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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

### Trig Integrals

In this learning module we learned how to compute integrals of the form:

 $$\int \sin^n(x) \cos^m(x)\,dx$$ If $n$ is odd or if $m$ is odd, then we can use the identity $\sin^2(x)+\cos^2(x)=1$ to manipulate the integral into the form $\int \sin^k(x) \cos(x)\,dx$ which can be solved with the substitution $u=\sin(x)$; or into the form $\int \sin(x) \cos^k(x) \,dx$ which can be solved with the substitution $u=\cos(x)$. If $n$ and $m$ are both even, though, this doesn't work and we need to use the double angle formulas $\sin^2(x) = \frac{1}{2} \Bigl(1-\cos(2x)\Bigr)$, or $\cos^2(x) = \frac{1}{2}\Bigl(1 + \cos(2x)\Bigr)$ to simplify the problem. $$\int \tan^n(x) \sec^m(x)\,dx,$$ If $n$ is odd or $m$ is even we can use the identity $\tan^2(x) + 1 = \sec^2(x)$ to manipulate the integral into the form $\int \tan^k(x) \sec^2(x) \,dx$ which can be solved with the substitution $u=\tan(x)$; or into the form $\int \tan(x) \sec^k(x) \,dx$ which can be solved with the substitution $u=\sec(x)$. If neither works, then there may be a way to simplify the problem using integration by parts, but it's likely to be hard. $$\int \sin(ax)\sin(bx)\,dx$$ There are two methods: one is to integrate by parts twice; the other is to use addition-of-angle formulas.