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#### The Fundamental Theorem of Calculus

Three Different Quantities
The Whole as Sum of Partial Changes
The Indefinite Integral as Antiderivative
The FTC and the Chain Rule

#### The Indefinite Integral and the Net Change

Indefinite Integrals and Anti-derivatives
A Table of Common Anti-derivatives
The Net Change Theorem
The NCT and Public Policy

#### Substitution

Substitution for Indefinite Integrals
Revised Table of Integrals
Substitution for Definite Integrals

#### Area Between Curves

The Slice and Dice Principle
To Compute a Bulk Quantity
The Area Between Two Curves
Horizontal Slicing
Summary

#### Volumes

Slicing and Dicing Solids
Solids of Revolution 1: Disks
Solids of Revolution 2: Washers
Volumes Rotating About the $y$-axis

#### Integration by Parts

Behind IBP
Examples
Going in Circles
Tricks of the Trade

#### Integrals of Trig Functions

Basic Trig Functions
Product of Sines and Cosines (1)
Product of Sines and Cosines (2)
Product of Secants and Tangents
Other Cases

#### Trig Substitutions

How it works
Examples
Completing the Square

#### Partial Fractions

Introduction
Linear Factors
Quadratic Factors
Improper Rational Functions and Long Division
Summary

#### Strategies of Integration

Substitution
Integration by Parts
Trig Integrals
Trig Substitutions
Partial Fractions

#### Improper Integrals

Type I Integrals
Type II Integrals
Comparison Tests for Convergence

#### Differential Equations

Introduction
Separable Equations
Mixing and Dilution

#### Models of Growth

Exponential Growth and Decay
Logistic Growth

#### Infinite Sequences

Close is Good Enough (revisited)
Examples
Limit Laws for Sequences
Monotonic Convergence

#### Infinite Series

Introduction
Geometric Series
Limit Laws for Series
Telescoping Sums and the FTC

#### Integral Test

Road Map
The Integral Test
When the Integral Diverges
When the Integral Converges

#### Comparison Tests

The Basic Comparison Test
The Limit Comparison Test

#### Convergence of Series with Negative Terms

Introduction
Alternating Series and the AS Test
Absolute Convergence
Rearrangements

The Ratio Test
The Root Test
Examples

#### Strategies for testing Series

List of Major Convergence Tests
Examples

#### Power Series

Radius and Interval of Convergence
Finding the Interval of Convergence
Other Power Series

#### Representing Functions as Power Series

Functions as Power Series
Derivatives and Integrals of Power Series
Applications and Examples

#### Taylor and Maclaurin Series

The Formula for Taylor Series
Taylor Series for Common Functions
Adding, Multiplying, and Dividing Power Series
Miscellaneous Useful Facts

#### Applications of Taylor Polynomials

What are Taylor Polynomials?
How Accurate are Taylor Polynomials?
What can go Wrong?
Other Uses of Taylor Polynomials

#### Partial Derivatives

Definitions and Rules
The Geometry of Partial Derivatives
Higher Order Derivatives
Differentials and Taylor Expansions

#### Multiple Integrals

Background
What is a Double Integral?
Volumes as Double Integrals

#### Iterated Integrals over Rectangles

One Variable at the Time
Fubini's Theorem
Notation and Order

#### Double Integrals over General Regions

Type I and Type II regions
Examples
Order of Integration
Area and Volume Revisited

### Revised Table of Integrals

 $\displaystyle \int u^n \frac{du}{dx} dx = \int u^n du = \frac{u^{n+1}}{n+1} + C$ as long as $n \ne -1$ $\displaystyle \int \frac{1}{u} \frac{du}{dx} dx =\int \frac{du}{u} = \ln\left(|u|\right) + C$ (Don't forget the absolute value!) $\displaystyle \int e^u \frac{du}{dx} dx =\int e^u du = e^u + C$ $\displaystyle \int \cos(u) \frac{du}{dx} dx =\int \cos(u) du = \sin(u) + C$ $\displaystyle \int \sin(u) \frac{du}{dx} dx =\int \sin(u) du = - \cos(u) + C$ $\displaystyle \int \sec^2(u) \frac{du}{dx} dx =\int \sec^2(u) du = \tan(u) + C$ $\displaystyle \int \sec(u)\tan(u) \frac{du}{dx} dx =\int \sec(u) \tan(u) du = \sec(u) + C$ $\displaystyle \int \csc^2(u) \frac{du}{dx} dx =\int \csc^2(u) du = - \cot(u) + C$ $\displaystyle \int \csc(u)\cot(u) \frac{du}{dx} dx =\int \csc(u)\cot(u) du = -\csc(u) + C$ $\displaystyle \int \frac{1}{1+u^2} \frac{du}{dx} dx =\int \frac{du}{1+u^2} = \tan^{-1}(u) + C$ $\displaystyle \int \frac{1}{\sqrt{1-u^2}} \frac{du}{dx} dx =\int \frac{du}{\sqrt{1-u^2}} = \sin^{-1}(u) + C$