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

Breaking a rational expression into a sum of simpler fractions whose denominators are the factors of the original denominator.

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: Split a rational function into simpler fractions whose denominators are the original's factors.

Common stuck point: The procedure for partial fraction decomposition is the easy part; the trap is decomposing an improper fraction directly. Asking "Is this a proper rational function whose denominator factors, that I need to break into a sum of simpler fractions?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Is this a proper rational function whose denominator factors, that I need to break into a sum of simpler fractions?

Worked Examples

Example 1

easy

Decompose

\dfrac{5}{(x+1)(x-2)}

and integrate.

Answer

\frac{5}{3}\ln\left|\frac{x-2}{x+1}\right| + C

First step

\frac{5}{(x+1)(x-2)} = \frac{A}{x+1}+\frac{B}{x-2}

Full solution

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

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

Example 3

medium

Decompose

\dfrac{4x+1}{(x-1)(x+3)}

Example 4

medium

Integrate

\displaystyle\int \frac{1}{x^2 - 9}\,dx

Example 5

hard

Decompose

\dfrac{x+4}{(x-1)^2}

Example 6

hard

Decompose

\dfrac{2x^2+3}{(x-1)(x^2+1)}

Example 7

medium

Integrate

\displaystyle\int \frac{1}{x(x+2)}\,dx

Example 8

hard

Compute

\dfrac{x^3 + 2}{x^2 - 1}

as a polynomial plus a proper fraction.

Example 9

medium

Integrate

\displaystyle\int \frac{3x+5}{(x+1)(x+2)}\,dx

Example 10

hard

Decompose

\dfrac{1}{x^3 - x}

Example 11

challenge

Use partial fractions to evaluate

\displaystyle\sum_{n=1}^{\infty} \frac{1}{n(n+1)}

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

Example 3

easy

What is the partial fraction form for

\frac{1}{(x+1)(x-2)}

Example 4

easy

Decompose

\frac{5x-1}{(x+1)(x-2)}

: find

A

for

\frac{A}{x+1}

Example 5

easy

Decompose

\frac{5x-1}{(x+1)(x-2)}

: find

B

for

\frac{B}{x-2}

Example 6

easy

Does

\frac{x^3}{x^2-1}

need long division before decomposing?

Example 7

easy

What is the form for

\frac{3}{(x-1)^2}

Example 8

easy

What numerator does the factor

x^2+1

require in a decomposition?

Example 9

easy

Combine

\frac{1}{x}+\frac{1}{x+1}

into a single fraction.

Example 10

easy

How many unknown constants appear in the decomposition of

\frac{1}{x(x-1)(x-2)}

Example 11

medium

Fully decompose

\frac{5x-1}{(x+1)(x-2)}

Example 12

medium

Decompose

\frac{1}{x^2-1}

Example 13

medium

Find

A

\frac{x+3}{(x-1)^2}=\frac{A}{x-1}+\frac{B}{(x-1)^2}

Example 14

medium

Decompose

\frac{2x}{(x^2+1)(x-1)}

: set up the equation for the constants.

Example 15

medium

After long division, write

\frac{x^2}{x-1}

in mixed form.

Example 16

medium

Use

\frac{1}{x^2-1}=\frac{1/2}{x-1}-\frac{1/2}{x+1}

to integrate

\int\frac{dx}{x^2-1}

Example 17

medium

Decompose

\frac{3x+5}{x^2+5x+6}

Example 18

medium

How many unknowns are needed for

\frac{1}{x(x^2+4)^2}

Example 19

medium

Decompose

\frac{x+4}{x^2+x}

Example 20

challenge

Decompose

\frac{x^2+1}{x(x-1)(x+1)}

Example 21

challenge

Decompose

\frac{x}{(x-1)^2(x+1)}

Example 22

challenge

Decompose

\frac{4}{x^3+x}=\frac{4}{x(x^2+1)}

Example 23

easy

Decompose

\dfrac{1}{(x-1)(x+1)}

Example 24

easy

Decompose

\dfrac{2x}{(x-1)(x+1)}

Example 25

easy

What is the partial-fraction form of

\dfrac{P(x)}{(x-1)(x-2)(x-3)}

Example 26

medium

Decompose

\dfrac{3x-2}{x^2 - x - 2}

Example 27

hard

Integrate

\displaystyle\int \frac{x+4}{(x-1)^2}\,dx

Example 28

medium

What is the proper partial-fraction form for

\dfrac{P(x)}{(x^2+1)(x-2)}

Example 29

medium

Decompose

\dfrac{1}{x(x+2)}

Example 30

medium

Before decomposing

\dfrac{x^3 + 2}{x^2 - 1}

, what must you do first?

Example 31

hard

Continue: decompose

\dfrac{x+2}{x^2-1}

Example 32

medium

Set up (do not solve) the partial-fraction decomposition of

\dfrac{1}{(x-2)^3(x+5)}

Example 33

medium

Combine into a single fraction:

\dfrac{2}{x-1} + \dfrac{3}{x+2}

Example 34

easy

Decompose

\dfrac{1}{(x-2)(x+3)}

Example 35

hard

Integrate

\displaystyle\int \frac{1}{x^3 - x}\,dx

Example 36

medium

Decompose

\dfrac{5}{(x+2)(x-3)}

Example 37

medium

Combine

\dfrac{3}{x-2} - \dfrac{2}{x+1}

into a single fraction.

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Related Concepts

Integral Long Division Improper Integrals

Background Knowledge

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

integrallong division