Math 152 – Spring 2016
Section 8.4
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Section 8.4 – Integration of Rational Functions by
Partial Fractions
Definition. Rational Functions are fractions where the numerator and denominator
are polynomials.
In this section we will integrate rational functions by rewriting the function as a sum
of simpler rational functions.
Example: Use the fact that
1
3(x + 1) + (x − 4)
4x − 1
3
+
=
= 2
x−4 x+1
(x − 4)(x + 1)
x − 3x − 4
to integrate
Z
4x − 1
dx
x2 − 3x − 4
Procedure: Suppose we wish to integrate f (x) =
tional functions.
P (x)
Q(x)
where P (x) and Q(x) are ra-
1. If the degree of P (x) (the highest power of x) is larger than the degree of Q(x),
then we use long division to divide P (x) by Q(x) and rewrite the function as
f (x) =
P (x)
R(x)
= S(x) +
Q(x)
Q(x)
where S(x) is a polynomial and the degree of R(x) (the remainder of the division)
is smaller than the degree of Q(x).
2. Factor the denominator Q(x) as much as possible.
3. Express the rational function
R(x)
Q(x)
as a sum of partial fractions of the form
A
Ax + B
or
(ax + b)i
(ax2 + bx + c)i
We will discuss four cases for how to rewrite
4. Integrate.
R(x)
Q(x) .
Math 152 – Spring 2016
Section 8.4
Case 1: The denominator Q(x) is a product of distinct linear factors.
Factor
Q(x) = (a1 x + b1 )(a2 x + b2 ) · · · (an x + bn )
Set
R(x)
A1
A2
An
=
+
+ ··· +
Q(x)
a1 x + b1
a2 x + b2
an x + bn
and solve for the constants A1 , A2 , . . . An .
Example 1.
R
x3 +x
x2 +2x−3
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Math 152 – Spring 2016
Section 8.4
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Case 2: Q(x) is a product of linear factors, some of which are repeated.
Factor Q(x) and suppose the first factor is (a1 x + b1 )r . Then intead of including the
R(x)
1
single term a1 A
x+b1 in the expression for Q(x) we must include
Ar
A2
A1
+ ··· +
+
a1 x + b1
(a1 x + b1 )2
(a1 x + b1 )r
Example 2.
R
x2
(x−3)(x+2)2
dx
Math 152 – Spring 2016
Section 8.4
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Case 3: Q(x) contains irreducible quadratic factors, none of which is repeated.
If Q(x) has the factor ax2 + bx + c with b2 − 4ac < 0 (this means it cannot be factored), then in addition to the partial fractions from the case 1 & 2 the expression for
R(X)/Q(x) will also include a term of the form
Ax + B
ax2 + bx + c
To integrate this term, complete the square in the denominator and split it into two
fractions. Then use u-substitution and the formula
Z
1
1
−1 x
dx
=
+C
tan
x2 + a2
a
a
R
x−1
Example 3. x3 +2x
2 +3x dx
Math 152 – Spring 2016
Section 8.4
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Case 4: Q(x) contains a repeated, irreducible quadratic factors.
If Q(x) has the factor (ax2 + bx + c)r with b2 − 4ac < 0, then the partial fraction
decomposition of R(x)/Q(x) contains the terms
An x + B n
A1 x + B 1
A2 x + B2
+ ··· +
+
ax2 + bx + c (ax2 + bx + c)2
(ax2 + bx + c)n
Example 4.
R
x4 +x+1
x(x2 +2)2
dx
Example 5. Write out the form of the partial fraction decomposition of the function
below, but do not solve for the constants.
x4 − 3x2 + 7
x3 (x − 8)(x + 2)(x2 + 1)2 (x2 + 2x + 6)