Home

#### 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

### How it works

Regular $u$-substitution works by setting $u=g(x)$ for some function $g$. We have to pick $u$ so that the integrand becomes a nice function of $u$ times $du$. In the video, we explore inverse substitutions.

With inverse substitutions, we let $x = g(\theta)$, where typically $g$ is a trig function. Then $dx = g'(\theta)\, d\theta$, and we can always rewrite the integrand in terms of $\theta$ and $d\theta$. The tricky part is converting everything back to $x$ at the end of the problem.

There are three very useful trig substitution:

 For integrals involving $x^2 + a^2$ or $\sqrt{x^2+a^2}$, use $x = a \tan(\theta)$. Then $dx = a\, \sec^2(\theta)\, d\theta$ and $x^2+a^2 = a^2 \sec^2(\theta)$. To convert back to $x$, draw a right triangle with opposite side $x$, adjacent side $a$ and hypotenuse $\sqrt{x^2+a^2}$ and use soh-cah-toa. For integrals involving $a^2 - x^2$ or $\sqrt{a^2-x^2}$, use $x=a \sin(\theta)$. Then $dx = a \cos(\theta) d\theta$ and $\sqrt{a^2-x^2}= a \cos(\theta)$. To convert back to $x$, draw a right triangle with opposite side $x$, hypotenuse $a$ and adjacent side $\sqrt{a^2-x^2}$ and use soh-cah-toa. For integrals involving $x^2-a^2$ or $\sqrt{x^2-a^2}$, use $x=a \sec(\theta)$. Then $dx = a \sec(\theta) \tan(\theta)\, d\theta$ and $\sqrt{x^2-a^2}= a \tan(\theta)$. To convert back to $x$, draw a right triangle with adjacent side $a$, hypotenuse $x$ and opposite side $\sqrt{x^2-a^2}$ and use soh-cah-toa.