Integration of Rational Functions by Partial Fractions

Definition

Partial fraction decomposition is an algebraic method for integrating rational functions $R(x) = \frac{P(x)}{Q(x)}$ by breaking them into a sum of simpler fractions that are easier to integrate.

• How to read: “The function R of x is equal to the ratio of P of x to Q of x.”
• Meaning: Any rational function (polynomial over polynomial) can be split into elementary fractions whose integrals are logs, powers, or arctangents.

Why It Matters

Complexity is often just a mask for simple components. By breaking rational functions into partial fractions, we transform “unsolvable” system-wide equations into manageable, elementary pieces that we can calculate with precision.

Core Concepts

• Denominator Factoring: The complexity of the decomposition depends on whether $Q(x)$ has distinct linear factors, repeated linear factors, or irreducible quadratic factors.
• Polynomial Division: If the degree of the numerator is $\geq$ the degree of the denominator, you must divide first to get a polynomial plus a proper rational function.
• Linear Independence: Each factor in the denominator contributes a specific type of term to the sum, with unknown constants that must be solved.
• Decomposition templates:
  • $\frac{1}{(x-a)(x-b)} = \frac{A}{x-a} + \frac{B}{x-b}$
    • How to read: “One divided by the quantity x minus a times the quantity x minus b is equal to A divided by the quantity x minus a plus B divided by the quantity x minus b.”
    • Meaning / when to use: Distinct linear factors each get a simple $\frac{A}{x-r}$ term; solve $A,B$ then integrate each to a log.
  • $\frac{1}{x^2+a^2}$ leads to $\arctan$ results.
    • How to read: “One divided by the quantity x squared plus a squared.”
    • Meaning: Irreducible quadratic denominators integrate to inverse tangent forms.
  • $\frac{1}{x-a}$ leads to $\ln$ results.
    • How to read: “One divided by the quantity x minus a.”
    • Meaning: Simple linear factors integrate directly to $\ln|x-a|$.

Connected Concepts

• Partial Fraction Decomposition
• The LIATE Rule
• Integration by Parts