Partial Fraction Calculator

Partial Fraction Decomposition Calculators

Partial fraction decomposition is a technique used to break down a rational function (a fraction where the numerator and denominator are polynomials) into a sum of simpler fractions. This is particularly useful in calculus for integrating rational functions, solving differential equations, or simplifying expressions. The process involves factoring the denominator and solving for coefficients in the partial fractions.

Your queries suggest you’re looking for online tools to perform this decomposition, ideally with step-by-step explanations, free access, and support for integrals. Based on reliable sources, here are some of the best free online partial fraction calculators. I’ve prioritized those that show detailed steps and are user-friendly. Most handle improper fractions by first performing polynomial long division.

Quick Technique Overview (Heaviside Cover-Up Method Example)

For a rational function like 5x−4×2−x−2\frac{5x-4}{x^2 – x – 2}x2−x−25x−4​, factor the denominator: $(x-2)(x+1)$. Assume 5x−4(x−2)(x+1)=Ax−2+Bx+1\frac{5x-4}{(x-2)(x+1)} = \frac{A}{x-2} + \frac{B}{x+1}(x−2)(x+1)5x−4​=x−2A​+x+1B​.

• Multiply through: $5x-4=A(x+1)+B(x-2)$.
• Cover up $x-2$: Set $x=2$, get A=5(2)−42+1=2A = \frac{5(2)-4}{2+1} = 2A=2+15(2)−4​=2.
• Cover up $x+1$: Set $x=-1$, get B=5(−1)−4−1−2=−3B = \frac{5(-1)-4}{-1-2} = -3B=−1−25(−1)−4​=−3.
• Result: 2x−2−3x+1\frac{2}{x-2} – \frac{3}{x+1}x−22​−x+13​.

For integrals, decompose first, then integrate each term (e.g., ∫2x−2dx=2ln⁡∣x−2∣\int \frac{2}{x-2} dx = 2\ln|x-2|∫x−22​dx=2ln∣x−2∣).

If you provide a specific rational function (e.g., x2+1×3−x\frac{x^2 + 1}{x^3 – x}x3−xx2+1​), I can compute and explain the decomposition step-by-step right here!