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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

Behind IBP
Examples
Going in Circles

#### 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
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

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

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

Not all functions are created equal. Some get much simpler when differentiated, others get a little simpler, others don't really change, and a few actually get more complicated. You always want to pick $u$ to be the part that gets a lot simpler when differentiated, and $dv$ to be the part that doesn't mind being integrated. Then $\int v\, du$ will be simpler than $\int u \,dv$.
 Functions like $\ln(x)$, $\frac{1}{x}$, $\tan^{-1}(x)$, $\frac{1}{1+x^2}$, $\sin^{-1}(x)$ and $\frac{1}{\sqrt{1-x^2}}$ almost never get grouped into $dv$. These functions either get a lot simpler when differentiated (like $\ln(x)$) or get a lot more complicated when integrated (like $1/x$). Functions like $x^n$ (with $n>0$) get a little simpler when differentiated and a little more complicated when integrated. They sometimes get grouped into $u$ and sometimes into $dv$, depending on what else is in the integrand. Functions like $\sin(x)$, $\cos(x)$ and $e^x$ stay more-or-less the same when differentiated or integrated. These get grouped into $dv$ more often than not. A few functions, like $\sec^2(x)$, $x e^{x^2}$ and $x^{-n}$ (with $n$ at least 2), actually get simpler when integrated. These almost always get grouped into $dv$. Don't forget that 1 is a function! You can always apply integration by parts to $\int f(x) \,dx$ by picking $u=f(x)$ and $dv= 1\, dx$. This gives the useful formula$$\int f(x) \,dx = x f(x) - \int x f'(x)\, dx$$that can be used to integrate $\ln(x)$ and $\tan^{-1}(x)$.