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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

### Examples

In this video, we work three examples, one with $x=a \tan(\theta)$, one with $x = a\sin(\theta)$, and one with $x = a \sec(\theta)$.

 Example 1: $$\int \bigl(4+x^2\bigr)^{-3/2}\, dx \overset{\fbox{ x\,=\,2\tan(\theta)}}{=}\int \bigl(4 \sec^2(\theta)\bigr)^{-3/2} 2 \sec^2(\theta) \,d\theta$$$$=\int \frac{d\theta}{4 \sec(\theta)}$$ $$\quad= \int \frac{1}{4}\cos(\theta)\, d\theta$$$$= \frac{\sin(\theta)}{4}+C.$$ We must then figure out what $\sin(\theta)$ is in terms of $x$. Draw a right triangle with opposite side $x$ and adjacent side $a$ (in this case 2), so that $\tan(\theta)=x/a$ and $x=a\tan(\theta)$. The hypotenuse is then $\sqrt{a^2+x^2}=\sqrt{4+x^2}$, so $$\sin(\theta)=\frac{{\text{opposite}}}{\text{hypotenuse}}=\frac{x}{\sqrt{x^2+4}},$$ and our final answer is $\displaystyle\int\bigl(4+x^2\bigr)^{-3/2}\,dx=\frac{x}{4\sqrt{x^2+4}}+C.$

 Example 2: $$\int \sqrt{9-x^2}\, dx\overset{ \fbox{ x\,=\,3 \sin(\theta)} }{=} \int \left(\sqrt{9-9\sin^2(\theta)}\right)3\cos(\theta)\,d\theta$$$$\hspace{-0.4 cm}= \int 9\, \cos^2(\theta) \,d\theta$$ $$\quad =\int 9\frac{1+\cos(2\theta)}{2}\,d\theta$$ $$\quad =\frac{9}{2} \left(\theta+\frac{\sin(2\theta)}{2}\right)+C$$ $$\quad\quad =\frac{9}{2}\bigl(\theta+\sin(\theta)\cos(\theta)\bigr)+C$$ Here we have used the methods of the last learning module to evaluate the trig inegral. Finally, we use soh-cah-toa to convert this to $x$. This gives us the final answer $$\int \sqrt{9-x^2}\, dx=\frac{9}{2} \sin^{-1} \left (\frac{x}{3} \right ) + \frac{x \,\sqrt{9-x^2}}{2} + C.$$

 Example 3: $$\int \frac{dx}{\sqrt{4x^2-1}} \overset{ \fbox{ x\,=\, \sec(\theta)/2} }{=} \int \frac{\frac{1}{2}\sec(\theta)\tan(\theta)\,d\theta}{\sqrt{4\frac{\sec^2{\theta}}{4}-1}}$$ $$\qquad\qquad\quad\overset{ \fbox{ \sec^2(\theta)-1\,=\,\tan^2(\theta)} }{=} \frac{1}{2}\int \frac{\sec(\theta)\tan(\theta)\,d\theta}{\tan(\theta)}$$ $$\qquad\qquad\qquad=\frac{1}{2} \ln\bigl\lvert\sec(\theta)+\tan(\theta)\bigr\rvert +C.$$ Converting to $x$ gives $$\int \frac{dx}{\sqrt{4x^2-1}} =\frac{1}{2} \ln \bigl\lvert 2x + \sqrt{4x^2-1}\bigr\rvert + C.$$