mac101_chap6

8/3/2019 MAC101_Chap6

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Advanced mathematics I: Calculus

Chapter 6

Essential Calculus, James StewartDepartment of Mathematics, FPT University

Techniques of Integration

8/3/2019 MAC101_Chap6


Introduction

Topics will be covered:

• Trigonometric Integrals and Substitutions

• Partial Fractions

• Approximate Integration

• Improper Integral

8/3/2019 MAC101_Chap6


Trigonometric Integrals

Trigonometric integrals of sin x and cos x

• If m or n is odd then use substitution u = sin x or u =cos x, and the identity

• If both m and n are even then use half-angleidentities

dx x xnm

cossin

1cossin22 x x

2

2cos1cos,

2

2cos1sin

22 x x

x x

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Trigonometric integrals of sin mx and cos nx

Use identities:

dxnxmxdxnxmxdxnxmx coscos,sinsin,cossin

])cos()[cos(2

1coscos

])cos()[cos(

2

1sinsin

])sin()[sin(2

1cossin

xnm xnmnxmx

xnm xnmnxmx

xnm xnmnxmx

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Example

Calculate

(a)

(b)

(c)

dx x )2(sin 4

dx x x 52 cossin

dx x x 5cos2sin

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Trigonometric substitution

Expression Substitution

x = a tan(θ ) ( –π /2 < θ < π /2)

x = a sin(θ ) ( –π/2 ≤ θ ≤ π /2) or

x = a cos(θ ) (0 ≤ θ ≤ π) x = a / cos(θ ) (θ [0,π /2)[π,3π /2))

22 xa

22 xa

22a x

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Example

Let x = 2sin(θ )

4 ?)4(

2

2 / 32

x

x

dx I

)(cos8))(sin1(4)4( 32 / 3

22 / 32 x

d dx )cos(2

C d d

I )tan(4

1

)(cos4

1

)(cos8

)cos(223

θ

x2

24 x

24)tan(

2)sin(

x

x x

C

x

x I

244

1

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Example

Calculate

(a)

(b)

dx x x 4

1

22

dx x 1

1

2

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Partial fractions

Theorem

•

If the function f(x) has the form

then ∫f (x )dx can be computed using partial fractions .

n

n

m

m

n

m

xa xaa

xb xbb

xQ

xP x f

...

...

)(

)()(

10

10

If m<n then

k k

n

m

cbx x

B Ax

a x

A

xQ

xP

)()()(

)(2

• If m≥n then

with degree R < degree T.

)(

)()()(

)(

xT

x R xS xQ

xP

n

m

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Partial fractions

Theorem

To find partial fractions for P m (x )/ Q n (x ) with m <n:

• Factorize Q n (x ) via (x –a )k and (x 2+bx +c )k

• Each factor (x –

a )

k

corresponds to

• Each factor (x 2+bx +c )k corresponds to

k

k

a x

A

a x

A

a x

A

)(...

)()(2

21

k

k k

cbx x

B x A

cbx x

B x A

cbx x

B x A

)(...

)()(222

22

2

11

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Partial fractions

Example

?)1( 2

x x

dx I

22 )1(1)1(

1

x

C

x

B

x

A

x x

x x x B x )1()1(1 2

11 C x

C x

x x

dx x x x

I

1

1|1|ln||ln

)1( 1111 2

Cx x x B x A )1()1(1 2

10 Ax

12 B x

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Partial fractions

Example

?)1( 2 x x

dx I

1)1(

12

221

2

x

B x A

x

A

x x

10 1 A x

00 B x

C x x x

xdx

x

dx I

|1|ln

2

1||ln

1

2

2

)()1(1 22

2

1 B x A x x A

)()1(1 22

2 B x A x x 22 B x A x

12 A

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Approximate integration

Left endpoint Method

b

a

n x f x f x f xdx x f )](...)()([)( 110

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Right endpoint Method

b

a

n x f x f x f xdx x f )](...)()([)(21

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Midpoint Method

f ( x)

∆x

1 x 2 x3 x

n x

)](...)()([)( 21 n

b

a

x f x f x f xdx x f

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Trapezoidal Method

)]()([2

...)]()([2

)( 110 nn

b

a

x f x f x

x f x f x

dx x f

)]()(2...)(2)([2

110 nn x f x f x f x f x

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Simpson Method

)]()(4)([3

...)]()(4)([3

)( 12210 nnn

b

a

x f x f x f x x f x f x f xdx x f

)]()(4...)(2)(4)([3

1210 nn x f x f x f x f x f x

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Example

2

1(1/ ) x dxApproximate the integral

with n = 8, using:

a. Left/Right endpoints

b. Midpoints

c. Trapezoidal method

d. Simpson method

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Estimate error for Midpoint and Trapezoidal method

• Suppose | f’’ (x ) | ≤ K for a ≤ x ≤ b .

• If E T and E M are the errors in the Trapezoidal andMidpoint Rules, then

3 3

2 2

( ) ( )and

12 24T M

K b a K b a E E

n n

i i i

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Estimate error for Simpson method

• Suppose | f(4)(x) | ≤ K for a ≤ x ≤ b .

• If E S is the error in the Simpson method, then

5

4( )180

s K b a E n

A i i i

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Example

How large should we take n in order to guarantee

that the Trapezoidal, Midpoint Rule, Simpson rule

approximations for

are accurate to within 0.0001?

2

1 (1/ ) x dx

A i i i

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| f’’ ( x ) | ≤ 2 for 1 ≤ x ≤ 2

Accuracy to within 0.0001 means that error < 0.0001Trapezoidal: Choose smallest n so that:

n = 41

Midpoint:

n = 30

Simpson:

3

2

2(1)0.0001

12n

3

2

2(1)

0.000124n (4)

5

24( ) 24 f x

x

5

4

24(1)0.0001

180n n = 7

I i l

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Improper integrals

Definition (Improper Integral of Type 1)

Suppose exists for any N ≥ a

The improper integral of f (x ) with x from a to +∞ is:

If this limit exists and is finite, we say that the integration

converges, otherwise it diverges.

N

a

dx x f )(

a

dx x f )( ,)(lim

N

a N

dx x f

I i t l

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Improper integrals

Example

Let a > 0. Investigate the convergence of the improper integrals:

aaax

dx

x

dx

x

dx,,

23 / 1

Theorem

converges if p>1 and diverges if p 1.

a

p x

dx

I i t l

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Improper integrals

Definition

Suppose exists for any –N ≤ a

Improper integral of f (x ) with x from -∞ to a is:

If this limit exists and is finite, we say that integrationconverges, otherwise, it diverges.

a

N

dx x f )(

a

dx x f )(

a

N N

dx x f )(lim

I i t l

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Improper integrals

Definition

Suppose that and exist for any N > 0.

The improper integral of f (x ) with x from -∞ to +∞ is:

In the case both limits above exist, we say that integration

converges, otherwise, it diverges.

0

)(N

dx x f

:)( dx x f

N

dx x f

0

)(

N

N M

M

dx x f dx x f

0

0

)(lim)(lim

I i t l

8/3/2019 MAC101_Chap6


Improper integrals

Example

Investigate the convergence of the improper integrals:

(a)

(b)

0

23

dxe xx

dxe xx3

2

I i t l

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Improper integrals

Definition (Improper Integral of type II )

Let f (x ) be a function such that:

• f(x) is unbounded at a,

• exists for all ε > 0.

The improper integral of f (x ) over [a, b] is the followinglimit:

If this limit exists and is finite, we say that integrationconverges, otherwise, it diverges.

b

a

dx x f

)(

b

t

at

b

a

dx x f dx x f )(lim)(

I i t l

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Improper integrals

Definition


• f(x) is unbounded at b,

• exists for all ε > 0

The improper integral of f (x ) over [a, b] is the followinglimit:

If this limit exists and is finite, we say that the integralconverges, otherwise it diverges.

b

a

dx x f )(

t

abt

b

a

dx x f dx x f )(lim)(

Improper integrals

8/3/2019 MAC101_Chap6


Improper integrals

Example

Let b> 2. Investigate the convergence of the improper integrals:

bbb

x

dx

x

dx

x

dx

22

2

2

3 / 12

,)2(

,)2(

Theorem

diverges if p 1 and converges if p<1.

b

a

p

a x

dx

)(

Improper integrals

8/3/2019 MAC101_Chap6


Improper integrals

Definition


• f(x) is unbounded at c [a , b ],

• and exist for all ε > 0.

The improper integral of f (x ) over [a, b] is:

If this limit exists and is finite, we say that integralconverges, otherwise it diverges.

c

a

dx x f )(

b

c

dx x f

)(

t

a

b

t ct ct

b

a

dx x f dx x f dx x f )(lim)(lim)(

Improper integrals

8/3/2019 MAC101_Chap6


Improper integrals

Example

Investigate the convergence of the improper integral:

3

01 x

dx

Improper integrals

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Improper integrals

Comparison Theorem

Suppose that f (x ), g (x ) are integrable functions over (a , ).If

f (x )≥g (x )≥0

for any x (a , ), we have:

• If converges then converges,

• If diverges then diverges.

a

dx x f )(

a

dx xg )(

adx x f )(

adx xg )(

Note: A similar theorem is true for Type II integrals

Improper integrals

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Improper integrals

Example

1

2|)cos(|

x dx x I Does converge?

We have,

and converges

→ I converges.

22

1|)cos(|

0 x x

x

1

2x

dx

Improper integrals

8/3/2019 MAC101_Chap6


Improper integrals

Example

Investigate the convergence of the improper integrals

(a)

(b)

0

2

dxex

0

1dx

x

ex

Summary

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Summary

We have studied:

-Trigonometric Integrals and Substitutions.-Partial Fractions-Approximate Integration-Improper Integral

Homework:

6.2: 14, 19, 33, 34, 41, 49, 54

6.3: 5, 23, 32, 34, 416.5: 5, 7, 18, 316.6: 2, 5, 7, 13, 25, 29, 41, 43, 45, 48,

mac101_chap6

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