Math 152: 8.4 Integration of Rational Functions by Partial Fractions
Warm-Up
1. a)
b)
c)
⎛
1
⎞
1
∫ ⎜⎝ 2x − 3 + x + 4 ⎟⎠ dx =
1
2x − 3
∫ 2x
+
1
x +4
=
3x + 1
2
+ 5x − 12
dx =
2. A quadratic expression ax 2 + bx + c has non-real zeroes if _______________. Such
a quadratic is called _________________.
3. Factoring and Special Products:
a) A quadratic expression ax 2 + bx + c
factors over the rational numbers if
_______________.
()
b) Rational Root Theorem: Find a rational root “a” of a polynomial p x , such that
()
p a = 0 , then divide/repeat until p is a product of irreducible factors.
Example: List the possible rational roots of x 3 + 3x 2 − 4 , identify an actual
rootè an actual factor, perform long division and lastly factor fully.
(
)(
)
c) Difference of Squares: a 2 − b 2 = a − b a + b . Example: Factor x 8 − 1
( ) = (a − bi ) (a + bi ) . Ex: Factor 25x
d) Sum of Squares: a 2 + b 2 = a 2 − bi
© Raelene Dufresne 2013
2
4
+ 1 over 
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Math 152: 8.4 Integration of Rational Functions by Partial Fractions
(
)(
)
e) Sum/Difference of Cubes: a 3 ± b 3 = a ± b a 2  ab + b 2 Ex: Factor 8x 6 − 27y 9
f) Substitution: Let u = x 2 . Ex: Factor x 4 − 5x 2 − 36 over 
4. The LCM (least common multiple) of 2 or more numbers is the ______________
number that ______________ evenly into each of the numbers.
5. The LCD (lowest common denominator) is the ______ of the ________________
of 2 or more _______________ that are being __________ or ______________.
6. Determine the LCM/LCD (if these expressions are denominators) of the following:
a) 2, 22 , 25
7. Add:
c) x 3 + 8 , x 3 + 3x 2 − 4 , 4x + 8
b) 72, 36, 48
2x
+
x −9
2
x −1
x + 6x + 9
2
+
4
x +x −6
2
8. The division algorithm states that
n
=q+
r
. The fraction is proper if the
d
d
numerator is smaller than the denominator, and improper otherwise. Convert
a)
7
3
to proper form, and
=
b) 5
4
7
to improper form.
=
9. Section 8.4 deals with the integration of rational functions (functions that are ratios
or quotients of polynomials) that can be expressed in either improper fraction form,
n x
( ) d ((x ))
f x =
where
()
()
deg ⎡⎢n x ⎤⎥ ≥ deg ⎡⎢d x ⎤⎥
⎣
⎦
⎣
⎦
or
proper
fraction
form,
( ) ( ) d ((x )) where deg ⎡⎣⎢r (x )⎤⎦⎥ < deg ⎡⎣⎢d (x )⎤⎦⎥ . Express:
x − 7x + 1
5
a) f ( x ) =
in proper form
b) f ( x ) = 3x + 1 −
in improper form
5x − 2
2x − 3
f x =q x +
r x
3
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Math 152: 8.4 Integration of Rational Functions by Partial Fractions
10. Integrate
∫
x 3 − 7x + 1
5x − 2
dx .
8.4 -The Method of Integration of Rational Functions by Partial Fraction Decomposition:
The process of working backwards to break apart (decompose) a fraction as the sum
and/or difference of its “partial” fractions (whose denominators are linear or
irreducible quadratic polynomials) is called Partial Fraction Decomposition. There are
FOUR CASES of partial fraction decomposition that we will focus on.
()
Let the rational function whose denominator is d x be given in proper fraction form.
Case I: The denominator is a product of linear factors with no repeats.
( ) 2x 3+x5+x1− 12 ,
a) decompose f ( x ) .
b) determine ∫ f ( x ) dx .
Exercise 1: Given f x =
2
Case II: The denominator is a product of linear factors with some repeats.
()
Exercise 2: Given f x =
a) Decompose
x2 − x
(2x − 1) (x + 1)
2
.
7
into a sum of two different fractions in reduced form, in three
8
different combinations. What do you notice about all of the denominators and their
numerators?
()
c) determine ∫ f ( x ) dx .
b) decompose f x .
© Raelene Dufresne 2013
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Math 152: 8.4 Integration of Rational Functions by Partial Fractions
Case III: The denominator includes irreducible quadratic factors with no repeats.
( ) (x + 1) (x + 2) ,
Exercise 3: Given f x =
x 3 − 4x 2 + 2
2
2
()
b) determine ∫ f ( x ) dx .
a) decompose f x .
Case IV: The denominator includes irreducible quadratic factors with some repeats.
Exercise 4: Write out the partial fraction decomposition in terms of its unsolved
()
numerators for the rational function f x =
x −1
Exercise 5: (a)
∫ x +2
Exercise 6: (a)
∫x
)(
)(
x x −1 x + x +1 x +1
2
(b)
∫
dx , where a > 0
2
(b)
∫x
−a
© Raelene Dufresne 2013
2
sin x cos2 x
dx
1
2
(
x3 + x2 + 1
5 + cos2 x
1
2
+ 2x − 3
)
3
.
dx
dx
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