9. Partial Fractions

If this page is confusing, read it briefly and then reread it after the examples.

a2. General Decompositions

On the previous page we were able to integrate the rational function \(\dfrac{x^4+2x-16}{x^3+4x}\) because we were able to rewrite it as \[ \dfrac{x^4+2x-16}{x^3+4x}=x-\dfrac{4}{x}+\dfrac{2}{x^2+4} \] where the first term is a polynomial, \(x\), and the remaining terms, \(-\dfrac{4}{x}\) and \(\dfrac{2}{x^2+4}\), are rational functions whose denominators, \(x\) and \(x^2+4\), are the factors of the original denominator, \(x^3+4x\). Each of the three terms, \(x\), \(\dfrac{4}{x}\) and \(\dfrac{2}{x^2+4}\) could then be integrated using previous techniques. The same process will work (in principle) for any rational function \(\dfrac{p(x)}{q(x)}\). Here are the steps:

Remove any Polynomial Part:
If the degree of \(p(x)\) is greater than or equal to the degree of \(q(x)\), then we can write the fraction as \[ \dfrac{p(x)}{q(x)}=r(x)+\dfrac{s(x)}{q(x)} \] where \(r(x)\) and \(s(x)\) are polynomials with the degree of \(s(x)\) being less than the degree of \(q(x)\). Here \(p(x)\) is called the dividend, \(q(x)\) is the divisor, \(r(x)\) is the quotient, and \(s(x)\) is the remainder. You can find \(r(x)\) and \(s(x)\) by Long Division of Polynomials. From now on we will assume this has been done so that the degree of \(p(x)\) is less than the degree of \(q(x)\).
Cancel Common Factors:
Factor \(p(x)\) and \(q(x)\) and cancel any common factors. From now on we will assume this has been done so that \(p(x)\) and \(q(x)\) have no common factors.
Factor the Denominator \(q(x)\):
The Unique Factorization Theorem from algebra says you can always factor a polynomial with real coefficients into a product of linear factors, \((x-a)\), and irreducible quadratic factors, \((x^2+rx+s)\) and one constant factor, all with real coefficients. The linear and quadratic factors may be repeated. If a factor is repeated \(n\) times, \(n\) is called the multiplicity of the factor.

A quadratic polynomial is irreducible if it cannot be written as the product of two linear factors with real coefficients. As a consequence, by completing the square, an irreducible quadratic polynomial \(x^2+rx+s\) can always be written as \((x-b)^2+c^2\).

To complete the square on \(x^2+rx+s\), let \(b=-\dfrac{r}{2}\) and add and subtract \(b^2\). Then \[\begin{aligned} x^2+rx+s&=x^2-2bx+s\\ &=x^2-2bx+b^2+s-b^2\\ &=(x-b)^2+e \end{aligned}\] where \(e=s-b^2\). Since \(x^2+rx+s\) is irreducible, \(e\) must be positive.

Why? Because, if \(e\) is negative or zero, we write it as \(e=-c^2\). Then \[\begin{aligned} x^2+rx+s&=(x-b)^2+e=(x-b)^2-c^2\\ &=(x-b-c)(x-b+c) \end{aligned}\] by factoring the difference of two squares. This expresses \(x^2+rx+s\) as a product of two linear factors which is impossible since it is irreducible. So \(e\) cannot be negative or zero.

Since \(e\) is positive, we write it as \(e=c^2\). Then \[ x^2+rx+s=(x-b)^2+e=(x-b)^2+c^2 \] as claimed.

It is useful to complete the square in these quadratic polynomials because
(1) it will help us complete the partial fraction expansion and
(2) it will help us compute the resulting integrals.

Write out the General Partial Fraction Expansion for \(\dfrac{p(x)}{q(x)}\):
The Partial Fraction Theorem from algebra says \(\dfrac{p(x)}{q(x)}\) can be uniquely written as a sum of terms which depend on the factors of \(q(x)\): Here are the types of factors of \(q(x)\) and the correspond terms in the expansion:
1. Non-Repeated Linear Factors:
  For each non-repeated linear factor \((x-a)\), include a term of the form: \[ \dfrac{A}{x-a} \]
2. Repeated Linear Factors:
  For each repeated linear factor \((x-a)^n\), include terms of the form: \[ \dfrac{A_1}{x-a}+\dfrac{A_2}{(x-a)^2}+\cdots+\dfrac{A_n}{(x-a)^n} \]
3. Non-Repeated Quadratic Factors:
  For each non-repeated quadratic factor \((x-b)^2+c^2\), include a term of the form: \[ \dfrac{B(x-b)+C}{(x-b)^2+c^2} \]
  
  We write the quadratic term in the form \(\dfrac{B(x-b)+C}{(x-b)^2+c^2}\) instead of \(\dfrac{Bx+C}{x^2+rx+s}\)for two reasons:
  (1) We will use the number \(x=b\) in solving for the coefficients \(B\) and \(C\), and
  (2) It is easier to integrate \(\dfrac{B(x-b)}{(x-b)^2+c^2}\) than \(\dfrac{Bx}{(x-b)^2+c^2}\).
4. Repeated Quadratic Factors:
  For each repeated quadratic factor \(\left((x-b)^2+c^2\right)^n\) include terms of the form: \[ \dfrac{B_1(x-b)+C_1}{(x-b)^2+c^2} +\dfrac{B_2(x-b)+C_2}{\left((x-b)^2+c^2\right)^2} +\cdots +\dfrac{B_n(x-b)+C_n}{\left((x-b)^2+c^2\right)^n} \]
For example, the general partial fraction decomposition of \(\dfrac{x^3-4x^2+1}{(x-5)(x-2)^3((x-1)^2+4)(x^2+9)^2}\) is: \[\begin{aligned} &\dfrac{x^3-4x^2+1}{(x-5)(x-2)^3((x-1)^2+4)(x^2+9)^2} \\ &\quad=\dfrac{A_1}{x-5}+\dfrac{A_2}{x-2}+\dfrac{A_3}{(x-2)^2}+\dfrac{A_4}{(x-2)^3} \\ &\qquad+\dfrac{B_1(x-1)+C_1}{(x-1)^2+4}+\dfrac{B_2x+C_2}{x^2+9}+\dfrac{B_3x+C_3}{(x^2+9)^2} \end{aligned}\] In practice, we don't use subscripts; instead we use different capital letters.
Find the Coefficients \(A_i\), \(B_i\) and \(C_i\):
There are various techniques for finding the coefficients \(A_i\), \(B_i\) and \(C_i\). Some are discussed in the examples below and more are discussed on the next two pages.

We are now ready for examples. The first shows how to remove the polynomial part. The others show how to handle each type of factor in the factorization of \(q(x)\) and a little about finding the coefficients.

On the next two pages we will summarize the techniques for finding the coefficients in a partial fraction expansion.

You can also practice finding General Partial Fraction Expansions by using the following Maplet (requires Maple on the computer where this is executed):

Partial Fractions: General Decomposition Rate It

A Maplet on finding the coefficients is on the next page.

9. Partial Fractions

a2. General Decompositions

Heading