TOPIC
TECHNIQUES OF INTEGRATION
TECHNIQUES OF INTEGRATION
1. Integration by parts
2. Integration by trigonometric substitution
3. Integration by miscellaneous substitution
4. Integration by partial fraction
TECHNIQUES OF INTEGRATION
4. Integration by partial fraction
DEFINITION
A rational function is a function which can be expressed as
the quotient of two polynomial functions. That is, a function H
is a rational function if
H x  
f x 
gx 
where both f(x) and g(x)
are polynomials. In general, we shall be concerned in
integrating expressions of the form:

f x 
g x 
dx
If the degree of f(x) is less than the degree of g(x), their
quotient is called proper fraction; otherwise, it is called
improper fraction. An improper rational function can be
expressed as the sum of a polynomial and a proper
rational function.
Thus, given a proper rational function:
x
2
3
 x
x
2
x 1
x 1
Every proper rational function can be expressed
as the sum of simpler fractions (partial fractions)
which may have a denominator which is of linear
or quadratic form.
The method of partial fractions is an algebraic
procedure of expressing a given rational function as a
sum of simpler fractions which is called the partial
fraction decomposition of the original rational function.
The rational function must be in its proper fraction form
to use the partial fraction method.
Four cases shall be considered.
Case 1. Distinct linear factor of the denominator
Case 2. Repeated linear factor of the denominator
Case 3. Distinct quadratic factor of the denominator
Case 4. Repeated quadratic factor of the denominator
Case 1. Distinct linear factor of the denominator
For each linear factor a i x  b i of the denominator, there
corresponds a partial fraction having that factor as the
denominator and a constant numerator.
f x 
That is,
gx 

A
a 1x  b 1

B
a2x  b2
 ... 
N
anx  bn
where A, B, …..N are constants to be determined
Thus,

f x 
dx 
gx 

A
a 1x  b 1
dx 

B
a2x  b2
dx  ... 

N
anx  bn
dx
EXAMPLE: Evaluate each integral.
1 .
x
3
x  4x  5x
3
2
dx
2 .
2
0
x  4x 1
2
 x  1  x
2
 2x  3

dx
Case 2. Repeated linear factor of the denominator
If the linear factor ax  b  appears as the denominator
of the rational function for each repeated linear factor
of the denominator, there corresponds a series of
partial fractions,
n
A
ax  b

B
ax
 b
2

C
ax
 b
3
 ... 
N
ax
 b
n
where A, B, C, …, N are constants to be determined.
The degree n of the repeated linear factor gives the
number of partial fractions in a series. Thus,

f(x)
g( x )
dx 

A
ax  b
dx 

B
ax
 b
2
dx 

C
ax
 b
3
dx  ... 

N
ax
 b
n
dx
EXAMPLE: Evaluate each integral.
1 .
y3
4y  4y  y
3
2
x 1
2
dy
2 .
x  3x  3x  x
5
4
3
2
dx
Case 3. Non-repeated quadratic factor of the
denominator
For each non-repeated irreducible quadratic factor
2
of the denominator ( ax  bx  c ) there corresponds
a partial fraction of the form.
f ( x)

A ( 2 a1 x1  b1 )  B
g ( x)
a1 x  b1 x  c1
2

C ( 2 a 2 x  b2 )  D
a 2 x  b2 x  c 2
2
 ... 
N ( 2 a n x  bn )  M
a n x  bn x  c n
2
where A, B, …..N are constants to be determined
Thus,

f ( x)
g ( x)
dx 

A ( 2 a1 x1  b1 )  B
a1 x  b1 x  c1
2


C ( 2 a 2 x  b2 )  D
a 2 x  b2 x  c 2
2
 ... 

N ( 2 a n x  bn )  M
a n x  bn x  c n
2
EXAMPLE: Evaluate each integral.
1 .
x
3
x  x 1
2
6x  3x  2
2
dx
2 .
( x  1)( x  x  1)
2
dx
Case 4. Repeated quadratic factor of the denominator
For each repeated irreducible quadratic factor
2
n
of the denominator ( ax  bx  c ) there corresponds
a partial fraction of the form.
f ( x)

A ( 2 ax  b )  B
ax  bx  c
2
g ( x)

C ( 2 ax  b )  D
( ax  bx  c )
2
2
N ( 2 ax  b )  M
 ... 
( ax  bx  c )
2
n
where A, B, …..N are constants to be determined
Thus,

f ( x)
g ( x)


A ( 2 ax  b )  B
ax  bx  c
2


C ( 2 ax  b )  D
( ax  bx  c )
2
2
 ... 

N ( 2 ax  b )  M
( ax  bx  c )
2
n
EXAMPLE: Evaluate each integral.
1 .
dx
x ( x  1)
2
2
x  2x  3x
5
2
2 .
3
( x  1)( x  1)
2
3
dx
Evaluate each integral.
1 .
2 .
12 x  18
( x  2 )( x  4 )( x  1)
( 2 x  1)
( x  2 )( x  3 )
2
dx
13 x  17
( 2 x  1)( x  4 )
2
2
4 .
x  2 x  3x
3
2
dx
x  x8
2
2
3 .
dx
6 x  23 x  9
dx
5 .
( 2 x  3 )( x  2 x  2 )
6 .. 
2
x2
( x  1)( x  1)
2
2
2
dx
dx