Maths A Level
Calculus III

Integration by Parts: Summary

Integration by parts is a trick for integrating many functions in the form f(x)g(x). It is a major part of integration at A Level.

 

Transcript

Lesson overview

Integrating More Functions

  1. Introduction to Integrating More Functions
  2. The One Algebraic Function that Got Away
  3. The Purpose of the Modulus
  4. What Went Wrong?
  5. What We Know So Far
  6. Integrating Exponential Functions
  7. Integrating Trig Functions
  8. Integrating sin(x)
  9. Integrating cos(x)
  10. Integrating tan(x)
  11. Integrating cot(x)
  12. Integrating cosec(x)
  13. Integrating sec(x)
  14. Integrating Four More Trig Functions
  15. Integrating sec^2(x)
  16. Integrating cosec^2(x)
  17. Integrating cosec(x)cot(x)
  18. Integrating sec(x)tan(x)
  19. Revisiting Definite Integrals
  20. The Area Under a Sine Curve

The Reverse Chain Rule

  1. Introduction to the Reverse Chain Rule
  2. What We Can Differentiate
  3. Liouville’s Vision
  4. The Fate of Integration
  5. Identifying f'(x) Divided by f(x)
  6. Integrating f'(x) Divided by f(x)
  7. Integrating kf'(x) Divided by f(x) Part 1
  8. Identifying kf’x Divided by f(x)
  9. Integrating kf'(x) Divided by f(x) Part 2
  10. Integration and Partial Fractions
  11. Identifying f'(x) Multiplied by (f(x))^n
  12. Identifying kf'(x) multiplied by (f(x))^n
  13. Integrating kf'(x) Multiplied by (f(x))^n
  14. The Reverse Chain Rule Part 1
  15. The Reverse Chain Rule Part 2
  16. Difficulty with the Reverse Chain Rule
  17. Making a Substitution
  18. Converting the Infinitesimal
  19. Integrating with Respect to u
  20. Integration by Substitution
  21. Non-Linear Substitutions
  22. Speeding Up the Process
  23. Substitutions Where x is the Subject
  24. Implicitly Defined Substitutions
  25. Finding Your Own Substitution
  26. Pick the Expression That’s Been Raised to a Power
  27. Why Did We Learn the First Two Tricks?
  28. Converting the Boundaries of a Definite Integral

Integration by Parts

  1. Introduction to Integration by Parts
  2. The One Up, One Down Game
  3. Mastering the One Up, One Down Game
  4. Changing the Rules of the Game
  5. When Both Expressions are Algebraic
  6. When One Expression is Logarithmic
  7. Integration by Parts 1
  8. Integration by Parts 2
  9. Completing Integration by Parts
  10. Integrating by Parts Multiple Times
  11. Integration by Parts and the Product Rule
  12. Integrating ln(x)
  13. Integration by Parts and Definite Integrals

Integrating More Trigonometric Functions

  1. Introduction to Integrating More Trigonometric Functions
  2. The Return of the Identities
  3. A Reciprocal Pythagorean Identity
  4. Another Reciprocal Pythagorean Identity
  5. The Original Pythagorean Identity
  6. The Double Angle Identity for Sine
  7. The Double Angle Identity for Cosine Part 1
  8. The Double Angle Identity for Cosine Part 2
  9. The Double Angle Identity for Cosine Part 3
  10. The Double Angle Identity for Tangent
  11. What Trig Functions Can We Now Integrate?

Finding More Areas Under Curves

  1. Introduction to Finding More Areas with Integration
  2. Finding the Area Between Two Curves
  3. Finding the Area Between Two Points of Intersection
  4. The Areas We Can’t Find Yet
  5. What is a Trapezium?
  6. Finding the Area of a Trapezium
  7. Splitting an Area into Right Trapeziums
  8. Finding the Width of the Strips
  9. Finding the Boundary Points
  10. Finding the Value of y at Each Boundary Point
  11. Finding the Area Under the Curve
  12. The Trapezium Rule Part I
  13. The Trapezium Rule Part II
  14. The Trapezium Rule Part III
  15. The Trapezium Rule Part IV
  16. Overestimating and Underestimating the Area

Integrating Parametric Equations

  1. Integration and Parametric Equations
  2. Convert to Cartesian Form then Integrate
  3. Integrate Without Converting to Cartesian Form
  4. Rewriting the Integral in Terms of x
  5. Rewriting the Boundaries in Terms of t
  6. Recapping the Strategies

Solving Differential Equations

  1. Introduction to Differential Equations
  2. Differential Equations in the Real World
  3. Difficulties with Differential Equations
  4. The Constant of Integration and Families of Curves
  5. Methods for Differential Equations
  6. Integrating as Normal
  7. Solving Differential Equations
  8. Why do we Use the Term ‘Solution’
  9. The Differential Equations We Can Solve So Far
  10. Recognising a Special Type of Differential Equation
  11. Separation of Variables Part 1
  12. Separation of Variables Part 2
  13. Finding Particular Solutions to Differential Equations
  14. Modelling with Differential Equations I
  15. Modelling with Differential Equations II
  16. Modelling with Differential Equations III

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