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Advanced Integration Techniques by Chtan FYHS-Kulai: Exploring Five Core Methods

In this chapter, Chtan FYHS-Kulai delves into five essential methods of integration crucial for mastering calculus. Explore integration through standard forms, partial fractions, substitution, parts, and reduction formulae. Each method is presented with examples and solutions, helping learners grasp complex concepts. Understand common substitution types and how they can simplify expressions for easier integration. This resource serves as a vital reference for students and educators aiming to enhance their calculus skills.

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Advanced Integration Techniques by Chtan FYHS-Kulai: Exploring Five Core Methods

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  1. Chapter 37 Further methods of integration By Chtan FYHS-Kulai

  2. “Life is either a daring adventure or nothing.” -- Helen Keller By Chtan FYHS-Kulai

  3. You must memorise the following learned formula [forever] : By Chtan FYHS-Kulai

  4. 1. By Chtan FYHS-Kulai

  5. 2. By Chtan FYHS-Kulai

  6. 3. By Chtan FYHS-Kulai

  7. 4. By Chtan FYHS-Kulai

  8. 5. By Chtan FYHS-Kulai

  9. 6. By Chtan FYHS-Kulai

  10. 7. By Chtan FYHS-Kulai

  11. 8. Take note By Chtan FYHS-Kulai

  12. 9. Take note By Chtan FYHS-Kulai

  13. 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. By Chtan FYHS-Kulai

  14. Method of substitution By Chtan FYHS-Kulai

  15. Integration by substitution: e.g. 1 Soln : Let By Chtan FYHS-Kulai

  16. By Chtan FYHS-Kulai

  17. e.g. 2 By using the substitution , evaluate . Soln : By Chtan FYHS-Kulai

  18. By Chtan FYHS-Kulai

  19. e.g. 3 Evaluate using the substitution cosx=c . Soln : By Chtan FYHS-Kulai

  20. When When By Chtan FYHS-Kulai

  21. Some common types of substitutions : (1) Of the form Let By Chtan FYHS-Kulai

  22. (2) Of the form or Let By Chtan FYHS-Kulai

  23. (3) Of the form or Let By Chtan FYHS-Kulai

  24. (4) Of the form or Let By Chtan FYHS-Kulai

  25. (5) Of odd powers of sine or cosine Let respectively By Chtan FYHS-Kulai

  26. (6) Of the form or Let By Chtan FYHS-Kulai

  27. (7) Of the form Let By Chtan FYHS-Kulai

  28. (8) Of the form Let By Chtan FYHS-Kulai

  29. e.g. 4 Evaluate , assuming . See slide #21 Soln : Use type 2 Let By Chtan FYHS-Kulai

  30. By Chtan FYHS-Kulai

  31. By Chtan FYHS-Kulai

  32. e.g. 5 Evaluate . Soln : See slide #25 Use type 6 By Chtan FYHS-Kulai

  33. Let Do you know how to get this? By Chtan FYHS-Kulai

  34. and By Chtan FYHS-Kulai

  35. When x=∏/2, t=1, when x=0, t=0 By Chtan FYHS-Kulai

  36. e.g. 6 Integrate the function . Soln : By Chtan FYHS-Kulai

  37. e.g. 7 Integrate the function . Soln : By Chtan FYHS-Kulai

  38. By Chtan FYHS-Kulai

  39. You can further simplify this answer, By Chtan FYHS-Kulai

  40. e.g. 8 Evaluate . By Chtan FYHS-Kulai

  41. Soln : Let By Chtan FYHS-Kulai

  42. By Chtan FYHS-Kulai

  43. By Chtan FYHS-Kulai

  44. x 1 By Chtan FYHS-Kulai

  45. Integration by parts By Chtan FYHS-Kulai

  46. In S1S, you have this : So, we have, By Chtan FYHS-Kulai

  47. Since So, and, By Chtan FYHS-Kulai

  48. Geometrically in terms of areas : u v 0 By Chtan FYHS-Kulai

  49. e.g. 9 Soln : Let By Chtan FYHS-Kulai

  50. Note : It is customary to drop the first constant of integration when determining v. By Chtan FYHS-Kulai

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