Partial Fraction Decomposition Formula

The Formula

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

When to use: 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.

Quick Example

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

Notation

\frac{P(x)}{Q(x)} = proper rational function (deg P < deg Q). A, B, C, ... are constants to be determined.

What This Formula Means

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.

Formal View

If \deg P < \deg Q and Q(x) = (x-a_1)^{m_1} \cdots (x-a_k)^{m_k}(x^2+b_1x+c_1)^{n_1} \cdots, then \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 A_{ij}, B_{ij}, C_{ij}.

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.

Common Mistakes

  • Forgetting to do long division when the degree of the numerator is greater than or equal to the degree of the denominator: \frac{x^3}{x^2-1} must be divided first.
  • Using the wrong form for repeated factors: (x-1)^2 in the denominator requires \frac{A}{x-1} + \frac{B}{(x-1)^2}, NOT just \frac{A}{(x-1)^2}.
  • Using the wrong form for irreducible quadratic factors: x^2 + 1 requires \frac{Ax + B}{x^2+1}, NOT \frac{A}{x^2+1}—you need a linear numerator.

Why This Formula Matters

Partial fractions turn the integration of any rational function into a collection of standard integrals (logarithms and arctangents). It's also used in Laplace transforms, differential equations, and signal processing.

Frequently Asked Questions

What is the Partial Fraction Decomposition formula?

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.

How do you use the Partial Fraction Decomposition formula?

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.

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

\frac{P(x)}{Q(x)} = proper rational function (deg P < deg Q). A, B, C, ... are constants to be determined.

Why is the Partial Fraction Decomposition formula important in Math?

Partial fractions turn the integration of any rational function into a collection of standard integrals (logarithms and arctangents). It's also used in Laplace transforms, differential equations, and signal processing.

What do students get wrong about Partial Fraction Decomposition?

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

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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