Partial Fraction Decomposition Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Partial Fraction Decomposition.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

A technique for breaking a rational expression \frac{P(x)}{Q(x)} into a sum of simpler fractions whose denominators are the factors of Q(x). This makes integration of rational functions possible.

Just as \frac{7}{12} can be split into \frac{1}{3} + \frac{1}{4}, a complex fraction like \frac{5x-1}{(x+1)(x-2)} can be split into \frac{A}{x+1} + \frac{B}{x-2}. The simpler pieces are each easy to integrate.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Every proper rational function can be decomposed into partial fractions. The form depends on the factors of the denominator: linear factors give \frac{A}{x-a}, repeated factors give \frac{A}{(x-a)^k}, and irreducible quadratics give \frac{Ax+B}{x^2+bx+c}.

Common stuck point: Make sure the fraction is proper (degree of numerator < degree of denominator) before decomposing. If it's improper, do polynomial long division first.

Sense of Study hint: Factor the denominator completely first, then write the template with unknowns A, B, C and use strategic x-values to solve for them quickly.

Worked Examples

Example 1

easy
Decompose \dfrac{5}{(x+1)(x-2)} and integrate.

Solution

  1. 1
    \frac{5}{(x+1)(x-2)} = \frac{A}{x+1}+\frac{B}{x-2}.
  2. 2
    x=2: B=5/3. x=-1: A=-5/3.
  3. 3
    \int = -\frac{5}{3}\ln|x+1| + \frac{5}{3}\ln|x-2| + C = \frac{5}{3}\ln\left|\frac{x-2}{x+1}\right|+C.

Answer

\frac{5}{3}\ln\left|\frac{x-2}{x+1}\right| + C
Strategic substitution: set x equal to each root of the denominator to isolate each constant.

Example 2

hard
Integrate \displaystyle\int \frac{x^2+2x-1}{x(x-1)^2}\,dx.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
Decompose \dfrac{3x+5}{(x+1)(x+2)}.

Example 2

medium
Integrate \displaystyle\int \frac{1}{x^2-4}\,dx.
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Background Knowledge

These ideas may be useful before you work through the harder examples.

integrallong division