1. When Partial Fractions Are Used
Partial fractions help when integrating rational functions where both numerator and denominator are polynomials. The idea is to break a complicated fraction into simpler fractions that are easy to integrate. In personal-notes style: turn one big messy fraction into a few small clean ones.
2. Condition for Applying Partial Fractions
The method works only when the rational function is proper. That means the degree of the numerator must be less than the degree of the denominator.
2.1. If Not Proper
If the numerator's degree is higher or equal, first perform polynomial division, then apply partial fractions to the remainder.
3. Types of Denominators and Their Decomposition
The form of the denominator determines how the fraction is split. These are the standard cases.
3.1. Case 1: Distinct Linear Factors
If the denominator is (x − a)(x − b), the split is:
\frac{1}{(x-a)(x-b)} = \frac{A}{x-a} + \frac{B}{x-b}
3.2. Case 2: Repeated Linear Factors
For (x − a)^n, the split becomes:
\frac{1}{(x-a)^n} = \frac{A_1}{x-a} + \frac{A_2}{(x-a)^2} + \cdots + \frac{A_n}{(x-a)^n}
3.3. Case 3: Irreducible Quadratic Factors
If the denominator includes a quadratic like (x² + px + q), the numerator is linear:
\frac{1}{x^2 + px + q} = \frac{Ax + B}{x^2 + px + q}
4. Steps to Decompose a Rational Function
- Factor the denominator completely.
- Write the partial fraction form based on the factor types.
- Multiply through by the denominator to remove fractions.
- Compare coefficients or substitute values to find constants.
- Integrate each simpler fraction.
5. Examples
These examples show how decomposition makes integration easier.
5.1. Example 1 — Distinct Linear Factors
Integrate:
\int \frac{1}{x^2 - 1} dx
Factor denominator:
x^2 - 1 = (x - 1)(x + 1)
Split:
\frac{1}{(x-1)(x+1)} = \frac{1}{2} \left(\frac{1}{x-1} - \frac{1}{x+1}\right)
Integrate:
\int \frac{1}{x^2 - 1} dx = \frac{1}{2} \ln|x-1| - \frac{1}{2} \ln|x+1| + C
5.2. Example 2 — Repeated Linear Factor
Integrate:
\int \frac{3x + 5}{(x + 2)^2} dx
Decompose:
\frac{3x + 5}{(x + 2)^2} = \frac{A}{x+2} + \frac{B}{(x+2)^2}
Find A and B, then integrate.
5.3. Example 3 — Irreducible Quadratic
Integrate:
\int \frac{2x}{x^2 + 4} dx
Since numerator is derivative of denominator, substitution works quickly, but partial fractions form is:
\frac{2x}{x^2 + 4} = \frac{Ax + B}{x^2 + 4}
Here, A = 2, B = 0.
Integral:
\int \frac{2x}{x^2 + 4} dx = \ln(x^2 + 4) + C
6. Why Partial Fractions Are Useful
This method breaks down otherwise difficult rational integrals into basic forms such as ln, arctan, or simple power formulas. It is one of the most reliable tools for integrating rational expressions.