Chapter Study Guide: Partial Fractions

Learning Objectives

After completing this study guide, you should be able to:

• Determine when partial fraction decomposition is the appropriate integration technique.
• Perform long division on improper rational functions before applying partial fractions.
• Factor denominators completely into linear and irreducible quadratic factors.
• Set up and solve algebraic equations to find the unknown constants in a partial fraction decomposition.
• Evaluate the final integrals using logarithmic, power, or inverse trigonometric integration formulas.

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Key Terms & Glossary

• Rational Function: A function that can be written as the ratio of two polynomials, $f(x) = \frac{P(x)}{Q(x)}$.
• Proper Rational Function: A rational function where the degree of the numerator $P(x) is strictly less than the degree of the denominator Q(x)$.
• Improper Rational Function: A rational function where the degree of the numerator is greater than or equal to the degree of the denominator.
• Irreducible Quadratic: A quadratic expression $ax^2 + bx + c that has no real roots (i.e., its discriminant b^2 - 4ac < 0$) and cannot be factored into real linear expressions.
• Partial Fraction Decomposition: The algebraic process of breaking down a complex rational function into a sum of simpler fractions.

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The "Big Idea"

In early algebra, you learned how to add two fractions by finding a common denominator:

$\frac{2}{x-1} + \frac{3}{x+2} = \frac{2(x+2) + 3(x-1)}{(x-1)(x+2)} = \frac{5x+1}{x^2+x-2}$

Partial Fractions is simply this process in reverse.

When we are asked to integrate a complex rational expression like $\int \frac{5x+1}{x^2+x-2} dx$, we do not have a direct integration rule for it. However, if we can algebraicly reverse-engineer the expression back into its "partial fractions" $\left( \frac{2}{x-1} + \frac{3}{x+2} \right)$, we can easily integrate the pieces using the natural logarithm rule: $\int \frac{1}{u} du = \ln|u| + C$.

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Formula / Concept Box

The form of your partial fraction setup depends entirely on the factors of the denominator $Q(x)$.

[!IMPORTANT] Always check if the rational function is improper (Degree of Numerator $\ge$ Degree of Denominator) before applying these formulas. If it is improper, you must perform Polynomial Long Division first!

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Hierarchical Outline

1. Analyze the Rational Function $\frac{P(x)}{Q(x)}$
   • Check the degrees of $P(x)$ and $Q(x)$.
   • If $\text{deg}(P) \ge \text{deg}(Q)$, perform long division to get a polynomial plus a proper remainder.
2. Factor the Denominator $Q(x)$
   • Break the denominator into the product of linear factors and irreducible quadratic factors.
3. Determine the Form of the Decomposition
   • Assign a constant (e.g., $A, B$) over each distinct linear factor.
   • Assign increasing powers for repeated factors.
   • Assign a linear numerator (e.g., $Ax+B$) over irreducible quadratic factors.
4. Solve for the Unknown Constants
   • Multiply both sides by the common denominator to clear the fractions.
   • Method 1: Substitute strategic values of $x$ (roots of the linear factors).
   • Method 2: Equate the coefficients of like terms on both sides of the equation.
5. Integrate the Decomposed Function
   • Apply basic integration rules (Natural log, power rule, or inverse tangent).

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Visual Anchors

The Partial Fraction Workflow

Visualizing Decomposition

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Definition-Example Pairs

• Term: Long Division Pre-requisite

  • Definition: The necessary step of dividing $P(x)$ by $Q(x)$ to extract polynomial terms when the numerator's degree is equal to or greater than the denominator's.
  • Real-World Example: $\int \frac{x^3}{x-1} dx$. Since $3 \ge 1$, we divide to get $x^2 + x + 1 + \frac{1}{x-1}$.

• Term: Irreducible Quadratic

  • Definition: A quadratic denominator that cannot be factored into real roots because $b^2 - 4ac < 0$.
  • Real-World Example: $x^2 + 4$. Setting $x^2 + 4 = 0 yields imaginary numbers, so it stays as x^2 + 4 in the denominator with an (Ax+B)$ numerator.

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Comparison Tables

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Worked Examples

Example 1: Distinct Linear Factors

Evaluate $\int \frac{dx}{x^2-1}$

Step 1: The integrand is proper. Factor the denominator. $x^2 - 1 = (x-1)(x+1)$

Step 2: Set up the partial fraction decomposition. $\frac{1}{(x-1)(x+1)} = \frac{A}{x-1} + \frac{B}{x+1}$

Step 3: Clear the denominators. $1 = A(x+1) + B(x-1)$

Step 4: Solve for $A$ and $B$.

• Let $x = 1$: $1 = A(2) + B(0) \implies A = \frac{1}{2}$
• Let $x = -1$: $1 = A(0) + B(-2) $\implies B = -\frac{1}{2}$$

Step 5: Integrate the decomposed function. $\int \left( \frac{1/2}{x-1} - \frac{1/2}{x+1} \right) dx = \frac{1}{2}\ln|x-1| - \frac{1}{2}\ln|x+1| + C$

▶Click to expand Example 2: Irreducible Quadratic Factor

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

Step 1: The integrand is proper. The denominator has a linear factor and an irreducible quadratic factor. $\frac{x-2}{(x+1)(x^2+4)} = \frac{A}{x+1} + \frac{Bx+C}{x^2+4}$

Step 2: Clear the denominator. $x - 2 = A(x^2+4) + (Bx+C)(x+1)$

Step 3: Solve for $A, B, C$.

• Let $x = -1$: $-3 = A(5)$\implies A = -\frac{3}{5}$$
• Expanding the rest to equate coefficients: $x - 2 = Ax^2 + 4A + Bx^2 + Bx + Cx + C$ $x - 2 = (A+B)x^2 + (B+C)x + (4A+C)$
• Since $A = -3/5$, and the $x^2$ coefficient on the left is 0: $0 = -\frac{3}{5} + B \implies B = \frac{3}{5}$
• For the $x$ coefficient: $1 = B + C \implies 1 = \frac{3}{5} + C \implies C = \frac{2}{5}$

Step 4: Integrate. $\int \frac{-3/5}{x+1} dx + \int \frac{\frac{3}{5}x + \frac{2}{5}}{x^2+4} dx$ Split the second integral into a u-sub portion and an arctan portion to finish.

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Checkpoint Questions

Test your active recall by answering these questions without looking at the material above:

1. What is the very first thing you must check before applying partial fraction decomposition to a rational function?
2. Write out the general partial fraction setup for the denominator expression: $Q(x) = (x-2)^2(x^2+9)$.
3. What makes a quadratic "irreducible", and how does its partial fraction numerator differ from a linear factor's numerator?
4. Once a partial fraction decomposition is completed, what are the three most common integration techniques used to evaluate the resulting integrals?

[!TIP] Self-Correction: If you forget to include a linear numerator (like $Bx+C$) over an irreducible quadratic, your algebraic system will likely have no solution, or lead you to an incorrect answer! Always double-check your initial setup.