Hard

Setting Up a Partial Fraction Decomposition

Simple Explanation

A rational function whose denominator factors into distinct linear factors, like 1/[(x−a)(x−b)], can be rewritten as a sum of simpler fractions A/(x−a) + B/(x−b) — finding the constants A and B is called partial fraction decomposition.

Why Do We Need It?

Complicated rational functions are often impossible to integrate directly, but each simple piece A/(x−a) integrates immediately to a logarithm — decomposition is the key that unlocks integration for a whole family of otherwise-difficult integrals.

Formula

Partial Fraction Decomposition (Distinct Linear Factors)

(px+q) / [(x−a)(x−b)] = A/(x−a) + B/(x−b)

Splits a rational function with two distinct linear factors in its denominator into a sum of simpler fractions, each easy to integrate on its own.

A, B: — constants found by clearing denominators and substituting x=a and x=b
a, b: — the two distinct roots of the denominator

When to use it: Whenever a rational function needs to be integrated and its denominator factors into distinct linear factors.

Worked Example

Decompose a rational function into partial fractions

Decompose 1/[(x−1)(x+2)] into partial fractions.

Why Does This Work?

Substituting the root of one factor (x=1, or x=−2) makes that factor exactly zero, so it wipes out every term that still contains it — leaving a simple equation that isolates just one unknown constant at a time.

Real-Life Example

Analyzing an electrical circuit's response

Circuit analysis often produces a rational function of a variable (like frequency), with a denominator that factors into distinct linear pieces corresponding to different circuit components.

Decomposing that rational function into partial fractions separates the combined response into simpler, individually-understandable pieces.

Practice

For 1/[(x−1)(x+1)] = A/(x−1) + B/(x+1), find the value of A.

Hard

Common mistake

Forgetting that the decomposition setup only works directly this way when the factors are distinct and linear — repeated or non-linear (e.g. irreducible quadratic) factors need a different, more elaborate setup.

Quick Review

For distinct linear factors: (px+q)/[(x−a)(x−b)] = A/(x−a) + B/(x−b).
Clear denominators, then substitute x=a and x=b to isolate A and B one at a time.
Only applies directly to distinct linear factors.