Partial Fraction Decomposition

Calculus 2

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Overview

Partial fraction decomposition (PFD) is a technique that helps us to integrate rational functions. Recall that rational functions are simply functions which have a polynomial numerator and denominator.

The goal of partial fraction decomposition is to turn a single rational function into a combination of rational functions, each of which we know how to integrate (using a u-substitution, a trig integral, or the natural log rule).

Which integral do you know how to calculate:

\int \frac{4x+1}{x^2+x} \ dx \ \ \ \ \text {or} \ \ \ \ \int \left( \frac{1}{x} + \frac{3}{x+1} \right) dx ?

It should be clear that the second integral is easier (they become natural logs).

But what if I told you that those two integrals were the same?

Adding Rational Functions

To get a better understanding of this technique, we begin by adding two simple rational functions.

Example 1

Add $\frac{1}{x}$ and $\frac{3}{x+1}$ .

View Answer

$\frac{4x+1}{x^2+x}$

View Solution

We find a common denominator:

\begin{aligned} \frac{1}{x} + \frac{3}{x+1} &= \frac{1}{x} \left( \frac{x+1}{x+1} \right) + \frac{3}{x+1}\left( \frac{x}{x} \right)\\ &= \frac{x+1+3x}{x(x+1)}\\ &= \frac{4x+1}{x^2+x} \end{aligned}

Now the question becomes, given $\frac{4x+1}{x^2+x}$ , how do we work backwards to get $\frac{1}{x} + \frac{3}{x+1}$ ?

Example 2

Decompose $\frac{4x+1}{x^2+x}$ .

View Answer

$\frac{1}{x} + \frac{3}{x+1}$

View Solution

Factor the denominator:
$\begin{aligned} \frac{4x+1}{x^2+x} &= \frac{4x+1}{x(x+1)} \end{aligned}$
Split the fraction into two:
$\begin{aligned} \frac{4x+1}{x(x+1)} &= \frac{A}{x} + \frac{B}{x+1} \end{aligned}$
Solve for the coefficients in the numerator:
$\begin{aligned} \frac{4x+1}{x(x+1)} &= \frac{A}{x} + \frac{B}{x+1}\\ &= \frac{A}{x} \left( \frac{x+1}{x+1} \right) + \frac{B}{x+1} \left( \frac{x}{x} \right)\\ &= \frac{A(x+1) + B(x)}{x(x+1)}\\\\ \Rightarrow 4x+1 &= A(x+1)+B(x) \ \ \ \ \text{(as denominators are equal)}\\ &= Ax+A+Bx\\ &= (A+B)x+A\\ \end{aligned}$

(3a) Polynomial Mirroring Method

\begin{aligned} 4x+1 &= A(x+1)+B(x) \end{aligned}

\begin{gathered} \underline{\text{Compare }x\text{ coefficients:}} & \ \ \ & \underline{\text{Compare constants:}}\\ 4x=(A+B)x & \ \ \ & 1=A\\ 4=A+B & & \boxed{A=1}\\ 4=1+B & & \\ \boxed{B=3} \end{gathered}

(3b) Heaviside "Cover-Up" Method

\begin{aligned} 4x+1 &= A(x+1)+B(x) \end{aligned}

\begin{gathered} \underline{\text{Let } x=-1\text{:}} & \ \ \ & \underline{\text{Let } x=0\text{:}}\\ 4(-1)+1=A(-1+1)+B(-1) & & 4(0)+1=A(0+1)+B(0)\\ -3=-B & & 1=A\\ \boxed{B=3} & & \boxed{A=1} \end{gathered}

Plug in coefficients
$\begin{aligned} \frac{4x+1}{x^2+x)} &= \frac{A}{x} + \frac{B}{x+1}\\ &= \frac{1}{x} + \frac{3}{x+1}\\ \end{aligned}$

Note that we had two options to use when finding the coefficients:

Polynomial Mirroring Method: comparing coefficients based on powers of $x$
Heaviside "Cover-Up" Method: strategically choosing values of $x$ to cancel terms out

Occasionally, we might benefit from using a combination of both methods to get our coefficients.

Proper and Improper Rational Functions

In order to perform a partial fraction decomposition, we need to make sure we are working with a proper rational function.

A proper rational function is defined as a rational function whose degree of the denominator is greater than that of the numerator.

An improper rational function is defined as a rational function whose degree of the numerator is greater than or equal to that of the denominator.

If we have an improper rational function, then we must perform polynomial long division first to turn it proper before attempting a partial fraction decomposition.

Example 3

Rewrite $\frac{2x^3-8x^2+1}{x^2-4x}$ using long division.

View Answer

$\frac{4x+1}{x^2+x}$

View Solution

Long division is not easily written in LaTeX unfortunately!

Example 4

Rewrite $\frac{x^4-x^2+4x+2}{x^2-1}$ using long division.

View Answer

$\frac{4x+1}{x^2+x}$

View Solution

Long division is not easily written in LaTeX unfortunately!

Non-Repeating Linear Factors

When working with a non-repeating linear factor in the denominator of the form $ax\pm b$ , we decompose it into the form $\frac{A}{ax\pm b}$ . If there are multiple factors to decompose, we repeat the process with different constants ( $A$ , $B$ , $C$ , etc.).

Example 5

Compute $\int \frac{x+14}{x^2+7x+10} \ dx$ .

View Answer

$\frac{4x+1}{x^2+x}$

View Solution

Solution coming soon.

Repeating Linear Factors

When working with a repeating linear factor in the denominator of the form $(ax\pm b)^n$ , we decompose it into the form $\frac{A}{ax\pm b} + \frac{B}{(ax\pm b)^2} + \cdots$ . If there are multiple factors to decompose, we repeat the process with different constants ( $A$ , $B$ , $C$ , etc.).

Example 6

Compute $\int \frac{5x-2}{(x+3)^2} \ dx$ .

View Answer

$\frac{4x+1}{x^2+x}$

View Solution

Solution coming soon.

Non-Repeating Quadratic Irreducible Factors

When working with a non-repeating quadratic irreducible factor in the denominator of the form $ax^2+bx+c$ , we decompose it into the form $\frac{Ax+B}{ax^2+bx+c}$ . If there are multiple factors to decompose, we repeat the process with different constants ( $A$ , $B$ , $C$ , etc.).

Example 7

Compute $\int \frac{-2x+4}{(x^2+1)(x-1)} \ dx$ .

View Answer

$\frac{4x+1}{x^2+x}$

View Solution

Solution coming soon.

Repeating Quadratic Irreducible Factors

When working with a repeating quadratic irreducible factor in the denominator, we combine the processes seen for a repeating linear factor and a quadratic irreducible factor.

We won’t do an example for this case, but you can see the general structure of the decomposition below.

Recap

Partial fraction decomposition is a great technique to use when the integrand is a rational function (polynomial in numerator and denominator) for which the denominator can be factored.

If the rational function is improper, meaning the degree of the numerator is greater than or equal to the degree of the denominator, long division should be performed before performing a partial fraction decomposition.

Practice Problems

Coming soon!

Integration Techniques

Integration by Parts