Partial Fractions

Partial fractions is an algebraic technique that breaks a complicated rational function — a ratio of two polynomials — into a sum of simpler fractions that are much easier to integrate. Just as you might add 1/2 + 1/3 to get 5/6, partial fractions works in reverse: it splits 5/6 back into 1/2 + 1/3. In calculus, this reverse process is essential because integrals of simple fractions like 1/(x − a) or 1/(x² + bx + c) are straightforward, while the combined form is not. The technique appears constantly in calculus courses, physics (Laplace transforms, circuit analysis), and engineering. Whenever you encounter an integrand of the form P(x)/Q(x) where the degree of P is less than the degree of Q and Q factors nicely, partial fractions is almost certainly the right tool. If the degree of the numerator is greater than or equal to that of the denominator, polynomial long division must be performed first to reduce the expression to a proper rational function before decomposition. There are three main cases to master: (1) the denominator has distinct linear factors, (2) the denominator has repeated linear factors, and (3) the denominator has irreducible quadratic factors. Each case produces a different form for the partial fraction decomposition. Once you have separated the fraction into its component parts, you integrate each piece using the natural logarithm rule or arctangent rule, producing a clean closed-form antiderivative.

Key takeaways

Check That the Fraction Is Proper
Factor the Denominator Completely
Write the Decomposition Template
Solve for the Unknown Constants
Integrate Each Partial Fraction

Check That the Fraction Is Proper

A rational function P(x)/Q(x) is proper when deg(P) < deg(Q). If deg(P) ≥ deg(Q), use polynomial long division to write P(x)/Q(x) = S(x) + R(x)/Q(x), where S(x) is a polynomial and R(x)/Q(x) is now proper. Integrate S(x) normally and apply partial fractions to R(x)/Q(x).

Factor the Denominator Completely

Factor Q(x) into linear factors (ax + b) and irreducible quadratic factors (ax² + bx + c, where b² − 4ac < 0) over the reals. For example, x³ − x = x(x − 1)(x + 1) gives three distinct linear factors, while x³ + x = x(x² + 1) gives one linear and one irreducible quadratic factor.

Write the Decomposition Template

For each distinct linear factor (ax + b), introduce a constant A in the numerator: A/(ax + b). For each repeated linear factor (ax + b)^n, introduce n terms: A₁/(ax + b) + A₂/(ax + b)² + … + Aₙ/(ax + b)ⁿ. For each irreducible quadratic factor (ax² + bx + c), introduce a linear numerator: (Bx + C)/(ax² + bx + c). Combine all these terms and set the result equal to the original fraction.

Solve for the Unknown Constants

Multiply both sides of the equation by Q(x) to clear all denominators. This gives a polynomial identity. You can find the constants by (a) substituting the roots of each linear factor — called the cover-up method — or (b) expanding and matching coefficients of like powers of x on both sides. Both methods must give the same answer.

Integrate Each Partial Fraction

Use these standard results: ∫ A/(ax+b) dx = (A/a) ln|ax+b| + C; ∫ A/(ax+b)ⁿ dx = A · (ax+b)^(1−n) / [a(1−n)] + C for n ≥ 2; ∫ (Bx+C)/(x²+px+q) dx requires completing the square in the denominator and splitting into a logarithmic part and an arctangent part. Each piece integrates to a recognizable elementary function.

The Key Equation

\frac{P(x)}{Q(x)} = \frac{A_1}{a_1 x + b_1} + \frac{A_2}{a_2 x + b_2} + \cdots + \frac{B x + C}{a x^2 + b x + c}

P(x) — Numerator polynomial (dimensionless): The polynomial in the numerator; its degree must be strictly less than the degree of Q(x) for the decomposition to apply directly.
Q(x) — Denominator polynomial (dimensionless): The fully factored denominator polynomial, written as a product of linear and irreducible quadratic factors.
A_i — Partial fraction constant (dimensionless): An unknown real constant associated with each linear factor; determined by the cover-up method or coefficient matching.
Bx + C — Linear numerator for quadratic factor (dimensionless): A linear polynomial placed in the numerator above each irreducible quadratic factor; both B and C are unknown constants to be determined.

Worked Example

Problem. Evaluate ∫ (3x + 5) / [(x − 1)(x + 3)] dx.

2\ln|x-1| + \ln|x+3| + C

Answer. 2 ln|x − 1| + ln|x + 3| + C

Figure

Diagram

Origins · 1 minGottfried Wilhelm Leibniz and how this was discovered

Gottfried Wilhelm Leibniz

1646–1716 · German

Leibniz co-invented calculus independently of Newton and developed much of the notation we use today, including the integral sign ∫ and dy/dx for derivatives. He also worked extensively on decomposing rational functions, laying early groundwork for systematic integration methods including what would become partial fractions. His correspondence with Johann Bernoulli in the 1690s explicitly discussed techniques for integrating rational functions by splitting them into simpler parts. Leibniz designed a mechanical calculator called the Stepped Reckoner in 1673 that could multiply, divide, add, and subtract — a remarkable feat of engineering centuries before electronic computers.

U-Substitution — U-substitution is the other major algebraic integration technique. Recognizing which method — substitution or partial fractions — applies to a given rational integrand is a key skill in integral calculus.
Completing the Square — When the denominator contains an irreducible quadratic factor, completing the square is necessary to convert it into a form suitable for the arctangent integration formula, making it an essential companion technique.
Polynomial Long Division — Polynomial long division must be applied before partial fractions whenever the rational function is improper (numerator degree ≥ denominator degree), making it a prerequisite step.
Natural Logarithm Integration Rule — The result ∫ 1/(ax+b) dx = (1/a) ln|ax+b| + C is the most commonly used formula after partial fraction decomposition; understanding it deeply is essential to completing these integrals.