Integration by Partial Fractions — Mathematics Knowledge System

Overview

Integration by partial fractions integrates rational functions (ratios of polynomials) by first decomposing them into a sum of simpler fractions — each with a linear or irreducible quadratic denominator — that can be integrated individually using basic log, arctan, or power rule antiderivatives.

Intuition

A complicated rational function like (3x+5)/[(x-1)(x+2)] is hard to integrate directly, but it's algebraically equal to a sum of simpler pieces, A/(x-1) + B/(x+2), each of which integrates to a natural log. Partial fraction decomposition is the algebra step that unlocks this; integration is comparatively easy once the decomposition is found.

Interactive Graph

Formal Definition

Definition

For a proper rational function P(x)/Q(x) with Q(x) factored into linear and irreducible quadratic factors, the standard antiderivatives used after decomposition are:

\int \frac{A}{x-a}\,dx = A\ln|x-a| + C

Linear factor term

\int \frac{A}{(x-a)^n}\,dx = \frac{-A}{(n-1)(x-a)^{n-1}} + C \quad (n>1)

Repeated linear factor term

\int \frac{Bx+C}{x^2+px+q}\,dx \ \text{(irreducible quadratic)}: \text{split into a log term and an arctan term}

Irreducible quadratic factor term

Applications

Inverse Laplace transforms in control systems routinely require decomposing a rational transfer function into partial fractions before integrating or looking up standard transform pairs.

Worked Examples

Decompose: (3x+5)/[(x-1)(x+2)] = A/(x-1) + B/(x+2). Solve for A, B: 3x+5=A(x+2)+B(x-1).
$3x+5 = A(x+2)+B(x-1)$
At x=1: 8=3A ⟹ A=8/3. At x=-2: -1=-3B ⟹ B=1/3.
$A=\frac{8}{3},\ B=\frac{1}{3}$
Integrate each term.
$\int \left[\frac{8/3}{x-1}+\frac{1/3}{x+2}\right]dx = \frac{8}{3}\ln|x-1| + \frac{1}{3}\ln|x+2| + C$

Answer: (8/3)ln|x-1| + (1/3)ln|x+2| + C

Practice Problems

Difficulty 6/10

Evaluate ∫ (x+1)/[x(x-2)] dx.

Common Mistakes

Common Mistake

Applying partial fraction decomposition to an improper rational function (degree of numerator ≥ degree of denominator) without first doing polynomial long division.

Partial fractions only decomposes a PROPER rational function. If the numerator's degree is not less than the denominator's, perform polynomial division first to write it as a polynomial plus a proper remainder fraction.

Common Mistake

Using a single constant A/(x-a) term for a repeated linear factor (x-a)².

A repeated factor (x-a)² requires TWO terms in the decomposition: A/(x-a) + B/(x-a)², not just one.

Summary

Partial fraction decomposition rewrites a proper rational function as a sum of simpler fractions with linear or irreducible quadratic denominators.
Linear factor terms A/(x-a) integrate to A ln|x-a|.
Irreducible quadratic terms typically split into a logarithmic piece and an arctangent piece.
Always check the rational function is proper first; if not, do polynomial long division before decomposing.

References

WebsiteWikipedia — Partial fraction decomposition