Integration By Substitution: Antiderivatives 
Integration By Substitution: Antiderivatives
by Robert Donley
Video Lecture 31 of 39
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Date Added: April 4, 2016

Lecture Description

Calculus: We introduce the technique of integration by substitution for indefinite integrals. This is an integration method that reverses the chain rule for derivatives.
Examples include (a) int x(1+x^2)^3 dx, (b) int x^6 cos(2x^7) dx, (c) int sin(x)cos(x) dx, and (d) int x(1+x)^2 dx. In problem (c), we recall an occasional problem with integrals, and, in problem (d), we show an instance where re-substitution is required.

Course Index

  1. Definition of Antiderivative
  2. Antiderivative of a Polynomial
  3. Antiderivative of (x-1)(x-2)/sqrt(x^3)
  4. Basic Trig Antiderivatives
  5. Antiderivative of tan^2(x)
  6. Antiderivative of sin(x)/[1-sin^2(x)]
  7. Visualizing an Antiderivative
  8. Graphing f(x) from f'(x)
  9. Antiderivative of a Piecewise-Defined Function
  10. Solving Differential Equations
  11. Equations of Motion
  12. Overview of Rectangular Approximation of Area
  13. Rectangular Approximation of Area
  14. Overview of Summation Formulas
  15. Limit Summation Formula
  16. General Method for Integer Power Sum Formula
  17. Riemann Sum with Signed Area
  18. Limit Process for Area
  19. Definition of Definite Integral
  20. Definite Integral as Area 1 - Using the Graph
  21. Definite Integral as Area 2 - Breaking Up the Region
  22. Definite Integral as Area 3 - Area Below the x-axis
  23. The First Fundamental Theorem of Calculus
  24. Area Under a Curve 1
  25. Area Under a Curve 2
  26. The Mean Value Theorem for Integrals
  27. Example of Mean Value Theorem for Integration
  28. The Second Fundamental Theorem of Calculus
  29. Example of 2nd Fundamental Theorem of Calculus 1
  30. Example of 2nd Fundamental Theorem of Calculus 2
  31. Integration By Substitution: Antiderivatives
  32. Integration by Substitution: Definite Integrals
  33. Example of Integration by Substitution 1: f(x) = (-x)/[(x+1)-sqrt(x+1)]
  34. Example of Integration by Substitution 2: f(x) = x^5/(1-x^3)^3
  35. Example of Integration by Substitution 3: f(x) = x^2(1+x)^4 over [0,1]
  36. Example of Integration by Substitution 4: f(x) = (2x+3)/sqrt(2x+1)
  37. Example of Integration by Substitution 5: f(x) = sin(x)/cos^3(x)
  38. Example of Trapezoid Rule with Error Bound
  39. Example of Simpson's Rule with Error Bound

Course Description

In this second chapter of a series of calculus lessons, Dr. Bob covers integral calculus, including topics such as: Antiderivatives, Area, Definite Integrals, Fundamental Theorems of Calculus, Integration by Substitution, Trapezoid and Simpson's Rules. This series has 39 lessons, and it is a continuation of Part I: Limits and Derivatives.


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