chapter 7 – techniques of integration 7.1 integration by parts 1erickson

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Chapter 7 – Techniques of Integration

7.1 Integration by Parts

7.1 Integration by Parts Erickson

7.1 Integration by Parts2

Introduction

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Until now we have learned how to integrate by using the antiderivatives of each function.

That however will not always be the case.

Sometimes we will have to use special integration techniques to obtain indefinite integrals.


Introduction

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In this section we will use the technique of integration by parts which corresponds to the product rule for differentiation.

For example, the product rule says:

df x g x f x g x g x f x

dx


Let’s integrate both sides.

Solve for this term

Introduction

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df x g x dx f x g x dx g x f x dx

dx

f x g x f x g x dx g x f x dx

f x g x dx f x g x g x f x dx


Formula for Integration by Parts

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OR

WHERE

udv uv vdu

( ) ( )

u f x v g x

du f x dx dv g x dx

f x g x dx f x g x g x f x dx


Formula 2Combining formula 1 with the Fundamental Theorem of

Calculus we have:

bbb

a aa

udv uv vdu

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Example 1 – Page 457 #2

Evaluate the integral using integration by parts with the indicated choices of u and dv.

cos , , cosd u dv d

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Choosing our function u A mnemonic device which is helpful for selecting u when

using integration by parts is the LIATE principle of precedence for u:

Logarithmic

Inverse Trigonometric

Algebraic

Trigonometric

Exponential If the integrand has several factors, we try to choose let u

be the highest function on the LIATE list.

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Example 2Evaluate the integral.

2. arctan 4 . sin 3 t dt e d a b

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Tabular Integration

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Example: Find Solution: with f (x) = x3, we list

f (x) and its derivatives g(x) and its integrals

Then we add the products of the functions connected by arrows, with every other sign changed, to obtain

3

23

6

6

0

x

x

x

sin

cos

sin

cos

sin

x

x

x

x

x

3 3 2sin cos 3 sin 6 cos 6sinx x dx x x x x x x C

3 sinx x dx

+

+

-

-


Tabular Integration

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NOTE: Tabular integration does not always work.

You need to be able to differentiate one of the factors to 0, so that the table ends. This fails when say, you have integrand of the form of product

of exponential and cosine or sine, neither which can be differentiated to zero.

Sometimes tabular integration is just too complicated.


Example 3Evaluate the integral.

3

0

31

20

cos

4

x x dx

rdr

r

a.

b.

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2 24

1

3 2

/2

ln

cos

x x dx

d

c.

d.


Find the error - 1It’s a beautiful Spring day. You leave your calculus class feeling sad and

depressed. You aren’t sad because of the class itself. On the contrary, you have just learned an amazing integration technique: Integration by Parts. You aren’t sad because it is your birthday. On the contrary, you are still young enough to actually be happy about it. You are sad because you know that every time you learn something really wonderful in calculus, a wild-eyed stranger runs up to you and shows you a “proof” that is false. Sure enough, as you cross the street, he is waiting for you on the other side.

“Good morning, Kiddo,” he says.

“I just learned integration by parts. Let me have it.”

“What do you mean?” he asks.

“Aren’t you going to run around telling me all of math is lies?”

“Well, if you insist,” he chuckles…and hands you a piece of paper:

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Find the error - continued

“Hey,” you say, “I don’t get it! You did everything right this time!”

“Yup!” says the hungry looking stranger.

“But…Zero isn’t equal to negative one!”

“Nope!” he says.

You didn’t think he could pique your interest again, but he has. Spite him. Find the error in his reasoning.

sin tan

cos1

sincostan sec cos

tan

tan 1 tan

0 1

xxdx dx

x

u dv xdxx

du x xdx v x

xdx uv vdu

xdx xdx

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Find the Error - 2What a wonderful day! You have survived another encounter with the wild-eyed

stranger, demolishing his mischievous pseudo-proof. As you leave his side, you can’t resist a taunt.

“Didn’t your mother tell you never to forget your constants?” It seemed a better taunt when you were thinking it than it did when you said it.

“Eh?” he says. You come up to him again.

“I was just teasing you. Just pointing out that when doing indefinite integration, those constants should not be forgotten. A silly, simple error, not worthy of you.” You look smug. You are the victor.

“Yup. Indefinite integrals always have those pesky constants.” For some reason he isn’t looking defeated. He is looking crafty.

“Right. Well, I’m going to be going now…”

“Of course, Kiddo, definite integrals don’t have constants, sure as elephants don’t have exoskeletons.”

“Yes. Well, I really must be going.”

Surprisingly quickly, he snatches the paper out of your hand, and adds to it. This is what it looks like.

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Find the Error 2 - Continued

“No constants missing here! Happy Birthday!” The stranger leaves, singing the “Happy Birthday” song in a minor key. Now there are no constants involved in the argument. But, the conclusion is the same: 0 = -1. Is the stranger right? Has he finally demonstrated that all you learned is suspect and contradictory? Or can you, using your best mathematical might, find the error in this new version of his argument?

/4 /4

/6 /6

/4 /4

/6 /6

sin tan

cos

1 sin

costan sec cos

tan

ta

xxdx dx

x

u dv xdxx

du x xdx v x

xdx uv vdu

/4 /4

/6 /6

n 1 tan

0 1

xdx xdx

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Resources

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Hippo Campus – Integration by Parts Integration by Parts: The Basics – A YouTube Video Interactive Math – Integration by Parts Visual Calculus – Integration by Parts

http://classic.hippocampus.org/course_locator?skinPath=http://classic.hippocampus.org/hippocampus.skins/default&course=General+Calculus+II&lesson=46&topic=1&topicTitle=Integration+by+parts+formula

http://youtu.be/815s-YB9X44



http://www.intmath.com/methods-integration/7-integration-by-parts.php

http://archives.math.utk.edu/visual.calculus/4/int_by_parts.3/int1.html

chapter 7 – techniques of integration 7.1 integration by parts 1erickson

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