Partial Fraction Decomposition Formula

Partial fraction decomposition is breaking a rational expression into a sum of simpler fractions whose denominators are the factors of the original.

The Formula

P(x)(x−a)(x−b)=Ax−a+Bx−b\frac{P(x)}{(x-a)(x-b)} = \frac{A}{x-a} + \frac{B}{x-b} Repeated: P(x)(x−a)2=Ax−a+B(x−a)2\frac{P(x)}{(x-a)^2} = \frac{A}{x-a} + \frac{B}{(x-a)^2}. Irreducible quadratic: Ax+Bx2+bx+c\frac{Ax+B}{x^2+bx+c}.

When to use: Just as 712\frac{7}{12} can be split into 13+14\frac{1}{3} + \frac{1}{4}, a complex fraction like 5x−1(x+1)(x−2)\frac{5x-1}{(x+1)(x-2)} can be split into Ax+1+Bx−2\frac{A}{x+1} + \frac{B}{x-2}. The simpler pieces are each easy to integrate.

Quick Example

∫5x−1(x+1)(x−2) dx\int \frac{5x-1}{(x+1)(x-2)}\,dx
Decompose: 5x−1(x+1)(x−2)=2x+1+3x−2\frac{5x-1}{(x+1)(x-2)} = \frac{2}{x+1} + \frac{3}{x-2}.
=2ln⁡∣x+1∣+3ln⁡∣x−2∣+C= 2\ln|x+1| + 3\ln|x-2| + C

Notation

P(x)Q(x)\frac{P(x)}{Q(x)} = proper rational function (deg PP < deg QQ). AA, BB, CC,... are constants to be determined.

What This Formula Means

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

Just as 712\frac{7}{12} can be split into 13+14\frac{1}{3} + \frac{1}{4}, a complex fraction like 5x−1(x+1)(x−2)\frac{5x-1}{(x+1)(x-2)} can be split into Ax+1+Bx−2\frac{A}{x+1} + \frac{B}{x-2}. The simpler pieces are each easy to integrate.

Formal View

If deg⁡P<deg⁡Q\deg P < \deg Q and Q(x)=(x−a1)m1⋯(x−ak)mk(x2+b1x+c1)n1⋯Q(x) = (x-a_1)^{m_1} \cdots (x-a_k)^{m_k}(x^2+b_1x+c_1)^{n_1} \cdots, then P(x)Q(x)=∑i=1k∑j=1miAij(x−ai)j+∑i=1∑j=1niBijx+Cij(x2+bix+ci)j\frac{P(x)}{Q(x)} = \sum_{i=1}^{k} \sum_{j=1}^{m_i} \frac{A_{ij}}{(x-a_i)^j} + \sum_{i=1} \sum_{j=1}^{n_i} \frac{B_{ij}x + C_{ij}}{(x^2+b_ix+c_i)^j} for unique constants Aij,Bij,CijA_{ij}, B_{ij}, C_{ij}.

Worked Examples

Example 1

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

Answer

53ln⁡∣x−2x+1∣+C\frac{5}{3}\ln\left|\frac{x-2}{x+1}\right| + C

First step

1
5(x+1)(x−2)=Ax+1+Bx−2\frac{5}{(x+1)(x-2)} = \frac{A}{x+1}+\frac{B}{x-2}.

Full solution

  1. 2
    x=2x=2: B=5/3B=5/3. x=−1x=-1: A=−5/3A=-5/3.
  2. 3
    ∫=−53ln⁡∣x+1∣+53ln⁡∣x−2∣+C=53ln⁡∣x−2x+1∣+C\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 xx equal to each root of the denominator to isolate each constant.

Example 2

hard
Integrate ∫x2+2x−1x(x−1)2 dx\displaystyle\int \frac{x^2+2x-1}{x(x-1)^2}\,dx.

Example 3

medium
Decompose 4x+1(x−1)(x+3)\dfrac{4x+1}{(x-1)(x+3)}.

Common Mistakes

  • Decomposing an improper fraction directly - do long division first so the leftover is proper (deg top < deg bottom).
  • Using a constant numerator over an irreducible quadratic - quadratics like x2+1x^2+1 need a linear numerator Ax+BAx+B.
  • Forgetting repeated-factor terms - (x−a)2(x-a)^2 needs both Ax−a\frac{A}{x-a} and B(x−a)2\frac{B}{(x-a)^2}, not just one.

Why This Formula Matters

Many rational functions can't be integrated as written but become trivial once split into Ax−a+Bx−b\frac{A}{x-a}+\frac{B}{x-b}, each integrating to a logarithm — so partial fractions is the bridge between algebra and the integral of rational functions. It also reveals the structure hidden by combining fractions. Recognizing it by "Is this a proper rational function whose denominator factors, that I need to break into a sum of simpler fractions?" — rather than by familiar numbers — is what lets a student tell it apart from combining fractions (common denominator) and polynomial long division and factoring in a mixed problem set.

Frequently Asked Questions

What is the Partial Fraction Decomposition formula?

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

How do you use the Partial Fraction Decomposition formula?

Just as 712\frac{7}{12} can be split into 13+14\frac{1}{3} + \frac{1}{4}, a complex fraction like 5x−1(x+1)(x−2)\frac{5x-1}{(x+1)(x-2)} can be split into Ax+1+Bx−2\frac{A}{x+1} + \frac{B}{x-2}. The simpler pieces are each easy to integrate.

What do the symbols mean in the Partial Fraction Decomposition formula?

P(x)Q(x)\frac{P(x)}{Q(x)} = proper rational function (deg PP < deg QQ). AA, BB, CC,... are constants to be determined.

Why is the Partial Fraction Decomposition formula important in Math?

Many rational functions can't be integrated as written but become trivial once split into Ax−a+Bx−b\frac{A}{x-a}+\frac{B}{x-b}, each integrating to a logarithm — so partial fractions is the bridge between algebra and the integral of rational functions. It also reveals the structure hidden by combining fractions. Recognizing it by "Is this a proper rational function whose denominator factors, that I need to break into a sum of simpler fractions?" — rather than by familiar numbers — is what lets a student tell it apart from combining fractions (common denominator) and polynomial long division and factoring in a mixed problem set.

What do students get wrong about Partial Fraction Decomposition?

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.

What should I learn before the Partial Fraction Decomposition formula?

Before studying the Partial Fraction Decomposition formula, you should understand: integral, long division.

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