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Section 2.1 Partial Fractions

In this section, we introduce another essential tool in the evaluation of integrals, namely the method of partial fractions. This method can be used to evaluate integrals of 'complicated' rational functions.

Remark 2.1.1.
Recall that a rational function is of the form \(\displaystyle \frac{P(x)}{Q(x)}\text{,}\) where both \(P(x)\)and \(Q(x)\) are polynomials.

Subsection 2.1.1 Motivation

An example of a 'complicated' rational function is \(\displaystyle \frac{1}{x^2-16}\text{,}\) and it is easy to verify that $$ \frac{1}{x^2-16} = \frac{1}{(x-4)(x+4)} = \frac{1}{8} \left[ \frac{1}{x-4} - \frac{1}{x+4} \right]. $$

Verify the above.
Hint
Work backwards, that is, start from the right-hand side and make a common denominator. Carefully keep track of negative signs.

While we cannot evaluate \(\displaystyle \int \frac{dx}{x^2-16}\) directly, we can use the above decomposition to evaluate the integral.

Show that $$\int \frac{dx}{x^2-16} = \frac{1}{8} \ln \left| \frac{x-4}{x+4} \right| + C.$$
Hint
Start with \(\displaystyle \int \frac{dx}{x^2-16} = \frac{1}{8} \left[ \int \frac{dx}{x-4} - \int \frac{dx}{x+4} \right]\text{,}\) and recall that \(\displaystyle \int \frac{dx}{x} = \ln|x| + C.\)
Solution
\begin{alignat*}{1} \int \frac{dx}{x^2-16} &= \frac{1}{8} \left[ \int \frac{dx}{x-4} - \int \frac{dx}{x+4} \right] \\ &= \frac{1}{8} \left[ \ln | x-4 | - \ln | x+4| \right] + C \\ &= \frac{1}{8} \ln \left| \frac{x-4}{x+4} \right| + C. \end{alignat*}

The key to making progress in this motivational example was to write a 'complicated' rational function (in this case \(\frac{1}{x^2-16})\) as a sum of simpler functions (in this case \(\frac{1/8}{x-4}\) and \(\frac{-1/8}{x+4}\)). This is called decomposition. Partial fraction decomposition, or simply 'partial fractions', is a systematic method to write a rational function as a sum of simpler functions.

Subsection 2.1.2 Proper Rational Functions

To be more precise, partial fraction decomposition is a systematic method to write a proper rational function as a sum of simpler functions.

Definition 2.1.4.
A proper rational function is a rational function \(\displaystyle \frac{P(x)}{Q(x)}\) such that the degree of the polynomial \(P(x)\) in the numerator is less than the degree of the polynomial \(Q(x)\) in the denominator.

For each of the following rational functions, determine whether it is proper or not.

  1. \(\displaystyle \displaystyle \frac{x}{x^3-1}\)
  2. \(\displaystyle \displaystyle \frac{x^3+2x-5}{x^3-1}\)
  3. \(\displaystyle \displaystyle \frac{x^4-3x}{x^3-1}\)
Solution
  1. \(\displaystyle \frac{x}{x^3-1}\) is a proper rational function, since the degree of the polynomial in the numerator (namely 1) is less than the degree of the polynomial in the denominator (namely 3).
  2. \(\displaystyle \frac{x^3+2x-5}{x^3-1}\) is not a proper rational function, since the degrees of the polynomials in the numerator and the denominator are the same (namely 3).
  3. \(\displaystyle \frac{x^4-3x}{x^3-1}\) is not a proper rational function, since the degree of the polynomial in the numerator (4) is greater than the degree of the polynomial in the denominator (3).

Subsection 2.1.3 Overview of Partial Fraction Decomposition

The steps to determine the partial fraction decomposition of a proper rational function are as follows:

  1. Factor \(Q(x)\) as far as possible.

    In MATH 136, we will restrict ourselves to the case where \(Q(x)\) is a product of distinct linear factors.

  2. Write the partial fraction decomposition using the rule that any linear factor \(ax+b\) in \(Q(x)\) gives rise to a partial fraction of the form $$\frac{A}{ax+b},$$ where \(A\) is an unknown to be determined in step 3.
    $$\frac{4x^2+14x-2}{(x-1)(x+1)(x+3)} = \frac{A}{x-1} + \frac{B}{x+1} + \frac{C}{x+3}.$$
  3. Solve for the unknowns (\(A\text{,}\) \(B\text{,}\) \(C\text{,}\) etc.).

Once you have solved for the unknowns, you can proceed with any integration if needed.

Subsection 2.1.4 One Example

In the next video, we will work through the example \(\displaystyle \int \frac{4x^2 + 14x - 2}{x^3 + 3x^2-x-3} \ dx\text{.}\) Note that the degree of the polynomial in the numerator is 2, and the degree of the polynomial in the denominator is 3. So the integrand is a proper rational function, and it is OK to proceed with partial fraction decomposition.

Figure 2.1.7. Video demonstrating how partial fractions works. NOTE: The video covers the introduction from above, so you may skip through the first 2 minutes or so and start viewing at 2:44.

Subsection 2.1.5 A Second Method to Solve for the Unknowns

In the previous video, we showed one method to solve for the unknowns (let's refer to it as 'Tollo's Method'). It is a method that works really well for the case that \(Q(x)\) is a product of distinct linear factors. In the next video, we show a second method to solve for the unknowns ('Gerla's Method').

Figure 2.1.8. Video demonstrating a second method for solving for the unknowns in the method of partial fractions.

All right, so now we have two methods that we can use to solve for the unknowns. You have a choice: you may use either Tollo's Method or Gerla's Method. Which do you like best?

Subsection 2.1.6 CAUTION! A Long Division Step May Be Needed Before Proceeding with a Partial Fraction Decomposition

Up to now, we have considered rational functions that are proper. This raises the question what to do when a given rational function is not proper.

The rational function $$\frac{4x^2+3x+2}{x^2+x-2}$$ is not proper. Since the degree in the numerator is larger than the degree in the denominator, we cannot proceed immediately with the partial fraction decomposition.

In these cases, it is necessary to take the preliminary step of using long division to divide \(Q(x)\) into \(P(x)\) and obtain $$ \frac{P(x)}{Q(x)} = S(x) + \frac{R(x)}{Q(x)}, $$ where \(S(x)\) is a polynomial and \(\displaystyle \frac{R(x)}{Q(x)}\) is a proper rational function. The partial fraction decomposition then can proceed for \(\displaystyle \frac{R(x)}{Q(x)}\text{.}\)

In the next video, we work through the example \(\displaystyle \int \frac{4x^2 + 3x+2}{x^2+x-2} \ dx\text{.}\)

Figure 2.1.10. Video demonstrating the method of long division.

Subsection 2.1.7 Summary

  • A proper rational function is a rational function such that the degree of the polynomial in the numerator is less than the degree of the polynomial in the denominator.
  • Partial fraction decomposition is a systematic method to write a proper rational function as a sum of simpler rational functions.
  • In general, the simpler rational functions are of the form $$ \frac{A}{(ax+b)^n} \ \ \ \textrm{or} \ \ \ \frac{Bx+C}{(ax^2 + bx + c)^n}, $$ where \(a\text{,}\) \(b\text{,}\) \(c\text{,}\) \(A\text{,}\) \(B\text{,}\) and \(C\) are constants and \(n\) is a positive integer, and the quadratic \(ax^2+bx+c\) is irreducible (cannot be factored further since \(b^2 - 4 a c \lt 0\)).

  • If a proper rational function is not already in one of these simpler forms, then the steps to determine the partial fraction decomposition are as follows:

    1. Factor \(Q(x)\) as far as possible.

      • In general, \(Q(x)\) will be the product of linear factors and/or irreducible quadratic factors.
      • In MATH 136, we work with the restriction that \(Q(x)\) is the product only of distinct linear factors.
    2. Under the restriction on \(Q(x)\) mentioned in Step 1, write the partial fraction decomposition using the rule that each linear factor \(ax+b\) in \(Q(x)\) gives rise to a partial fraction of the form $$\frac{A}{ax+b},$$ where \(A\) is an unknown to be determined in step 3.
    3. Solve for the unknowns (\(A\text{,}\) \(B\text{,}\) \(C\text{,}\) etc.).
  • If a given rational function \(\displaystyle \frac{P(x)}{Q(x)}\) is not proper, then it is necessary to use long division first and obtain $$ \frac{P(x)}{Q(x)} = S(x) + \frac{R(x)}{Q(x)}, $$ where \(S(x)\) is a polynomial and \(\displaystyle \frac{R(x)}{Q(x)}\) is a proper rational function.

Subsection 2.1.8 Don't Forget

Don't forget to return to eClass to complete the pre-class quiz.

Subsection 2.1.9 Further Study

Remember that the notes presented above only serve as an introduction to the topic. Further study of the topic will be required. This includes working through the pre-class quizzes, reviewing the lecture notes, and diligently working through the homework problems.

As you study, you should reflect on the following learning outcomes, and critically assess where you are on the path to achieving these learning outcomes:

The following references provide a good start for review and further study:

Learning Outcome Video Textbook Section
1 2.E1 5.6
2 2.E3 5.6
3 2.E1, 2.E2, and 2.E3 5.6
4 2.E1, 2.E2, and 2.E3 5.6