Partial Fractions Decomposition: The Complete Workflow

Partial fraction decomposition is the algebraic skill that lets you integrate any rational function on the planet. Instead of fighting one ugly fraction, you split it into pieces that are easy to integrate term by term. This guide walks through every case you'll meet.

The setup

A rational function is $\frac{P(x)}{Q(x)}$ where $P, Q$ are polynomials. Partial fractions only works when the degree of $P$ < degree of $Q$. If not, do polynomial long division first to peel off the polynomial part.

After dividing, factor $Q(x)$ completely over the reals. Every factor falls into one of four categories.

The four cases

Case 1: distinct linear factors

If $Q(x) = (x - a)(x - b)$, write:

$\frac{P(x)}{(x-a)(x-b)} = \frac{A}{x-a} + \frac{B}{x-b}$

Example. Decompose $\frac{5x - 1}{(x - 1)(x + 2)}$.

Multiply through: $5x - 1 = A(x + 2) + B(x - 1)$.

Plug $x = 1$: $4 = 3A \Rightarrow A = 4/3$.
Plug $x = -2$: $-11 = -3B \Rightarrow B = 11/3$.

So $\frac{5x-1}{(x-1)(x+2)} = \frac{4/3}{x-1} + \frac{11/3}{x+2}$.

Case 2: repeated linear factor

For $(x - a)^k$, you need one term per power up to $k$:

$\frac{A_1}{x-a} + \frac{A_2}{(x-a)^2} + \dots + \frac{A_k}{(x-a)^k}$

Case 3: irreducible quadratic factor

For each irreducible $x^2 + bx + c$, use a numerator with two unknowns:

$\frac{Bx + C}{x^2 + bx + c}$

Case 4: repeated irreducible quadratic

Same idea as case 2, but each power gets a $Bx + C$ form.

Integration application

Once decomposed, integrate term by term:

• $\int \frac{1}{x - a} dx = \ln|x - a| + C$
• $\int \frac{1}{(x - a)^k} dx = \frac{-1}{(k-1)(x-a)^{k-1}} + C$ for $k > 1$
• $\int \frac{Bx + C}{x^2 + bx + c} dx$ splits into a $\ln$ part and an $\arctan$ part.

Common mistakes

• Forgetting to do long division first when degree of $P$ ≥ degree of $Q$.
• Skipping repeated terms — $(x - 1)^3$ requires three separate fractions.
• Trying to factor irreducible quadratics — check the discriminant before forcing real roots.

Try with the AI Integral Solver

The Integral Solver automatically does partial fraction decomposition when needed and shows every step.

Related references:

• Factor Calculator — for breaking $Q(x)$ apart
• Polynomial Calculator — for the long-division setup
• Limit Calculator — used in some PFD verification tricks