CHAPTER 4 INTEGRATION - · PDF fileBA201 ENGINEERING MATHEMATICS 2 2012 87 CHAPTER 4 INTEGRATION 4.1 INTRODUCTION TO INTEGRATION Why do we need

BA201 ENGINEERING MATHEMATICS 2 2012

87

CHAPTER 4 INTEGRATION

4.1 INTRODUCTION TO INTEGRATION

Why do we need to study Integration?

The Petronas Towers

of Kuala Lumpur

Often we know the relationship involving the rate of change of two variables, but we may need to know the direct relationship between the two variables. For example, we may know the velocity of an object at a particular time, but we may want to know the position of the object at that time.

To find this direct relationship, we need to use the process which is opposite to differentiation. This is called integration (or antidifferentiation).

The processes of integration are used in many applications.

The Petronas Towers in Kuala Lumpur experience high forces due to winds. Integration was used to design the building for strength.


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Sydney Opera House

The Sydney Opera House is a very unusual design based on slices out of a ball. Many differential equations (one type of integration) were solved in the design of this building.

Wine cask

Historically, one of the first uses of integration was in finding the volumes of wine-casks (which have a curved surface).

Other uses of integration include finding areas under curved surfaces, centres of mass, displacement and velocity, fluid flow, modelling the behaviour of objects under stress, etc.


89

4.1.1 Antiderivatives and The Indefinite Integral

Integration is the inverse process of differentiation.

'dy

f xdx

'f x dx y c

Two types of integrals

Indefinite integral

Definite integral

22d

xdx

22x c 4x

4xdx

C is arbitrary constant

integrand

The term dx identifies

x as the variable of

integration

integral


90

4.2 Indefinite Integral

4.2.1. Integration Of Constants nax

1

, 1 ( )1

nn ax

ax dx c n power rulen

Examples

Evaluate the following equation;

i. 24x dx

ii. 3 dp

iii. s ds

iv. 3

1dx

x


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Exercise

a) 1 dx

b) dx c) 3

4dx

d) 3

3dx

x

e) 2x dx f) 2

31

3x dx

g) 1

33x

h) 3

8dx

x

i) 10 x dx

4.2.2. Integration Of Summation & Subtraction

n m n max bx dx ax dx bx dx

1 1

1 1

n max bx

n m c


92

Example Find the derivative for each of the following functions.

a) 24 4 5x x dx

b) 26 9x x x dx

c) 2

2

46t dt

t

d) 2

x dxx

e) 2

1x dx

x

f) 1

2p dpp

g) 2

4

43x dx

x

h) 2x x dx i)

4

2 4x xdx

x


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4.2.3. INTEGRATION BY SUBSTITUTION

n

ax b dx , n

Step 1 : Let “u”

Step 2 : Differentiate “u”

Step 3 : Make as a subject

Step 4: Integrate the “u” equation

1

1

1

1

1

nn

n

n

duax b dx u

a

uc

n a

uc

n a

Step 5 : Substitute back “u”

So the final answer is

1

1

nax b

n a

Examples:

Evaluate the following equation:

Find 4

3 5 2x dx


94

Example

Find the following integration by substitution:

a). dxx 5)72( b). dx

x 4)85(

1

C). dxx23 d).

dxx 3

2

)76(


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4.2.4 INTEGRATION OF 1

xOR LOGARITHM FUNCTION

11dx x dx

x

, n=-1

ln xc

dx

dx

1 1

lndx ax b cax b a

Example 1:

i. 1

4dx

x

ii. 1

12 4 9x dx

iii. 1

3 2dt

t

iv. 128 6 2 1x x dx


96

Exercise:

4.2.5. INTEGRATION OF EXPONENTIAL FUNCTION

x xe dx e c

1ax axe dx e ca

ax bax b e

e dx cd

ax bdx

a) 1

5dx

x

b) 1

2 1dx

x c) 2

2

3

xdx

x d) 2

31

xdx

x


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Example:

a) 5xe dx

b) 1

2x

e dx c) dxe x2

4

d) 3 13

xe dx

e) 3 2 23te t dt f)

21 x

x

edx

e

4.2.5. INTEGRATION OF TRIGONOMETRIC FUNCTION

2

cossin

sincos

tansec

ax bax b dx c

dax b

dx

ax bax b dx c

dax b

dx

ax bax b dx c

dax b

dx


98

2

1sin cos

1cos sin

1sec tan

ax dx ax ca

ax dx ax ca

ax dx ax ca

Example:

a) sin 3 3x dx

b) 2 2cos 2 4 sec 2x x dx c) sin 2x dx

d) 22sec 3x dx

e) cos 2sinx x dx f) ∫


99

Challenge:

1. 2

33

1

3

pdp

p p

2. 7

2 31k k dk

3. tan x dx

4.

3

23

5

1

x

x

edx

e


100

Solve the integration below by using substitution:

a). 1

4 6dx

x b).

3

5

4 5 4dx

x c).

43 xx e dx

d). 5

4

x

xdx

e

e). sin 2 cos2x x dx f). 3 42 sinx x dx


101

Integrate for 2sin mx dx and 2cos mx dx

2 1sin 1 cos 2

2mx dx mx dx

2 1cos 1 cos 2

2mx dx mx dx

Example:

a) 2sin 2x dx

b) 2cos 3x dx


102

c) dxx

3sin4 2

4.3 DEFINITE INTEGRAL

4.3.1 Definite Integral

If f x dx F x c

So b

a

f x dx F b F a

Examples:

1. dx2

0

5


103

2.

2

1

2 )32( dxx

3. 4

4 2

2

xe dx

4. 4

2

3

sec 3z dz


104

5. 5

3

3

2 4dx

x

Exercise:

1. 4

2

0

4 3x dx

2. 4

0

3 4x dx

3. 9

1

3 xdx

x

4. 2

3

2

1 2x x dx


105

4.3.2 DEFINITE INTEGRAL BY SUBSTITUTION METHOD

1. Find the value of

a) 4

3

2

4 2x dx

b) 32

6 42

0

2x x

x e dx


106

c) 4

2

0

sec 3x dx

d) 2

0.2

0

3 te dt


107

Exercise:

1. 1

2

2

4 1x dx

2. 2

52

1

2 3x x dx

3.

1 2

23

0 1

xdx

x


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4.4 PROPERTIES OF THE DEFINITE INTEGRAL

a) b b

a a

kf x dx k f x dx

b) 1 2 1 2

b b b

a a a

f x f x dx f x dx f x dx

c) b a

a b

f x dx f x dx

d) b c c

a b a

f x dx f x dx f x dx

Examples:

i. Given that 6

3

4f x dx , find the value of:

a) 6

3

3 f x dx

b) 6

3

2f x dx


109

c) 3

6

f x dx

d) 4 6

3 4

f x dx f x dx

e) k if 6

3

12f x kx dx


110

Exercise:

Given that 2

1

4f x dx and 3

2

6f x dx , evaluate each of following

a) 1

2

f x dx

b) 2

1

5 f x dx

c) 3

1

f x dx

d) 2

1

4 3f x dx


111

POLITEKNIK KOTA BHARU JABATAN MATEMATIK, SAINS DAN KOMPUTER

BA 201 ENGINEERING MATHEMATICS 2

PAST YEAR FINAL EXAMINATION QUESTIONS

JULAI 2010

Integrate this following:

i. ∫

ii. ∫( )

iii. ∫

iv. ∫

( )

v. ∫

vi. ∫

( )

JANUARI 2010

a) Integrate this following:

i. ∫( ) ( )

ii. ∫

iii. ∫

b) Evaluate the integrals:

i. ∫ ( )

ii. ∫ (

)

JULAI 2009

Integrate:

a) ∫( )

b) ∫

c) ∫( )

d) ∫ ( )

e) ∫ ( )( )

JANUARI 2009

a) Solve the following of indefinite

integrals:

i. ∫

ii. ∫

iii. ∫

iv. ∫ ( )

b) Solve the following of definite

integrals:

i. ∫ ( )

ii. ∫ ( )

JULAI 2008

Integrate:

i. ∫

ii. ∫

iii. ∫ ( )

iv. ∫ ( )( )

v. ∫ ( )


112

JANUARI 2006

a) Solve the following of indefinite

integrals:

i. ∫( )

ii. ∫( )

iii. ∫

iv. ∫ ( )

b) Solve the following of definite

integrals:

i. ∫ ( )

ii. ∫ ( )

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CHAPTER 4 INTEGRATION - · PDF fileBA201 ENGINEERING MATHEMATICS 2 2012 87 CHAPTER 4 INTEGRATION 4.1 INTRODUCTION TO INTEGRATION Why do we need