Advanced Integration Techniques - Partial Fractions, Integration by Parts, and Recurrence Relations

Advanced Integration Techniques - Partial Fractions, Integration by Parts, and Recurrence Relations

Eddie Woo via YouTube Direct link

Integrating Rational Polynomials (Example)

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Integrating Rational Polynomials (Example)

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Classroom Contents

Advanced Integration Techniques - Partial Fractions, Integration by Parts, and Recurrence Relations

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  1. 1 Integrating Rational Polynomials (Example)
  2. 2 Partial Fractions (example with cubic denominator)
  3. 3 Partial Fractions for Integrating Rational Polynomials
  4. 4 Partial Fractions w/ Perfect Square Denominators
  5. 5 Rearranging Rational Polynomials w/ Division Transformation
  6. 6 Deriving Integration by Parts
  7. 7 Integration by Parts - Example 1: x cos x
  8. 8 Integration by Parts - Example 2: x*e^x & x²*e^x
  9. 9 Integration by Parts - Example 3: ln x
  10. 10 Integration by Parts - Example 4: e^x * sin x
  11. 11 An unusual integral with two very different solutions?
  12. 12 Integration by Parts - Example 5: sin¯¹(x)
  13. 13 Integrals that lead to Logarithmic Functions
  14. 14 Introduction to Recurrence Relations
  15. 15 Recurrence Relation Example: Integrating (sin x)^n
  16. 16 Recurrence Relations without IBP (1 of 2): (tan x)^n
  17. 17 Recurrence Relations without IBP (2 of 2): (x^n)/(x+1)
  18. 18 Sometimes integration by parts is a bad idea...
  19. 19 Integrating with t-results (1 of 3): Example integral through identities
  20. 20 Integrating with t-results (2 of 3): Changing the variable of integration
  21. 21 Integrating with t-results (3 of 3): Further example
  22. 22 Why do logarithmic functions have absolute value signs after integration?
  23. 23 Integrating √[x/(1-x)] by Trigonometric Substitution
  24. 24 Correction of Wrong Solution from Fitzpatrick (integration question)
  25. 25 Tricky Recurrence Relation with Unexpected Algebraic Manipulation
  26. 26 Evaluating Recurrence Relations
  27. 27 Proving Properties of Definite Integrals (1 of 2)
  28. 28 Proving Properties of Definite Integrals (2 of 2)
  29. 29 3 Definite Integrals Evaluated by Trigonometric Substitution
  30. 30 Integrating √(16-x²)/x² by Trigonometric Substitution
  31. 31 Integrating √[(1+x)/(1-x)] by Trigonometric Substitution
  32. 32 Evaluating Definite Integral w/ t-results
  33. 33 Integral of sin(x)tan²(x): Three Approaches
  34. 34 Proving Inequality from an Increasing Function
  35. 35 Proving Inequality from Prior Integral
  36. 36 Proving that a function is increasing
  37. 37 Using Properties of Definite Integrals: Trigonometric Integrand (1 of 2)
  38. 38 Determining Algebraic Recurrence Relation
  39. 39 Expressing I(n) without Recursion
  40. 40 Using Properties of Definite Integrals: Trigonometric Integrand (2 of 2)
  41. 41 Harder (Ext 2) Integral Requiring Substitution
  42. 42 Harder (Ext 2) Integral Question (1 of 2: Manipulating the Recurrence Relation)
  43. 43 Harder (Ext 2) Integral Question (2 of 2: Evaluating Related Limiting Sum)
  44. 44 Partial Fractions (1 of 3: Introducing an identity to simplify a quadratic denominator)
  45. 45 Partial Fractions (2 of 3: An alternative method to find introduced variables)
  46. 46 Partial Fractions (3 of 3: How to use Partial Fractions to split fractions with Quadratic Numerator)
  47. 47 Partial Fractions: Quadratic Factors (1 of 2: Issues with non-linear denominator factors)
  48. 48 Partial Fractions: Quadratic Factors (2 of 2: Problems with Quadratic Factors and Solution)
  49. 49 Partial Fraction: Repeated Linear Factors (Technique for Breaking down into partial fractions)
  50. 50 Integration of Square Root Function (1 of 2: Using a Trig Substitution to help with integration)
  51. 51 Integration of Square Root Function (2 of 2: What is this kind of integral solving for?)
  52. 52 Further Integration (1 of 2: Brief Overview of Extension II Integration)
  53. 53 Further Integration (2 of 2: Integrating Trig Functions without given substitution)
  54. 54 Further Integration [Continued] (1 of 3: Using Double Angle & Completing the Square to integrate)
  55. 55 Further Integration [Continued] (2 of 3: Adding and subtracting a constant to simplify integral)
  56. 56 Further Integration [Continued] (3 of 3: 'Breaking Apart' Integrals to simplify for integration)
  57. 57 Partial Fractions & Integration (Using Partial Fractions to simplify an integral for evaluation)
  58. 58 Harder Reverse Chain Rule Integral (Using a Substitution and Restrictions to evaluate)
  59. 59 Proving a Result from the Standard Integrals (1 of 3: Differentiate, Hence Integrate Proof)
  60. 60 Proving a Result from the Standard Integrals (2 of 3: Substituting tan to simplify the integral)
  61. 61 Proving a Result from the Standard Integrals (3 of 3: Using Partial Fractions to simplify sec)
  62. 62 Integration with t-results (1 of 2: Changing the variable of integration)
  63. 63 Integration with t-results (2 of 2: Dealing with the integral in t)
  64. 64 Integration by Parts (1 of 3: Deriving the Formula)
  65. 65 Integration by Parts (2 of 3: How to choose u & dv)
  66. 66 Integration by Parts (3 of 3: Integral of sin¯¹(x))
  67. 67 Integration by t results (Explanation of Harder Question)
  68. 68 Integration by Parts (What to choose for u and dv for integration by parts?)
  69. 69 Integration with t Results (1 of 2: Changing the variable to solve with t results)
  70. 70 Integration with t Results (2 of 2: Simplifying the integral before applying t results)
  71. 71 Recurrence Relations (1 of 4: Introduction to Recurrence relations with introductory examples)
  72. 72 Recurrence Relations (2 of 4: Considering if the question consists of some arbitrary power of n)
  73. 73 Recurrence Relations (3 of 4: Applying to Trigonometric Functions raised to power of n)
  74. 74 Recurrence Relations (4 of 4: Integration by parts with definite integrals)
  75. 75 Evaluating Recurrence Relations (1 of 4: When do you apply Recurrence Relations?)
  76. 76 Evaluating Recurrence Relations (2 of 4: Using Integration of parts to build recurring pattern)
  77. 77 Evaluating Recurrence Relations (3 of 4: Using Recurring Pattern to group Relation in a series)
  78. 78 Evaluating Recurrence Relations (4 of 4: Finding a term free of integrals)
  79. 79 Extension 2 Exam Review (1 of 7: Integration by Substitution, Partial Fractions)
  80. 80 Extension 2 Exam Review (2 of 7: Recurrence Relation, Graph Transformations)
  81. 81 Mathematics Extension 1 Exam Review (1 of 3: Integration by substitution)
  82. 82 Integration by Parts (2 of 2: When the integrand doesn't look like a product)
  83. 83 Integration by Parts (1 of 2: Arranging the integral with DETAIL)
  84. 84 Integration with Quadratic Denominators (3 of 3: Rationalising the numerator)
  85. 85 Integration with Quadratic Denominators (2 of 3: Distinguishing characteristics)
  86. 86 Integration with Quadratic Denominators (1 of 3: Introduction - why are they challenging?)
  87. 87 Partial Fractions (3 of 3: Three solution methods)
  88. 88 Partial Fractions (2 of 3: What are they?)
  89. 89 Partial Fractions (1 of 3: Review questions)
  90. 90 Integration by Unspecified Substitution (3 of 3: Trigonometric example)
  91. 91 Integration by Unspecified Substitution (2 of 3: Square root example)
  92. 92 Integration by Unspecified Substitution (1 of 3: Introductory example)
  93. 93 Intro to Further Integration (2 of 2: Foundational examples)
  94. 94 Intro to Further Integration (1 of 2: Expanding on prior learning)
  95. 95 Recurrence Relations (3 of 3: Trigonometric examples)
  96. 96 Recurrence Relations (2 of 3: Exponential example)
  97. 97 Recurrence Relations (1 of 3: Introduction & logarithmic example)
  98. 98 Recurrence Relation - worked example (1 of 2: Integration by parts)
  99. 99 Recurrence Relation - worked example (2 of 2: Connecting the integrals)
  100. 100 Logarithmic integral with recurrence relation (Exam Question 4 of 10)
  101. 101 Integral via partial fractions (Exam Question 9 of 10)
  102. 102 Reverse chain rule integral (Exam Question 7 of 10)
  103. 103 Recurrence Relation (1 of 6: Setting up integration by parts)
  104. 104 Recurrence Relation (2 of 6: Relating consecutive integrals)
  105. 105 Recurrence Relation (3 of 6: Establishing the pattern)
  106. 106 Integrating sec⁸θ
  107. 107 Inequality proof from integration by parts
  108. 108 Recurrence Relation (4 of 6: Expressing without integrals)
  109. 109 Recurrence Relation (5 of 6: Trigonometric substitution)
  110. 110 Recurrence Relation (6 of 6: Proving an inequality)
  111. 111 Integration: how to choose boundaries with trigonometric substitution
  112. 112 Integration by substitution (two ways)

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