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

### Integration by Parts

After $u$-substitution, integration by parts is by far the most important technique to learn. It converts a hard integral $\int u\,dv$ into an easier integral $uv - \int v\,du$. The tricky thing is figuring out what to pick for $u$ and $dv$. We want to choose $u$ so that $du$ is simpler than $u$, and $dv$ so that $v$ isn't much more complicated than $dv$. Remember that:

 Some functions get a lot simpler when differentiated or a lot more complicated when integrated. These almost always want to be part of $u$. Some functions get a little simpler when differentiated. These are a lower priority to include in $u$. Some functions, like $e^x$ and $\sin(x)$, don't get simpler at all. These have still lower priority, and usually are part of $dv$. Some functions actually get simpler when integrated and are almost always part of $dv$.

If we can pick $u$ from higher on the list and $dv$ from lower on the list, then $\int v\,du$ will be simpler than $\int u\,dv$.

If we have two terms at the same level, then we can try using some algebra to relate $\int v\,du$ to $\int u\,dv$. We may have to integrate by parts twice to make this work.

 Example: Compute $$\int\frac{\ln(x)}{x^3}\,dx.$$ Solution: Since $\displaystyle\Bigl(\ln(x)\Bigr)'=\frac{1}{x}$ is simpler than $\ln(x)$ and $\displaystyle\int x^{-3}\, dx$ is simpler than $x^{-3}$, we take $u=\ln(x)$ and $\displaystyle dv=\frac{1}{x^3}\,dx$. Then $$du=\frac{1}{x}, \quad \text{ and }\quad v=-\frac{1}{2x^2}.$$ Hence $$\int\frac{\ln(x)}{x^3}\,dx=-\frac{\ln(x)}{2x^2}-\left(\int -\frac{1}{2x^3}\,dx\right)=-\frac{\ln(x)}{2x^2}-\frac{1}{4x^2}+C.$$