Partial Fractions: Learn It 1

  • Break down and integrate rational functions using partial fractions
  • Identify and work with simple linear factors in rational functions
  • Handle repeated linear factors when using partial fractions
  • Work with quadratic factors in rational functions

What Is Partial Fraction Decomposition?

Partial fraction decomposition is a technique that lets you rewrite complicated rational functions as sums of simpler fractions. Think of it as “reverse engineering” fractions—instead of adding fractions together, you’re splitting them apart.

For example, you can rewrite [latex]\frac{3x}{{x}^{2}-x - 2}[/latex] as [latex]\frac{1}{x+1}+\frac{2}{x - 2}[/latex].

Why does this matter? Because integrating simpler fractions is much easier than tackling the original complex fraction.

You already know how to integrate basic rational functions:

  • [latex]\displaystyle\int \frac{du}{u}=\text{ln}|u|+C[/latex]
  • [latex]\displaystyle\int \frac{du}{{u}^{2}+{a}^{2}}=\frac{1}{a}{\tan}^{-1}\left(\frac{u}{a}\right)+C[/latex]

But what about something like [latex]\displaystyle\int \frac{3x}{{x}^{2}-x - 2}dx[/latex]? This doesn’t fit our basic patterns.

However, if we can decompose it into simpler parts:

[latex]\displaystyle\int \frac{3x}{{x}^{2}-x - 2}dx = \displaystyle\int \left(\frac{1}{x+1}+\frac{2}{x - 2}\right)dx[/latex]

Now we can integrate each piece separately:

[latex]\displaystyle\int \left(\frac{1}{x+1}+\frac{2}{x - 2}\right)dx=\text{ln}|x+1|+2\text{ln}|x - 2|+C[/latex]

Check Your Work: You can verify that [latex]\frac{1}{x+1}+\frac{2}{x - 2}=\frac{3x}{{x}^{2}-x - 2}[/latex] by finding a common denominator.

When Can You Use This Method?

Partial fraction decomposition works only when the degree of the numerator is less than the degree of the denominator.

degree requirement for partial fraction decomposition

For a rational function [latex]\frac{P\left(x\right)}{Q\left(x\right)}[/latex], you can use partial fractions only if [latex]\text{deg}\left(P\left(x\right)\right)<\text{deg}\left(Q\left(x\right)\right)[/latex].

What if the degree of the numerator is greater than or equal to the degree of the denominator?

You need to use polynomial long division first. This rewrites [latex]\frac{P\left(x\right)}{Q\left(x\right)}[/latex] as [latex]A\left(x\right)+\frac{R\left(x\right)}{Q\left(x\right)}[/latex], where [latex]\text{deg}\left(R\left(x\right)\right)<\text{deg}\left(Q\left(x\right)\right)[/latex].

Then you can apply partial fraction decomposition to [latex]\frac{R(x)}{Q(x)}[/latex]

Recall: Polynomial Long Division

  1. Set up the division problem as the numerator divided by the denominator
  2. Determine the first term of the quotient by dividing the leading term of the dividend by the leading term of the divisor.
  3. Multiply the answer by the divisor and write it below the like terms of the dividend.
  4. Subtract the bottom binomial from the top binomial.
  5. Bring down the next term of the dividend.
  6. Repeat steps 2–5 until reaching the last term of the dividend.
  7. If the remainder is non-zero, express as a fraction using the divisor as the denominator.

Visit this website for a review of long division of polynomials.

The following example, although not requiring partial fraction decomposition, illustrates our approach to integrals of rational functions of the form [latex]\displaystyle\int \frac{P\left(x\right)}{Q\left(x\right)}dx[/latex], where [latex]\text{deg}\left(P\left(x\right)\right)\ge \text{deg}\left(Q\left(x\right)\right)[/latex].

Evaluate [latex]\displaystyle\int \frac{{x}^{2}+3x+5}{x+1}dx[/latex].