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CHAPTER
4
4.1
4.2
4.3
4.4
CHAPTER
5
5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
CHAPTER
6
6.1
6.2
6.3
6.4
6.5
6.6
6.7
CHAPTER 7
7.1
7.2
7.3
7.4
7.5
CHAPTER 8
8.1
8.2
8.3
8.4
8.5
Contents
The
Chain
Rule
Derivatives by the Chain Rule
Implicit Differentiation and Related Rates
Inverse Functions and Their Derivatives
Inverses of Trigonometric Functions
Integrals
The Idea of the Integral
177
Antiderivatives
182
Summation vs. Integration
187
Indefinite Integrals and Substitutions
195
The Definite Integral
201
Properties of the Integral and the Average Value
206
The Fundamental Theorem and Its Consequences
213
Numerical Integration
220
Exponentials
and
Logarithms
An Overview
228
The Exponential
ex
236
Growth and Decay in Science and Economics
242
252
Separable Equations Including the Logistic Equation
259
Powers Instead of Exponentials
267
Hyperbolic Functions
277
Techniques
of Integration
Integration by Parts
Trigonometric Integrals
Trigonometric Substitutions
Partial Fractions
Improper Integrals
Applications
of
the
Integral
Areas and Volumes by Slices
Length of a Plane Curve
Area of a Surface of Revolution
Probability and Calculus
Masses and Moments
8.6
Force, Work, and Energy
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View Full DocumentCHAPTER
7
Techniques
of
Integration
Chapter
5
introduced the integral as a limit of sums. The calculation of areas was
startedby
hand or computer. Chapter 6 opened a different door. Its new functions
ex
and In x led to differential equations. You might say that all along we have been
solving the special differential equation dfldx
=
v(x). The
solution is
f
= 1
v(x)dx. But
the step to dyldx
=
cy was a breakthrough.
The truth is that we are able to do remarkable things.
Mathematics has a language,
and you are learning to speak it. A
short time ago the symbols dyldx and J'v(x)dx
were a mystery. (My own class was not too sure about v(x) itselfthe
symbol for a
function.) It is easy to forget how far we have come, in looking ahead to what is next.
I do want to look ahead. For integrals there are two steps to takemore
functions
and more applications.
By using mathematics we make it live.
The applications are
most complete when we know the integral. This short chapter will widen (very much)
the range of functions we can integrate.
A
computer with symbolic algebra widens it
more.
Up to now, integration depended on recognizing derivatives. If v(x)
=
sec2x then
f(x)
=
tan x. To integrate tan x we use a substitution:,
I!&dx.=
1"


In
u
=

In cos x.
U
What we need now ,are
techniques for other integrals,
to change them around until
we can attack them. Two examples are
j
x cos x dx and
5
,/
dx, which are not
immediately recognizable. With integration by parts, and a new substitution, they
become simple.
Those examples indicate where this chapter starts and stops. With reasonable effort
(and the help of tables, which is fair) you can integrate important functions. With
intense effort you could integrate even more functions. In older books that extra
exertion was madeit
tended to dominate the course. They had integrals like
which we could work on if we had to.
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 Spring '09
 fdfdf
 The Land

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