x + C 16 16 - x2 -1 + C 2 16 - x2 -3 + C 2(16 - x2)5/2 x 3 arcsin + C 4 None of these dx . (9 - x2)3/2 Find the indefinite integral: a. -2 + C 9 - x2 2 + C 3 9 - x2 x + C 9 - x2 x + C 9 9 - x2 None of these b. c. d. e. Page 14 math2414reviewtest3 43. Find the indefinite integral: a. x + C 81 81 - x2 x + C 81 - x2 2 + C 9 81 - x2 -2 + C 81 - x2 None of these dx . (81 - x2)3/2 b. c. d. e. 44. If a trigonometric substitution in the variable is used to solve 5/ 3 4x2 - 25 determine dx, the lower and upper limits of 5/2 integration for . a. lower limit, 0; upper limit, 3 b. lower limit, 0; upper limit, 6 c. lower limit, ; upper limit, 6 3 d. lower limit, ; upper limit, 4 3 e. None of these Page 15 math2414reviewtest3 45. If a trigonometric substitution in the variable is used to solve 5/2 25 - 4x2 dx, determine the lower and upper limits of 5/4 integration for . a. 5 5 lower limit, arcsin ; upper limit, arcsin 4 2 b. lower limit, ; upper limit, 4 c. lower limit, ; upper limit, 6 2 d. 1 lower limit, ; upper limit, arctan 4 2 e. None of these x3 dx. x2 - 1 46. Evaluate the integral: a. b. c. d. e. 1 tan x + tan3x + C 3 1 x5 + C 5 1 x2 - 1 + (x2 - 1)3/2 + C 3 4x cos + C 4 None of these Page 16 math2414reviewtest3 47. Determine which integral is obtained when the substitution x = sin is made in the integral: 1 1 - x2 dx. 0 a. /2 sin cos d 0 b. /2 cos2 d 0 c. cos d 0 d. cos2 d 0 e. None of these 4 16 - x2 dx -4 48. Evaluate the definite integral: a. b. c. d. e. 49. 4 0 32 3 8 None of these Find the area of the region bounded by the graph of y = 2 x2 + 1, the coordinate axes, and the line x = 1. a. 2 + ln 2 + 1 b. c. d. e. 4 2 - ln 2 - 1 2 + ln 2 None of these Page 17 math2414reviewtest3 50. Find the partial fraction decomposition: a. b. c. d. e. 51. Find the partial fraction decomposition: a. b. c. d. e. 52. Find the partial fraction decomposition: a. b. c. d. e. 1 + x+1 10 x+1 2 x+1 1 x+1 None of 3 (x + 1)2 4 (x + 1)2 1 (x + 1)2 3 (x + 1)2 these 1 - x-2 15 - x-2 1 - x-2 5 + x-2 -1/5 + x-3 1 + x-3 1 x-3 -1/5 + x-3 None of 2 4/5 + x+1 x+1 -4x + 1 x2 + 1 4 1 + x+1 x+1 (6/5)x - (2/5) x2 + 1 these -9 . (x + 1)2(x - 2) 2 1 7 - + x x2 x+1 20 1 11 - - x x2 x+1 9 1 1 + - x x2 x+1 -1 9 + x2 x+1 None of these x2 - 4x + 1 . (x - 3)(x2 + 1) 9x2 + x - 1 . x2(x + 1) Page 18 math2414reviewtest3 53. Find the partial fraction decomposition: a. b. c. d. e. 54. Find the partial fraction decomposition: a. 5x2 19x 28 - - - x3 4x2 4x 2 5x 19x 28 - - - x x+2 (x + 2)2 2 5 16 - + x x+2 (x + 2)2 7 2 5 - + + x x+2 (x + 2)2 None of these 5x2 - 9x + 12 . (x - 2)(x2 + x + 1) 3 4 7 + + x x+3 x2 + 9 3 4x + 7 - x x2 + 9 2 3x + 5 + x x2 + 9 2 3 5 + - x x+3 x2 + 9 None of these -5x2 - 19x - 28 . x3 + 4x2 + 4x -x2 - 7x + 27 . x(x2 + 9) b. c. d. e. 55. Find the partial fraction decomposition: a. b. c. d. e. 5x2 9x + 12 - x-2 x2 + x + 1 4/9 22/9 26/3 - + - x-2 x+1 (x + 1)2 2 3x - 5 + x-2 x2 + x + 1 5 2x - 7 + x-2 x2 + x + 1 None of these Page 19 math2414reviewtest3 56. 2 A B C Let = + + . (x - 1)2(x - 3) x-1 (x - 1)2 x-3 a. 1 1 A = - , B = -1, C = 2 2 b. A = 1, B = 2, C = -1 c. 1 1 A = , B = -1, C = 2 2 d. A = 1, B = 2, C = 3 e. None of these Ax + B C 3x2 - 6x + 9 Let = + . (x2 + 9)(x - 3) x2 + 9 x-3 a. A = 3, B = -6, C = 9 b. A = 2, B = 2, C = 1 c. A = 3, B = 0, C = -1 d. A = 2, B = 0, C = 1 e. None of these Evaluate: a. b. c. d. e. 59. Evaluate the integral: a. Find A, B, and C. 57. Find A, B, and C. 58. 2 dx. (x + 2)2(2 - x) 1 1 1 ln x + 2 + ln(x + 2)2 - ln 2 - x + C 8 2 8 1 x + 2 1 ln - + C 8 2 - x 2(x + 2) 2 - + ln 2 - x + C (x + 2) 1 1 - - ln 2 - x + C 2(x + 2) 8 None of these x+1 dx. x2 - 5x + 6 b. c. d. e. -x2 - 2x + 11 + C (x2 - 5x + 6)2 (x - 3)4 ln + C (x - 2)3 ln (x - 3)4(x - 2)3 + C ln (x - 3)3(x - 2)4 + C None of these Page 20 math2414reviewtest3 60. Evaluate the integral: a. b. c. sin2 x sin3 x - 2 3 ln sin x - sin2 sin x ln + 1 - sin x csc2 x csc3 x - 2 2 None of these cos x dx. sin x(1 - sin x) +C x + C C +C d. e. 61. 2 ex - 1 Find lim . x 0 2x2 a. 0 b. 1 2 c. d. Does not exist e. None of these x2 Find lim . x ln x a. 0 b. 1 c. d. Does not exist e. None of these Evaluate the limit using L'H pital's Rule, if applicable. 8ex/8 - (8 + x) lim x 0 x2 a. 1 16 b. c. 1 2 d. 0 e. None of these 62. 63. Page 21 math2414reviewtest3 64. Evaluate the limit using L'H pital's Rule, if applicable. 3ex/3 - (3 + x) lim x 0 x2 a. 0 b. c. 1 6 d. 1 2 e. None of these Evaluate the limit using L'H pital's Rule, if applicable. 7ex/7 - (7 + x) lim x 0 x2 a. 1 14 b. 0 c. 1 2 d. e. None of these Evaluate the limit using L'H pital's Rule, if applicable. sin x - x lim x 0 x3 a. 1 6 b. 0 c. 1 - 6 d. e. None of these Evaluate the limit using L'H pital's Rule, if applicable. 1 1 lim - x 0 x x2 a. 0 b. 1 2 c. -1 d. The limit does not exist. e. None of these 65. 66. 67. Page 22 math2414reviewtest3 68. Evaluate the limit using L'H pital's Rule, if applicable. lim e-x ln x x a. 0 b. c. 1 d. The limit does not exist. e. None of these Evaluate the limit using L'H pital's Rule, if applicable. sin x lim x 0 x a. 0 b. 1 c. d. The limit does not exist. e. None of these Find the limit: 71. ln x lim . x x3 69. 70. 1 Calculate the area under the curve y = above the x-axis on the x2 interval [1, ]. a. b. 1 c. 0 d. 2 e. None of these e-x dx. 1 72. Evaluate the integral or determine that it diverges: a. b. c. d. e. 1 e 0 1 - e None of these Page 23 math2414reviewtest3 73. Evaluate the improper integral: a. b. c. d. e. 74. Evaluate the improper integral: a. b. c. d. e. 75. 2 0 -2 The integral diverges. None of these The integral diverges. 2 0 -2 None of these -7 dx . -9 (x + 8)2 4 dx . 2 (x - 3)2 Identify the improper integral: a. 1 x dx 0 3 - 2x b. 2 x dx 0 3 - 2x c. 4 x dx 2 3 - 2x d. 1 1 dx 0 3 - 2x e. None of these Evaluate the improper integral: xe-x/2dx. 0 a. Diverges to - b. Diverges to c. Converges to -2 d. Converges to 4 e. None of these Evaluate the improper integral: ln(x) dx. 0 a. Diverges to - b. Diverges to c. Converges to 0 d. Converges to 2 e. None of these 76. 77. Page 24 math2414reviewtest3 78. Determine the divergence or convergence of the following improper integral and evaluate it if it converges. 1 dx 2 (x - 1)2 a. Diverges b. Converges to 1 c. Converges to -1 d. Converges to 0 e. None of these (ln x)2 Evaluate the improper integral: dx. 0 x a. Diverges to - b. Diverges to c. 2 Converges to 3 d. Converges to -3 e. None of these 1 Evaluate the improper integral: dx. 0 e2x a. Diverges to - b. Diverges to c. Converges to 0 d. 1 Converges to 2 e. None of these 79. 80. Page 1 Calculus 2 Review for Test 3 math2414reviewtest3 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50. 51. 52. c b b c d d a b b c c a b d c d a c c d c a b a a d d c a b b a c c a b c c b b a d a b c c b d a a d a Page 2 math2414reviewtest3 53. 54. 55. 56. 57. 58. 59. 60. 61. 62. 63. 64. 65. 66. 67. 68. 69. 70. 71. 72. 73. 74. 75. 76. 77. 78. 79. 80. b d c a d b b c b c a c a c d a b 0 b b a d b d b b b d

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