2. 6/10/2016By Chtan FYHS-Kulai1

3. 6/10/2016By Chtan FYHS-Kulai2 “Life is either a daring adventure or nothing.” -- Helen Keller

4. 6/10/20163By Chtan FYHS-Kulai You must memorise the following learned formula [forever] :

5. 6/10/20164By Chtan FYHS-Kulai 1.

6. 6/10/20165By Chtan FYHS-Kulai 2.2.

7. 6/10/20166By Chtan FYHS-Kulai 3.3.

8. 6/10/20167By Chtan FYHS-Kulai 4.4.

9. 6/10/20168By Chtan FYHS-Kulai 5.5.

10. 6/10/20169By Chtan FYHS-Kulai 6.6.

11. 6/10/201610By Chtan FYHS-Kulai 7.7.

12. 6/10/201611By Chtan FYHS-Kulai 8.8.

13. 6/10/201612By Chtan FYHS-Kulai 9.

14. 6/10/201613By Chtan FYHS-Kulai 5 basic methods of integration 1. Integration by means of standard forms. 2. Integration by the use of partial fractions. 3. Integration by substitution. 4. Integration by parts. 5. Integration by use of reduction formulae.

15. 6/10/2016By Chtan FYHS-Kulai14 Method of substitution

16. 6/10/201615By Chtan FYHS-Kulai Integration by substitution : e.g. 1 Soln : Let

17. 6/10/201616By Chtan FYHS-Kulai

18. 6/10/201617By Chtan FYHS-Kulai e.g. 2 By using the substitution, evaluate. Soln :

19. 6/10/2016By Chtan FYHS-Kulai18

20. 6/10/2016By Chtan FYHS-Kulai19 e.g. 3 Evaluate using the substitution cosx=c. Soln :

21. 6/10/2016By Chtan FYHS-Kulai20 When

22. 6/10/2016By Chtan FYHS-Kulai21 Some common types of substitutions : (1) Of the form Let

23. 6/10/2016By Chtan FYHS-Kulai22 (2) Of the form Let or

24. 6/10/2016By Chtan FYHS-Kulai23 (3) Of the form Let or

25. 6/10/2016By Chtan FYHS-Kulai24 (4) Of the form Let or

26. 6/10/2016By Chtan FYHS-Kulai25 (5) Of odd powers of sine or cosine Let respectively

27. 6/10/2016By Chtan FYHS-Kulai26 (6) Of the form or Let

28. 6/10/2016By Chtan FYHS-Kulai27 (7) Of the form Let

29. 6/10/2016By Chtan FYHS-Kulai28 (8) Of the form Let

30. 6/10/2016By Chtan FYHS-Kulai29 e.g. 4 Evaluate, assuming. Soln : Let See slide #21

31. 6/10/2016By Chtan FYHS-Kulai30

32. 6/10/2016By Chtan FYHS-Kulai31

33. 6/10/2016By Chtan FYHS-Kulai32 e.g. 5 Evaluate. Soln : See slide #25

34. 6/10/2016By Chtan FYHS-Kulai33 Let Do you know how to get this?

35. 6/10/2016By Chtan FYHS-Kulai34 and

36. 6/10/2016By Chtan FYHS-Kulai35 When x=∏/2, t=1, when x=0, t=0

37. 6/10/201636By Chtan FYHS-Kulai e.g. 6 Integrate the function. Soln :

38. 6/10/2016By Chtan FYHS-Kulai37 e.g. 7 Integrate the function. Soln :

39. 6/10/2016By Chtan FYHS-Kulai38

40. 6/10/2016By Chtan FYHS-Kulai39 You can further simplify this answer,

41. 6/10/2016By Chtan FYHS-Kulai40 e.g. 8 Evaluate.

42. 6/10/2016By Chtan FYHS-Kulai41 Soln : Let

43. 6/10/2016By Chtan FYHS-Kulai42

44. 6/10/2016By Chtan FYHS-Kulai43

45. 6/10/2016By Chtan FYHS-Kulai44 x 1

46. 6/10/2016By Chtan FYHS-Kulai45

47. 6/10/2016By Chtan FYHS-Kulai46 In S1S, you have this : So, we have,

48. 6/10/2016By Chtan FYHS-Kulai47 Since So, and,

49. 6/10/2016By Chtan FYHS-Kulai48 Geometrically in terms of areas : v u 0

50. 6/10/2016By Chtan FYHS-Kulai49 e.g. 9 Soln : Let

51. 6/10/2016By Chtan FYHS-Kulai50 Note : It is customary to drop the first constant of integration when determining v.

52. 6/10/2016By Chtan FYHS-Kulai51 e.g. 10 Soln :

53. 6/10/2016By Chtan FYHS-Kulai52 Sometimes an integration by parts must be repeated to obtain an answer. Check the following example…

54. 6/10/2016By Chtan FYHS-Kulai53 e.g. 11 Soln :

55. 6/10/2016By Chtan FYHS-Kulai54 Then let

56. 6/10/2016By Chtan FYHS-Kulai55

57. 6/10/2016By Chtan FYHS-Kulai56 Obtain a reduction formula for

58. 6/10/2016By Chtan FYHS-Kulai57

59. 6/10/2016By Chtan FYHS-Kulai58 Let Now,

60. 6/10/2016By Chtan FYHS-Kulai59 This allows us to reduce the exponent on cosx by 2, and is a very useful formula.

61. 6/10/2016By Chtan FYHS-Kulai60 e.g. 12 Soln : and with n=2

62. 6/10/2016By Chtan FYHS-Kulai61

63. 6/10/2016By Chtan FYHS-Kulai62 Similarly, we have :

64. 6/10/2016By Chtan FYHS-Kulai63 Miscellaneous Examples

65. 6/10/2016By Chtan FYHS-Kulai64 Remember these ?

66. 6/10/2016By Chtan FYHS-Kulai65 e.g. 13

67. 6/10/2016By Chtan FYHS-Kulai66 e.g. 14

68. 6/10/2016By Chtan FYHS-Kulai67