Partial Fraction Decomposition Calculator
Decompose rational functions P(x)/Q(x) into partial fractions with linear and repeated roots using Heaviside’s cover-up method for calculus integration.
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What Is a Partial Fraction Decomposition Calculator?
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.
How to Use This Calculator
- Enter numerator coefficients for P(x) = p₁x + p₀.
- Enter the roots of the factored denominator Q(x) = (x - r₁)(x - r₂).
- Review the decomposed partial fraction expression and the step-by-step Heaviside cover-up calculations.
Distinct Linear Factors Decomposition & Heaviside Method
\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.
Worked Example
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).
Tips & Key Notes
- Heaviside’s method lets you find coefficients without setting up or solving simultaneous linear equations.
- Always verify that your rational function is proper (degree of numerator strictly less than degree of denominator).
- Integrating the decomposed result yields simple logarithmic terms: -8 ln|x - 1| + 11 ln|x - 2| + C.
Frequently Asked Questions
Why is partial fraction decomposition used in calculus?
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|.
What is Heaviside’s Cover-Up Method?
It is an algebraic shortcut developed by Oliver Heaviside to find coefficients of non-repeated linear factors without solving simultaneous linear equations.
What if the degree of the numerator is greater than or equal to the denominator?
If deg(P) ≥ deg(Q), polynomial long division must be performed first to produce a polynomial quotient plus a proper rational remainder.
How are repeated roots handled?
For a factor (x - r)², the decomposition includes two terms: A / (x - r) + B / (x - r)², solved by equating polynomial coefficients.
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