Integration of Rational Functions by Partial Fractions
Motivation: Evaluating the integral
x
2
x5
dx would be much easier if we knew
 x2
ahead of time that
x5
2
1


x  x  2 x 1 x  2
2
P( x)
as a sum of simpler
Q( x)
fractions provided the degree of P is less than the degree of Q.
In general, it is possible to express a rational function f ( x) 
A Preliminary Example. Integrate the function
x3  x
 x  1 dx .
Some Algebraic Background
Any polynomial Q can be factored as a product of linear factors (of the form ax + b) and
irreducible quadratic factors (of the form ax2 + bx + c where the discriminant b2 – 4ac <
0). For example, factor x4 – 16.
Case 1: The denominator Q(x) is a product of distinct linear factors. That is,
Q( x)  (a1 x  b1 )( a2 x  b2 ) (ak x  bk )
where no factor is repeated. Then the partial fraction theorem states that there exist
constants A1, A2, …, Ak such that
Ak
A1
A2
P( x)



Q( x) a1 x  b1 a2 x  b2
ak x  bk
Examples
x 2  2x  1
 2 x 3  3x 2  2 x dx
x
2
dx
 a2
Case 2: Q(x) is a product of linear factors, some of which are repeated. If, for example,
the linear factor (a1x + b1) is repeated r times, i.e. (a1x + b1)r appears in factorization of
Q(x), then instead of the single term A1/(a1x + b1) in the expansion, we have
A1
A2
Ar


2
a1 x  b1 (a1 x  b1 )
(a1 x  b1 ) r
Example. Write the form of the partial fraction decomposition of
x3  x  1
.
x 2 ( x  1) 3
x 4  2x 2  4x  1
Example. Evaluate the integral  3
dx .
x  x2  x 1
Case 3: Q(x) contains irreducible quadratic factors, none of which is repeated. If Q(x)
has a factor ax2 + bx + c (irreducible), then the partial fraction expansion will have a term
of the form
Ax  B
ax  bx  c
2
where A and B are constants to be determined.
Example. Write the form of the p. f. decomposition of
x
.
( x  2)( x  1)( x 2  4)
A useful integration rule follows:
x
2
dx
1
 x
 arctan    C
2
a
a
a
2
Examples
2x 2  x  4
 x 3  4 x dx
4 x 2  3x  2
 4 x 2  4 x  3 dx
Case 4: Q(x) contains a repeated irreducible quadratic factor. If Q(x) has the factor (ax2 +
bx + c)r, then the following expression occurs in the partial fraction decomposition:
A1 x  B1
A2 x  B2
Ar x  Br


2
2
2
ax  bx  c (ax  bx  c)
(ax 2  bx  c) r
Example. Write the form of the partial fraction decomposition of the function
x3  x2  1
.
x( x  1)( x 2  x  1)( x 2  1) 3
Examples
1  x  2x 2  x3
 x( x 2  1) 2 dx

x4
dx
x