Decompose rational functions P(x)/Q(x) into partial fractions with linear and repeated roots using Heaviside’s cover-up method for calculus integration.
Did this calculation save you time?
Student ProjectHi! I'm a student developer building Calculat in my spare time. I was tired of searching for basic math tools and having to click through 10 spammy popups, loan ads, and cookie trackers.
I keep this website 100% free, private, and ad-free. If this helped you with your homework, project, or finances today, bookmarking this page or telling a friend helps me keep building more free tools!
The Partial Fraction Decomposition Calculator decomposes proper rational functions P(x) / Q(x) into a sum of simpler fractions. This algebraic technique is an essential prerequisite for computing indefinite integrals and inverse Laplace transforms in calculus and engineering.
\frac{P(x)}{(x - r_1)(x - r_2)} = \frac{A}{x - r_1} + \frac{B}{x - r_2}, \quad A = \frac{P(r_1)}{r_1 - r_2}, \quad B = \frac{P(r_2)}{r_2 - r_1}Heaviside’s Cover-Up Method solves for unknown coefficients directly by substituting the roots of each denominator factor.
Scenario: Decomposing (3x + 5) / [(x - 1)(x - 2)]
Form: A / (x - 1) + B / (x - 2).
Evaluate A at x = 1: A = (3(1) + 5) / (1 - 2) = 8 / (-1) = -8.
Evaluate B at x = 2: B = (3(2) + 5) / (2 - 1) = 11 / 1 = 11.
Decomposition: -8 / (x - 1) + 11 / (x - 2).
Direct integration of complex rational expressions is often difficult. Decomposing into partial fractions turns the integrand into standard logarithmic forms: ∫ [A / (x - r)] dx = A · ln|x - r|.
It is an algebraic shortcut developed by Oliver Heaviside to find coefficients of non-repeated linear factors without solving simultaneous linear equations.
If deg(P) ≥ deg(Q), polynomial long division must be performed first to produce a polynomial quotient plus a proper rational remainder.
For a factor (x - r)², the decomposition includes two terms: A / (x - r) + B / (x - r)², solved by equating polynomial coefficients.
Solve linear equations (ax+b=0), quadratic equations (ax²+bx+c=0 with discriminant Δ and complex roots), and 2x2 systems with step-by-step solutions.
Advanced online scientific calculator. Trigonometric functions (sin, cos, tan), natural and common logarithms, powers, roots, factorials, and memory.
Calculate the principal real cube root of positive and negative numbers. Identifies perfect cubes, nearest integer roots, and complex conjugate roots.