### 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

- Definition of Antiderivative
- Antiderivative of a Polynomial
- Antiderivative of (x-1)(x-2)/sqrt(x^3)
- Basic Trig Antiderivatives
- Antiderivative of tan^2(x)
- Antiderivative of sin(x)/[1-sin^2(x)]
- Visualizing an Antiderivative
- Graphing f(x) from f'(x)
- Antiderivative of a Piecewise-Defined Function
- Solving Differential Equations
- Equations of Motion
- Overview of Rectangular Approximation of Area
- Rectangular Approximation of Area
- Overview of Summation Formulas
- Limit Summation Formula
- General Method for Integer Power Sum Formula
- Riemann Sum with Signed Area
- Limit Process for Area
- Definition of Definite Integral
- Definite Integral as Area 1 - Using the Graph
- Definite Integral as Area 2 - Breaking Up the Region
- Definite Integral as Area 3 - Area Below the x-axis
- The First Fundamental Theorem of Calculus
- Area Under a Curve 1
- Area Under a Curve 2
- The Mean Value Theorem for Integrals
- Example of Mean Value Theorem for Integration
- The Second Fundamental Theorem of Calculus
- Example of 2nd Fundamental Theorem of Calculus 1
- Example of 2nd Fundamental Theorem of Calculus 2
- Integration By Substitution: Antiderivatives
- Integration by Substitution: Definite Integrals
- Example of Integration by Substitution 1: f(x) = (-x)/[(x+1)-sqrt(x+1)]
- Example of Integration by Substitution 2: f(x) = x^5/(1-x^3)^3
- Example of Integration by Substitution 3: f(x) = x^2(1+x)^4 over [0,1]
- Example of Integration by Substitution 4: f(x) = (2x+3)/sqrt(2x+1)
- Example of Integration by Substitution 5: f(x) = sin(x)/cos^3(x)
- Example of Trapezoid Rule with Error Bound
- 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.