Integration Problems to Work on in Preparation for the Technique Test UPDATED October 10, 2002 1. R x sec2 x dx 27. R √ x x2 +1 2. R 4x+4 2x+1 dx 28. R ex e2x −4 dx 3. R 7xex dx 29. R e2x e2x −4 dx 4. R dx x2 +6x+9 30. R ln 3 31. R x+5 x2 −2x−3 32. R x2 +3 x2 +1 33. R x2 ex dx 34. R 3 tan(2x) dx 35. R etan x sec2 x dx 36. R cos x sin(2 sin x) dx 37. R − ln(1+x) (1+x)2 dx 38. R 2 x2 +2x+1 dx 39. R 1 x(x+2) dx 40. R x2 (1−9x2 )(3/2) 41. R 2 5. R 10 6. R9√ 7. R 8. R 9. R 10. R 11. R 12. R 13. R 14. R 15. R cos(x2 ) dx 16. R cos2 x dx 0 dx x √ ex e +4 dx 3 −10 sin(x ) dx 4 + 3x dx √ ex 1 − e2x dx 1 ex sin x dx x ln x dx dx dx 3 2 sin (7x) dx 4x3 +3x2 +2x+1 x4 +x3 +x2 +x+1 dx 6x+3 (x−1)(x−2)(x−3) dx 2 x cos(x ) dx 2 x cos x dx (Use series.) dx 3x cos x dx √ √ (x x2 + 2x + x2 + 2x )dx 17. R x3 cos x2 dx 42. R 18. R cos3 x dx 43. Re 19. Re 44. R sin(ln(x)) dx 45. R 3 x2 (x2 +9) dx 46. R x3 x2 +1 dx 47. R dx x3 +x 1 ln x dx (ln x)2 dx 1 sin(ln(x)) dx x 20. R 21. R 22. R e3 23. R x ln(x2 + 1) dx 48. R sin(x)dx cos2 (x)−5 cos(x)+4 24. R x2 −1 x2 +1 dx 49. R e√ x dx x 25. R x2 +1 x2 −1 dx 50. R√ 26. R x x4 +1 dx 51. R (ln x)2 x e dx 1 x ln x dx √ 1 9 − 4x2 dx x+1 x2 +1 dx 2 2 52. R 2xe(x+1) + 2e(x+1) dx 53. R cos2 (x) − sin2 (x) dx 54. R 55. R x sin((x + 1)(x − 1)) dx 56. R 57. R 58. R π/2 59. Re 60. R sin x cos x dx 61. R dx x ln x ln(ln x) 62. R dx (1−x2 )3/2 √ x dx 1−x2 R 84. R1 √ R x5 −1 x−1 dx 86. R 2x((x2 + 1)2 + 1)2 dx 87. 1 88. 3 √x dx 1−x2 89. x2 ln x dx 90. 91. sin3 (2x) dx R √ 64. ln x dx R 65. xx+1 2 −4 dx R 1 66. (x2 +1) 3/2 dx 63. R 67. R ee 68. R cos(x) 1+sin(x) 69. R x3 +1 x2 −1 70. R 1 x(x2 +a2 ) 71. R1 72. R∞ 73. R x (x2 −a2 )3/2 74. R x(ln x)2 dx 75. R1 76. R 77. R 78. R1 0 √ 1 dx 1−x2 79. R1 x5 ln x dx 80. R2 81. R √ x dx x2 +3 82. R eln x dx x +x 1/2 x2 1 dx dx dx dx √1 1−x2 ln(x) x2 1 0 1+x 13 dx dx dx dx tan2 x dx 1 2 ln 3 0 0 1 1 ex +e−x dx 1 e− x x2 1 − x2 x2 dx 0 85. −π/2 (x−1)dx (x2 +1)(x+1) 83. dx 2 cos(x) ln(sin(x)) dx √ x dx x+1 R √ ( ln x/x) dx R 2 x x e dx R (sin x)3 dx R ln(x2 + 1) dx R ln(ln x) dx x