Extension 2 Mathematics: Coroneos - 100 Integrals

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NEW SOUTH WALES

HIGHER SCHOOL CERTIFICATE Mathematics Extension 2 Exercise 40/67 by James Coroneos* Find the following integrals. 1. 6. 11.

R

R

R

16.

R

21.

R

26. 31. 35. 39. 43. 46. 50. 54.

R

R

R

R

x dx 5x+2 √x dx 3. sin x cos3 x dx 5. sin x sec3 x dx x2 +4 2. x2 −4 dx 4. x2 +4 R R R R 3 dx cos2 x2 dx 7. x sin x dx 8. x sec2 2x dx 9. tan−1 2x dx 10. xx2 +1 R (x−1)(x+1) dx R R x3 dx R (1+x) dx dx x dx √ 13. (2x−1) (x+2)(x+4) 12. (x−2)(x−3) x2 +2x+3 14. 2x−1 15. 1−x−x2 R R R R dx dx dx dx x dx √ √ √ √ 18. x a2 −x2 19. x x2 −a2 20. 1 17. x+1 x a2 +x2 x2 (1−x2 ) 2

Rq

R

R

R

−1 cos x x+1 dx dx √ dx 22. sec4 3x dx 25. x2 (1−x) x−1 dx 23. x(log x)3 24. 1−x2 R R dx R R R dx 27. (1+x tan3 x dx 29. 5+3dxcos x 30. 3+5dxcos x 2 )2 28. x2 (1+x2 ) R sin x dx R R R 2 dx dx 32. 33. x sin x dx x x 34. 2 2 2 5+3 cos x 1+cos x cos 2 −sin 2 R R R 3 5x4 −7 R 2 x x dx e dx dx 36. 37. 38. x e dx 2 x 2 e −1 3 sin x+5 cos x R R R R (x−1)(x−2)(x−3) (3x+2) dx dx log x3 dx 42. ex +e−x x5 log x dx 40. x(x+1)3 41. R R R 2 3 dx (5x3 + 7x − 1) 2 .(15x2 + 7) dx 44. (x2 +1)(x (x + x − +1)−1 dx 2 +4) 45. R x R R R 1 e sin 2x dx 47. (x2 + x − 1)−1 dx 48. (x2 − x)− 2 dx 49. 1−2x 3+x dx R 3 R R R 2 1 2x dx x dx dx x (4 + x2 )− 2 dx 51. 3 cossin 2 x+4 sin2 x 52. 1−x4 53. sin x cos x R R dx R R √ 2 (x+1) dx x dx log x − 1 dx 55. ex −1 56. tan2sec 57. 1 x−3 tan x+2 (x2 −3x+2) 2

R R x dx R R 58. sin 2x cos x dx 59. 1+x x tan−1 x dx 61. (1 + 3x + 2x2 )−1 dx 3 60. R R R R 1 1 1 62. (9 − x2 ) 2 dx 63. (9 + x2 ) 2 dx 64. x(9 + x2 ) 2 dx 65. sec2 x tan3 x dx R R R R 2 66. x2 e−x dx 67. xex dx 68. sin x tan x dx 69. sin4 x cos3 x dx √ R R R (x3 +1) dx dx 71. log(x + x2 − 1) dx 72. 70. 1 x3 −x (x+1) 2 +(x+1)

Evaluate the following definite integrals, leaving results in irrational form. 73. 78. 82.

R 2 log x R1 R 2 (x+1) dx dx −1 √ 75. dx 76. cos x dx 77. 2 x 1 x(1+x ) 1 0 1 −2+3x−x2 √ R π2 R R R 1 4 2 dx dx 79. 0 x 1 − x2 dx 80. 2 x log x dx 81. 1 x2 +5x+4 2 sin2 x R0π2 cos x+2 R R R 1 2 −x 1 (7+x) dx 1 e−2x dx 1 −1 (1 + sin x) dx 83. x e dx 84. 85. 2 3 2 0 0 1+x+x +x 0 e−x +1 0

R4

x dx √ 0 x+4

74.

R2

*Other resources by James Coroneos are available. Write to P.O. Box 25, Rose Bay, NSW, 2029, Australia, for a catalogue. Typeset by AMS-TEX 1

2

86. 90. 93. 96. 100.

R

a 2

R a (a−x)2 dx R 1 (x+3) dx R 1 x2 dx y dy 87. 88. 89. 2 2 2 a +x 0 0 (x+2)(x+1) 0 x6 +1 0 a−y Rπ R π2 R a2 2 √ 2 cos mx dx, m integral 91. π x sin 2x dx 92. 0 x a2 − x2 dx 0 4 R1 R2 R π4 1 2 2 2 dx 95. sec x tan x dx 94. (x + 2)(x + 4x + 5) x(log x)2 dx 1 R04 x2 +4 R 4 x2 +40 R π2 cos x dx R 1 dx dx 97. 1 x(x+2) dx 98. 0 5−3 sin x 99. 0 3 3 x2 −1 (4−x2 ) 2 R

π 2

0

2 sin θ cos θ(3 sin θ − 4 sin3 θ) dθ