Integration of Rational Functions by Partial Fraction Decomposition
The Big Promise
Every rational function — that is, every quotient $\dfrac{P(x)}{Q(x)}$ of two polynomials — can be integrated in elementary terms (logarithms, arctangents, and rational expressions). This is the one large class of functions where integration is an algorithm, not an art. No guessing, no tricks — just follow the steps.
1. The Algorithm Roadmap
The entire method follows four stages, executed in order:
Properness Check
If $\deg P \ge \deg Q$, perform polynomial long division to write $\frac{P}{Q} = A(x) + \frac{R(x)}{Q(x)}$ with $\deg R < \deg Q$.
Factor the Denominator
Factor $Q(x)$ into linear factors $(ax+b)$ and irreducible quadratic factors $(ax^2+bx+c)$, each possibly repeated.
Set Up the Template
Write the partial fraction sum dictated by the factor types (constants over linears, linear numerators over quadratics).
Find Coefficients & Integrate
Solve for the unknown constants, then integrate each simple fraction using logarithms, power rules, or arctangent.