3.1 Integration of rational functions (using partial fractions)

polynomial
polynomial

MOOC MAT438/ UiTM

What is a • A polynomial function is a function which involves only non-negative
polynomial integer powers or only positive integer exponents of a variable.
functions ??
• It is defined by its degree.

• The degree of a Polynomial with one variable is the largest exponent
of that variable.

The most common types of polynomial functions are:

Polynomial or Not? Degree Name Example
0 constant 7
Exponents : 0, 1, 2, … 3 −2 2 1
2 3 − 2 + 3 − 1 2 Linear 2x - 3
terms + 1 3 Quadratic x2 − 2x + 1
4 2x3 + x2 − 1
Cubic x4− 3x + 2
Quartic

MOOC MAT438/ UiTM

What is a • A rational function is the ratio of two polynomials P(x)
Rational and D(x) like this
functions ??
Except that D(x) cannot be zero (and anywhere that
D(x)=0 is undefined)
• It is just like a fraction, but with polynomials..

Rational or Not?

+ 3 numerator − 2 The top is not a polynomial
− 2 denominator (a square root of a
+ 1 variable is not allowed)

because it is a “ratio”
of two polynomials

MOOC MAT438/ UiTM

If the degree of No need to do
< a long division

Long Division the degree of
≥

Express into the sum denominator with denominator with denominator with
of Partial Fraction distinct linear irreducible repeated factors
factors
quadratic factors

Integrate using useful integration
formulas (or u-substitution or
inverse tangent)

MOOC MAT438/ UiTM

MOOC MAT438/ UiTM

If the degree of No need to do
< a long division

Long Division the degree of
≥

Express into the sum denominator with denominator with denominator with
of Partial Fraction distinct linear irreducible repeated factors
factors
quadratic factors

Integrate using useful integration
formulas (or u-substitution or
inverse tangent)

MOOC MAT438/ UiTM

2 4 − 7 2 + 7 + 1 2
+ 2 − 1 2 =
3 + − 3 + 2 2 4 + − 7 2 + 7 + 1

− ( 2 4 + − 6 2 + 4 )

− 2 + 3 +1

MOOC MAT438/ UiTM

At the end of this session, the students should
be able to divide the polynomial and write into
the form of


+

MOOC MAT438/ UiTM

when we need
to use long
division??

Degree 4

2 4 − 7 2 + 7 + 1
+ 2 − 1 2 Degree 3

Improper Fraction : Need long
The degree of the top is division!
greater than or equal to the

degree of the bottom

MOOC MAT438/ UiTM

2 4 − 7 2 + 7 + 1 2 4 − 7 2 + 7 + 1 Arrange the indices in
+ 2 − 1 2 = 3 − 3 + 2 descending order !

Expand the use a zero to
factors fill in the

missing term

Degree 3 quotient

2 4 − 7 2 + 7 + 1 divisor 2
+ 2 − 1 2 =
3 + − 3 + 2 2 4 + − 7 2 + 7 + 1 dividend

Stop division when the degree of − ( 2 4 + − 6 2 + 4 ) 2 4
remainder less than the degree of divisor 3 = 2
− 2 + 3 +1
remainder

2 4 − 7 2 + 7 + 1 quotient remainder Degree 2
+ 2 − 1 2 =
+ Proper Fraction :
2 4 − 7 2 + 7 + 1 The degree of the top is
+ 2 − 1 2 = 2 + divisor less than the degree of

− 2 + 3 + 1 Degree 2 the bottom
+ 2 − 1 2 Degree 3

MOOC MAT438/ UiTM

MOOC MAT438/ UiTM

when we need
to use long
division??

Degree 2
16 2

2 − 1 2 Degree 2

Improper Fraction : Need long
The degree of the top is division!
greater than or equal to the

degree of the bottom

MOOC MAT438/ UiTM

16 2 16 2
2 − 1 2 = 4 2 − 4 + 1
Arrange the indices in
descending order !

Expand the use a zero
factors to fill in the
missing term

16 Degree 2 divisor 1 4 quotient 16 2
2 − 16 2 + + dividend 4 2 = 4
2 4 2 − 4 +

1 2=

− (16 2 − 16 + 4 )

Stop division when the degree of remainder 16 − 4 Degree 1
remainder
less than the degree of divisor

16 2 remainder
2 − 1 2 =
quotient +

divisor

16 2 4+ 16 − 4 Degree 1 Proper Fraction :
2 − 1 2 = 2 − 1 2 Degree 2 The degree of the top is
less than the degree of

the bottom

MOOC MAT438/ UiTM

MOOC MAT438/ UiTM

when we need
to use long
division??

Degree 4

4 − 3 − − 1
3 − 2 Degree 3

Improper Fraction : Need long
The degree of the top is division!
greater than or equal to the

degree of the bottom

MOOC MAT438/ UiTM

Arrange the indices in use a zero to
descending order ! fill in the

Degree 3 divisor quotient missing term
dividend
4 − 3 − − 1 = 3 − 2 4 − 3 + − − 1 4
3 =
3 − 2 − ( 4 − 3)

− − 1 remainder

Stop division when the degree of Degree 1
remainder less than the degree of divisor

4 − 3 − − 1 = quotient + remainder
divisor
3 − 2

4 − 3 − − 1 = − − 1 Degree 1 Proper Fraction :
+ 3 − 2 The degree of the top is
3 − 2 less than the degree of
Degree 3
the bottom

MOOC MAT438/ UiTM

MOOC MAT438/ UiTM