Integration by Parts

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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

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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
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Horizontal Slicing
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Volumes

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Volumes Rotating About the

$y$ -axis

Integration by Parts

Behind IBP
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Going in Circles
Tricks of the Trade

Integrals of Trig Functions

Basic Trig Functions
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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
Quadratic 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

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Infinite Sequences

Close is Good Enough (revisited)
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Infinite Series

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Integral Test

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Convergence of Series with Negative Terms

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Alternating Series and the AS Test
Absolute Convergence
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The Ratio and Root Tests

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List of Major Convergence Tests
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Power Series

Radius and Interval of Convergence
Finding the Interval of Convergence
Other Power Series

Representing Functions as Power Series

Functions as Power Series
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Taylor and Maclaurin Series

The Formula for Taylor Series
Taylor Series for Common Functions
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Applications of Taylor Polynomials

What are Taylor Polynomials?
How Accurate are Taylor Polynomials?
What can go Wrong?
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Partial Derivatives

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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
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Order of Integration
Area and Volume Revisited

Going in Circles

The main strategy for integration by parts is to pick $u$ and $dv$ so that $v du$ is simpler to integrate than $u \,dv$ . Sometimes this isn't possible. In those cases we look for ways to relate $\int u\, dv$ to $\int v \,du$ algebraically, and then use algebra to solve for $\int u \,dv$ .

Example: Find

$\int \sin^2 x\, dx$ .

Solution: We set

$u =\sin(x)$ and

$dv = \sin(x)\, dx$ , so

$v= -\cos(x)$ and

$du=\cos(x)$ . Applying integration by parts gives

$\int \sin^2(x) \,dx = -\sin(x)\cos(x) + \int \cos^2(x) \,dx.$ But

$\cos^2(x)=1-\sin^2(x)$ , so

$\begin{eqnarray}\int \sin^2(x)\, dx &=& - \sin(x)\cos(x) + \int dx - \int \sin^2(x) \,dx \cr 2 \int \sin^2(x)\, dx &=& x - \sin(x)\cos(x) + C \cr \int \sin^2(x) \,dx &=& \frac{x - \sin(x)\cos(x)}{2} + C\end{eqnarray}$

This method is especially good for integrals involving products of $e^x$ , $\sin(x)$ and $\cos(x)$ . Sometimes you need to integrate by parts twice to make it work.