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DUM20132 ENGINEERING MATHEMATICS 2
UNIT 4: INTEGRATION

(g) ∫( 3 + 2)(2 − 1) (h) ∫ 3 3−3 2+4
2

2. Find the following indefinite integral.

(a) ∫ 3 (b) ∫ 1
2

(c) ∫ − 5 (d) ∫ 7
2

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DUM20132 ENGINEERING MATHEMATICS 2
UNIT 4: INTEGRATION

3 (f) ∫ −5
+1
(e) ∫ 2

3

(g) ∫ 4 (h) ∫ −10
3 +2 6−5

3. Find the following indefinite integral. (b) ∫( − 9)8
(a) ∫( + 1)3

99

(c) ∫(4 + 7)5 DUM20132 ENGINEERING MATHEMATICS 2
UNIT 4: INTEGRATION

(d) ∫(3 − 8)6

(e) ∫ 2(4 − 3 )−6 (f) ∫ 5
(4−3 )3

100

4. Find the following indefinite integral. DUM20132 ENGINEERING MATHEMATICS 2
(a) ∫ 4 UNIT 4: INTEGRATION

(b) ∫ −5

(c) ∫ 31 (d) ∫ 4 −41

(d) ∫(2 2 − −3 ) (e) ∫( + 1 −2 )
2

101

5. Find the following indefinite integral. DUM20132 ENGINEERING MATHEMATICS 2
(a) ∫ 3   UNIT 4: INTEGRATION

(b) ∫ 3 4  

(c) ∫ 25 (d) ∫ 2
3

(d) ∫ 1 6 (e) ∫ −4 4  
3

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DUM20132 ENGINEERING MATHEMATICS 2
UNIT 4: INTEGRATION

6. Find the following indefinite integral.

(a) ∫ 3 2+2 (b) ∫ 9 +
3+2 −7 + 9

7. Find the following indefinite integral.

(a) ∫ 1 (b) ∫ 5
9+ 2 2+4

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DUM20132 ENGINEERING MATHEMATICS 2
UNIT 4: INTEGRATION

(c) ∫ 1 (d) ∫ 3
√36− 2 √16− 2

(e) ∫ 1 (f) ∫ 1
25− 2 81− 2

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DUM20132 ENGINEERING MATHEMATICS 2
UNIT 4: INTEGRATION

UNIT LEARNING After completing the unit, students should be able to:
OUTCOMES 3. Integrate composite functions byusing substitution method.
4. Integrate product of functionsusing integration by part.
5. Integrate quotient of functions using partial fraction method.

4.3 INTEGRATION

To integrate functions which are not in standard form, we need to use a few other
techniques to make the process of integration easier. There are three techniques:

4.3.1 Integration by Substitution
4.3.2 Integration by Part
4.3.3 Integration by Partial Fraction

4.3.1 INTEGRATION BY SUBTITUTION

∫ ( ( )) ′( ) = ( ( )) +

OR

∫ ( ) = ( ) +

STEP: where = ( ) and = ′( )

1. Choose u = g(x)
Find du = g'(x)
2.
dx
3. Substitute u = g(x), du = g ‘(x) dx
4. Solve the integration
5. Substitute u with g(x) therefore the final answer is in terms of x.

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DUM20132 ENGINEERING MATHEMATICS 2
UNIT 4: INTEGRATION

Example :

Evaluate the following using the appropriate substitution:

(a) ∫ 2 2

Solution :

Let = 2 ⇒ = 2
∴ ∫ 2 2 = ∫ 22 = ∫ = + = 2 +

 ( )(b) 3x2 x3 − 1 dx

Solution :

Let u = (x3 − 1)  du = 3x 2 dx

 3x2 (x3 − 1)3 dx = u du

= u2 + c +c
2

= (x3 − 1)2
2

(c) ∫

Solution :

Let = ⇒ =
∴ ∫ = ∫ ∫ = + = +

(d) ∫


Solution :

Let = ⇒ = 1



2 ( )2
∴ ∫ = ∫ = 2 + = 2 +

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DUM20132 ENGINEERING MATHEMATICS 2
UNIT 4: INTEGRATION

(e) ∫(4 + 5)7

Solution :

From the formula (Substitution Type I): ∫(4 + 5)7 = (4 +5)8 = (4 +5)8 +
4(8) 32

or using,

INTEGRATION BY SUBSTITUTION

= 4 + 5 ⇒ = 4 ⇒ =
4
8 (4 + 5)8
∴ ∫(4 + 5)7 = ∫ 7 = 1 (8) + = 32 +
4 4

(f) ∫ 1
4 −5

Solution :

From the formula (Substitution Type II): ∫ 1 = 1 |4 − 5| +
4 −5 4

or using,

INTEGRATION BY SUBSTITUTION

= 4 − 5 ⇒ = 4 ⇒ =
4
1 1 1 1
∴ ∫ 4 − 5 = ∫ ( ) 4 = 4 + = 4 |4 − 5| +

(g) ∫
+

Solution :

From the formula (Substitution Type III):

Let ( ) = 1 + ⇒ ′( ) =

∴ ∫ = |1 + | +
1+

or using,

INTEGRATION BY SUBSTITUTION

= 1 + ⇒ =

1 + = |1 + | +
∴ ∫ 1 + = ∫ =

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DUM20132 ENGINEERING MATHEMATICS 2
UNIT 4: INTEGRATION

EXERCISE 4.3.1

1. Evaluate the following using the appropriate substitution.
(a) ∫ 3 2 3  

(b) ∫ 2  

(c) ∫ 2 ( 2 + 3)4  
(d) ∫ ( 2 − 3)5

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DUM20132 ENGINEERING MATHEMATICS 2
UNIT 4: INTEGRATION

(e) ∫ 2√ 3 + 5

(f) ∫ 2  
√ 2+1

(g) ∫ 3 2  
√ 3−2

(h) ∫ 2

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DUM20132 ENGINEERING MATHEMATICS 2
UNIT 4: INTEGRATION

(i) ∫ +2  
2+4 +5

(j) ∫ 5  
5 −3

(k) ∫
(1− )3

(l) ∫ 1 2


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DUM20132 ENGINEERING MATHEMATICS 2
UNIT 4: INTEGRATION

TUTORIAL 4.3.1

1. Evaluate the following using the appropriate substitution.

(a) ∫ 9 2 (1 − 3 1

)2

(b) ∫ 2(1 + 3)4

(c) ∫
√1−4 2

1

(d) ∫ (1 − 2)2

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DUM20132 ENGINEERING MATHEMATICS 2
UNIT 4: INTEGRATION

(e) ∫ 4 (2 2 − 3)6

(f) ∫
3−

(g) ∫ 4 +6
( 2+3 +7)4

(h) ∫ 3 (5 4)

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DUM20132 ENGINEERING MATHEMATICS 2
UNIT 4: INTEGRATION

4.3.2 INTEGRATION BY PARTS

∫ = − ∫

The priority of choosing u :

L = Logarithmic function such as ln(x), log(x)
I = Inverse Trigonometric Function such as sin-1(x), cos-1(x), tan-1(x)
A = Algebraic Function such as x, x2, x3
T = Trigonometric Funtion such as sin(x), cos(x), tan (x)
E = Exponential Function such as ex, 3x

STEP: Choose , differentiate ⇒ and find

1.
2.
3. Choose , integrate and find
4. Substitute into the formula : − ∫

Simplify and solve the integration.

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DUM20132 ENGINEERING MATHEMATICS 2
UNIT 4: INTEGRATION

Example :

Evaluate the following.

(a) ∫ −3 = and = −3

Solution :

Choose:

= 1 ⇒ = ∫ 1 = ∫ −3 ⇒ = − −3
3


∴ ∫ = − ∫

= (− −33 ) − ∫ − −3
3

= − −3 − −3 +

39

(b) ∫

Solution :

Choose: = and =

= 1 ⇒ = 1 ∫ 1 = ∫ ⇒ = 2
2

∴ ∫ = − ∫

= ( 2) − ∫ 2 (1 )
2
2
= 2 − ∫
22

= 2 − 2 +
24

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DUM20132 ENGINEERING MATHEMATICS 2
UNIT 4: INTEGRATION

(c) ∫

Solution :

Choose: = and =
∫ 1 = ∫ ⇒ =
= 1 ⇒ =



∴ ∫ = − ∫
= − ∫
= − +

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DUM20132 ENGINEERING MATHEMATICS 2
UNIT 4: INTEGRATION

EXERCISE 4.3.2

1. Evaluate the following using the appropriate substitution.
(a) ∫ 2

(b) ∫ 2

(c) ∫ −

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DUM20132 ENGINEERING MATHEMATICS 2
UNIT 4: INTEGRATION

(d) ∫
(e) ∫ 4
(f) ∫ −2
(g) ∫ 2

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DUM20132 ENGINEERING MATHEMATICS 2
UNIT 4: INTEGRATION

TUTORIAL 4.3.2

1. Evaluate the following using the appropriate substitution.
(a) ∫

(b) ∫ −2

(c) ∫ 3

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DUM20132 ENGINEERING MATHEMATICS 2
UNIT 4: INTEGRATION

(d) ∫ 2

(e) ∫ 3

(f) ∫ 2

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DUM20132 ENGINEERING MATHEMATICS 2
UNIT 4: INTEGRATION

4.3.3 INTEGRATION BY PARTIAL FRACTION

Denominator Expression Form of Partial Fractions
Linear factor
( )
( + )( + ) ( + ) + ( + )

Example :

Find

(a) ∫ 1
( −2)( −3)

Solution :

1 = +
( −2)( −3) −2 −3

= ( −3)+ ( −2)

( −2)( −3)

∴ 1 = ( − 3) + ( − 2)

Using cover-up method,
Let = 3; 1 = (3 − 3) + (3 − 2)

1 = 0 +
= 1

Let = 2; 1 = (2 − 3) + (2 − 2)
1 = −
= −1

∫ 1 = ∫ −1 + 1
( −2)( −3) −2 −3

= − | − 2| + | − 3| +

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DUM20132 ENGINEERING MATHEMATICS 2
UNIT 4: INTEGRATION

(b) ∫ 3
2−4

Solution :

∫ 3 = ∫ 3
2−4 ( −2)( +2)

3 = +

( −2)( +2) −2 +2

= ( +2)+ ( −2)

( −2)( +2)

∴ 3 = ( + 2) + ( − 2)

Using cover-up method,

Let = 2; 3(2) = (2 + 2) + (2 − 2)

6 = 4
= 3

2

Let = −2; 3(−2) = (−2 + 2) + (−2 − 2)

−6 = −4
= 3

2

∫ 3 = ∫ 3 + 3
( −2)( +2) 2( −2) 2( +2)
= 3 | − 2| + 3 | + 2| +

22

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DUM20132 ENGINEERING MATHEMATICS 2
UNIT 4: INTEGRATION

EXERCISE 4.3.3

Express the functions in each of the following integrals in partial fractions and hence perform the integration.

(a) ∫ 1
( +5)( +6)

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DUM20132 ENGINEERING MATHEMATICS 2
UNIT 4: INTEGRATION

(b) ∫ 10
( −4)(2+5 )

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DUM20132 ENGINEERING MATHEMATICS 2
UNIT 4: INTEGRATION

(c) ∫ 3
2− −2

124

DUM20132 ENGINEERING MATHEMATICS 2
UNIT 4: INTEGRATION

(d) ∫ 2 +3
2−9

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DUM20132 ENGINEERING MATHEMATICS 2
UNIT 4: INTEGRATION

TUTORIAL 4.3.3

Express the functions in each of the following integrals in partial fractions and hence perform the integration.

(a) ∫ 6
( +1)( −2)

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DUM20132 ENGINEERING MATHEMATICS 2
UNIT 4: INTEGRATION

(b) ∫ 3 +2
( −1)(1−2 )

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DUM20132 ENGINEERING MATHEMATICS 2
UNIT 4: INTEGRATION

(c) ∫ +1  
2−25

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DUM20132 ENGINEERING MATHEMATICS 2
UNIT 4: INTEGRATION

(d) ∫ 2 +1
2+5 +6

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DUM20132 ENGINEERING MATHEMATICS 2
UNIT 4: INTEGRATION

UNIT LEARNING After completing the unit, students should be able to:
OUTCOMES 6. Evaluate the definite integraIs.

4.4 DEFINITE INTEGRAL

https://www.mathsisfun.com/calculus/integration-definite.html



∫ ( ) = ( ) − ( )



SOME PROPERTIES OF DEFINITE INTEGRAL

ba

(i)  f (x) dx = − f (x) dx

ab

a

(ii)  f (x) dx = 0

a

mb b

(iii)  f (x) dx +  f (x) dx =  f (x) dx

am a

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DUM20132 ENGINEERING MATHEMATICS 2
UNIT 4: INTEGRATION

Example :

Evaluate.

(a) 3

Solution : ∫ ( 2 + 3)

1

∫13( 2 + 3) = [ 3 + 3

3 3 ]

1
= [33 + 3(3)] − [13 + 3(1)]
33

27 1
= [ 3 + 9] − [3 + 3]

= 18 + 10

3

= 21 1 @ 64

33

(b)

Solution : ∫  

− 2

∫− 2   = [−cos ]− 2
= [−cos( )] − [−cos (− )]

2

= [1] − [0]
=1

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DUM20132 ENGINEERING MATHEMATICS 2
UNIT 4: INTEGRATION

EXERCISE 4.4
Evaluate the following.
(a)

2

∫ 6 2

0

(b)

3

1
∫ 2

2

(c)

25

∫ √

1

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DUM20132 ENGINEERING MATHEMATICS 2
UNIT 4: INTEGRATION

(d)

2

∫(1 + 2 + 2)

−1

(e)

3

∫ ( 4 + 6 2 + 9) 

1

(f)

6

∫ ( 2 + 1)

−1

(g)


2

∫ 2  

− 2

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DUM20132 ENGINEERING MATHEMATICS 2
UNIT 4: INTEGRATION

TUTORIAL 4.4
Evaluate the following.
(a)

2

∫ (3 − 3) 

0

(b)

2

∫ ( + 2 − 2) 

−1

(c)

9

∫ 2 √  

4

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DUM20132 ENGINEERING MATHEMATICS 2
UNIT 4: INTEGRATION

(d)

0

∫ ( + 1)( + 2)

−2

(e)

3 2 − 1
∫ 2

1

(f)

⁄2

∫ 2

0

135