Engineering Fluency OS

Why This Matters

When you need to integrate a fraction whose numerator and denominator are both polynomials, direct methods often fail. The integral of 1/(x^2 - 1) does not match any standard formula, but if you rewrite it as 1/(2(x - 1)) minus 1/(2(x + 1)), each piece integrates easily to a logarithm. This rewriting process is partial fraction decomposition.

The technique works by expressing a complex rational integrand as a sum of simpler fractions whose denominators are linear or irreducible quadratic factors. The decomposition process involves factoring the denominator, setting up unknown coefficients for each partial fraction, and solving a system of equations to find those coefficients. Once decomposed, each fraction integrates using basic rules.

Partial fractions appear in Laplace transforms, transfer functions in control theory, and probability generating functions. Any time a rational expression arises in applied math or engineering, partial fractions is the standard tool for breaking it into manageable pieces.

Define Terms

Visual Model

Rational IntegrandP(x) / Q(x)

Check Degreesdeg(P) < deg(Q)?

Long DivisionIf deg(P) >= deg(Q)

Factor DenominatorLinear and quadratic factors

Set Up Partial FractionsA/(x-r) + B/(x-s) + ...

Solve for CoefficientsSystem of equations

Integrate Each Fractionln, arctan, etc.

Final AnswerSum of simple integrals

The full process at a glance. Click Start tour to walk through each step.

Partial fraction decomposition breaks a complex rational integrand into simple fractions that integrate individually.

Code Example

Code

// Partial Fraction Decomposition
// Decompose (2x + 3) / (x^2 - 1) = A/(x-1) + B/(x+1)

// Solve: 2x + 3 = A(x+1) + B(x-1)
// x = 1:  2(1)+3 = A(2) => A = 5/2
// x = -1: 2(-1)+3 = B(-2) => B = -1/2

const A = 5 / 2;
const B = -1 / 2;
console.log("A =", A);   // 2.5
console.log("B =", B);   // -0.5

// Verify: A/(x-1) + B/(x+1) should equal (2x+3)/(x^2-1)
function original(x) {
  return (2 * x + 3) / (x * x - 1);
}

function decomposed(x) {
  return A / (x - 1) + B / (x + 1);
}

// Test at several points
for (const x of [2, 3, 5, 10]) {
  const orig = original(x).toFixed(6);
  const dec = decomposed(x).toFixed(6);
  console.log(`x=${x}: original=${orig}, decomposed=${dec}`);
}

// Integrate from 2 to 5
// integral = A*ln|x-1| + B*ln|x+1| evaluated from 2 to 5
function antideriv(x) {
  return A * Math.log(Math.abs(x - 1)) + B * Math.log(Math.abs(x + 1));
}
const integral = antideriv(5) - antideriv(2);
console.log("Integral from 2 to 5:", integral.toFixed(6));

Interactive Experiment

Try these exercises:

Decompose 1/(x^2 - 4) into partial fractions. Factor the denominator as (x-2)(x+2) and solve for the coefficients.
What happens when the denominator has a repeated factor, like 1/(x-1)^2? How does the partial fraction setup change?
Try decomposing (x^2 + 1)/(x^3 - x). You need three terms. Factor the denominator completely first.
Verify your decomposition numerically by evaluating the original and decomposed forms at several x values.
Use the decomposition to integrate 1/(x^2 - 4) from 3 to 10. Compare the exact answer with numerical integration.

Quick Quiz

Coding Challenge

Partial Fraction Decomposition

Write a function called `decompose` that takes three numbers: the coefficients a, b of the numerator (ax + b) and two roots r1, r2 of the denominator (x - r1)(x - r2). It should return a string in the format 'A=___, B=___' where A and B are the partial fraction coefficients rounded to 2 decimal places. Use the cover-up method: A = (a*r1 + b)/(r1 - r2) and B = (a*r2 + b)/(r2 - r1).

Loading editor...

Real-World Usage

Partial fraction decomposition appears throughout applied mathematics and engineering:

Laplace transforms: Inverting Laplace transforms in control theory and circuit analysis requires partial fractions to decompose transfer functions into simpler terms.
Signal processing: Z-transforms in digital signal processing are inverted using partial fractions to find the impulse response of a discrete-time system.
Probability: Generating functions for discrete distributions are often rational, and partial fractions help extract individual probability terms.
Differential equations: Solving linear ODEs with constant coefficients via Laplace transforms hinges on partial fraction decomposition.
Computer algebra systems: CAS tools like Mathematica and SymPy implement partial fractions as a core simplification routine used throughout symbolic computation.

Partial Fractions