EEM3213

Topic 2: Integration

Learning Outcome
At the end of the chapter, students are able to apply various integration methods in
solving the single integration and multiple integration problems.

2.1 Indefinite and Definite Integrals

Basic Principles
b
Given that k and c are constants, the indefinite -  f ( x)dx is called a definite
integrals are as follow: a
integral
1.  kdx = kx + c
where a is called lower limit
2.  kf ( x)dx = k  f ( x)dx b is called upper limit
ax n +1
3.  ax dx =
n
+ c , n  −1
n +1
b

 f ( x)dx = F ( x)a = F (b) − F (a )

b

4.  (ax + b ) dx =
n (ax + b )n+1 + c , n  −1 a

(n + 1)a = - F (a ) − F (b)
5.   f ( x)  g ( x)dx =  f ( x)dx   g ( x)dx a
= -  f ( x)dx
b

Example 1
 x 3 − 3x 
 (x − 4) dx x (x + 1)dx
4
3 c)   dx 
2
1. Find a) b) dx d)
x  x 

1 3

 (2 x + 1)(x + 3)dx  6x + 2x
2
2. Evaluate a) b) dx
0 −3

1
 2.2 Integration of Trigonometric, Exponential and Logarithmic Functions
1. Integration of Trigonometric Functions

cos(mx + b )
•  sin (mx + b)dx = − m
+c
sin (mx + b )
•  cos(mx + b)dx = m + c
•
sec(mx + b )
•  sec(mx + b) tan(mx + b)dx = m
+c
cot (mx + b )
•  cos ec (mx + b)dx = − m + c
2

 sin (mx + b)dx,  cos (mx + b)dx

2 2

 convert trigonometric function to double angle form before integrate.

 tan (mx + b)dx

2

 convert trigonometric function to identity form before integrate.

Example 2
1. Integrate the following with respect to x:
a) cos(2 − 5 x )
cos x 1
b) 2
c) sin 2 x d) cos 2 x e) tan 2 2 x
sin x 2

2. Evaluate the following:

 
3
  3
(sin 2 )2 d
a)  sin  3x + dx
  6
b) 
0
4
6

2
 2. Integration of Exponential Functions and Logarithmic Functions

e mx +b f ' ( x)
 e dx = 
mx + b
+c dx = ln f ( x) + c
m f ( x)

Example 3
Find the following:
0
1. a)  e 1− x
dx b)  2e
x ln 3
dx
−3

4 4
1 4 1
2. a)  2 x dx b)  − 1 − 2 x dx c) 2 − x − 2dx d)  tan 2 x dx

6
5
x x3
e)  x + 5 dx f)  x + 1dx
1

3
2.3 Integration Methods

- Integration by partial fractions

- Integration by parts
- Integration by substitution

2.3.1 Integration by partial fractions

Partial Fractions

Factors of the denominatorExample Partial Fractions

1. Linear 2x + 1 A B
+
(x + 3)(x − 1) x + 3 x −1
2.Quadratic that can’t be 2x + 1 A Bx + C
+ 2
factorized (x + 1)(x 2 + 3) x +1 x + 3
3. Repeated 2x + 1 A B C
+ +
(x + 1)(x + 3)2 x + 1 x + 3 ( x + 3)2

Remark
• The method of partial fractions only applies to proper rational functions where proper
rational functions exists when the degree of the numerator, P(x) , is less than that of
the denominator, Q(x) .
• Improper rational function can be integrated by performing a long division and
decompose the proper rational function into partial fractions.

Example 4
Evaluate exactly the following integrals
2x + 4 x3 + x2 + x −1
3
3
1. 
2
(2 x + 1)( x − 1)
dx 2.  3
x − 2 x 2
dx 3.  x 2 + 2 x + 2 dx

4
 2.3.2 Integration by parts

 dv   du   dv   du 
b b
   u dx = uv −   v dx a  u dx dx =uva − a  v dx dx
b
or
 dx   dx 

The priority of u is given to the function whose category occurs earlier in the “LIATE” list.
Logarithmic

Inverse trigonometric

Algebraic

Trigonometric

Exponential

Alternative method is tabular method for repeated integration by parts.

A Repeated Method For Repeated Integration By Parts

Integrals of the form
 p( x) f ( x)dx
where u = p(x)

Tabular Integration By Parts

Step 1: Differentiate p(x) repeatedly until you obtain 0, and list the results in the first column.
Step 2: Integrate f (x) repeatedly and list the results in the second column.
Step 3: Draw an arrow from each entry in the first column to the entry that is one row down in the
second column.
Step 4: Label the arrows with alternating + and – signs, starting with a +.
Step 5: For each arrow, form the product of the expressions at its tip and tail and then multiply that
product by +1 or -1 in accordance with the sign on the arrow. Add the results to obtain the
value of the integral.

Example 5
Solve the following integrals.

3

 (5x + 4)e dx 2.  ln (3x − 1) dx  x e dx  x sec

x 2 3x 2
1. 3. 4. x dx
0

5.

5
2.3.3 Integration by substitution

Example 6
Find the following integrals.

9
sin 2 x x2
1.  (cos 2 x )(3 − cos 2 x )dx; u = cos 2 x 2. 
6 x−5
dx

e2
1
3.  x(ln x )
e
2
dx 4.  cos 4 x sin xdx

6
2.4 Multiple Integrals

Integrals of functions of two or three variables are called multiple Integrals.

Suppose f ( x, y ) is continuous over the rectangle R where R = {( x, y ) : a  x  b, c  y  d } ,

The volume , V , of the solid bounded above by the graph z = f ( x, y ) over the rectangle R in
the xy - plane is denoted by V =  f ( x, y) dA .
R

Suppose f ( x, y ) is continuous on the region R . If R is the vertically simple region given by

b y2 ( x )
a  x  b, y1 ( x)  y  y 2 ( x) , then  f ( x, y) dA =   f ( x, y) dy dx
R a y1 ( x )

If R is the horizontally simple region given by c  y  d , x1 ( y )  x  x2 ( y ) , then

d x2 ( y )

 f ( x, y) dA =   f ( x, y) dx dy
R c x1 ( y )

7
Example 7
1. Evaluate the following.
 
2 2 1 x 3 y
a)   (cos x sin y ) dy dx b)   ( y - x ) dy dx c)  y 2 + 16 dx dy
0 0 0 x4 0 0

1 4
d)   sec x dx dy
0 tan −1 y

 (6 x + 2 y ) dA , where R is the region bounded by the parabola x = y 2 and

2
2. Evaluate
R
the straight line x + y = 2

2 4− 2 x 4− 2 x − y
3. Evaluate    6 xy dz dy dx .
0 0 0