chapter 7 – techniques of integration 7.5 strategy for integration 7.6 integration using tables...

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Chapter 7 – Techniques of Integration

7.5 Strategy for Integration

7.6 Integration Using Tables and Computers

7.5 Strategy for Integration 7.6 Integration Using Tables and Computers

Erickson

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When do we use it?


Differentiation is an easier straightforward process as opposed to Integration.

Integration is a more challenging process.

When integrating we will have to use algebra manipulation, substitution, integration by parts , partial fractions, and many times all of the above.

It is important that you memorize the Table of Integration Formulas on page 495 in your book.

Erickson

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Strategy for integration


Step 1: Simplify the integrand if possible.

Step 2: Look for an obvious substitution.

Step 3: Classify the Integrand according to its form

Step 4: Try again (substitution, parts, manipulations)

The following slides will illustrate some examples of these steps.

Erickson

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Step 1: Simplify the integrand Use algebraic manipulations to simplify the integrand

Examples

52 3 22x x x dx x x dx

2

cot cos

csc sin

x xdxx x

2sin cos sinx dx x xdx

22 4 3 23 7 6 14 42 14x x dx x x x x dx Erickson7.5 Strategy for Integration

7.6 Integration Using Tables and Computers

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Step 2: Use SubstitutionTry to find a function whose derivative occurs in the integrand. Examples

32

3 2

5 5 133 1 1 1

5 5 13 5 55(3 1)

u x xx du dudx

x x u udu x dx

3 3 3coscos sin

sin

u xx xdx u du u du

du x dx

Erickson7.5 Strategy for Integration 7.6 Integration Using Tables and Computers

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Step 3: Classify and Integral by Its Form


Trig functions - If our function is a product of powers of the trig functions then use trig substitution.

Rational functions - If our function is a rational functions then use partial fractions.

Integration by parts - If our function is a product of a power of x (or a polynomial) and a transcendental function then use integration by parts.

Radicals – If our function is a radical, we have certain options If occurs we use trig substitution

If occurs we use the rationalizing substitution

2 2x a

n ax b

nu ax b

Erickson

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Step 4: Try Again If the first three steps have not produced the

answer, remember that there are basically only two methods of integration: substitution and parts.

Try substitution. Even if no substitution is obvious (Step 2), try again.

Try parts. Although integration by parts is used most of the time on products (step 3), we can use it on single functions that are inverse functions.

Manipulate the integrand. Try rationalizing the denominator or trig identities.

Relate the problem to previous problems. Use several methods. Sometimes two or three methods

are needed to evaluate an integral.


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Can we integrate all continuous functions?


No. There are some functions that we can’t integrate in terms of functions we know. We will learn in chapter 11 how to express these functions as an infinite series. Some integrals we can’t evaluate.

2 3

2

1

1

lnsin

sin

cos

x

x

x

e dx x dx

edx dxx x

xx dx dx

x

e dx

Erickson

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Examples – Page 499 Evaluate the integral.

NIB.

14.

NIB.

20.

34.

44.

3tan d 3

21

xdx

x

322

20 1

xdx

x

2e dx

/2

/4

1 4cot

4 cot

xdxx

1 xe dx


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Examples – Page 500 Evaluate the integral.

NIB.

75.

1

2

tan xdx

x

2

1

2 4dx

x x


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Examples – Page 504

NIB.

11.

12.

24.

26.

30.

Use the table of integrals on Reference Pages 6-10 to evaluate the integral. State the formula you used.3

2 22

1

4 7dx

x x

02

1

tt e dt

2 3csc h 1x x dx

6sin 2x dx

14

0

xx e dx

2 2

2

sec tan

9 tand


chapter 7 – techniques of integration 7.5 strategy for integration 7.6 integration using tables...

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