Decompose Rational Functions
Decompose rational functions into simpler partial fractions. Our calculator breaks down complex rational expressions into a sum of simpler fractions, essential for integration and Laplace transforms in calculus.
Enter a rational function to perform partial fraction decomposition.
Partial fraction decomposition is a technique used to break down rational functions into a sum of simpler fractions. This is especially useful for integrating rational functions in calculus.
Consider the rational function:
The partial fraction decomposition would be:
Where the original rational function is expressed as a sum of simpler fractions.
Partial fraction decomposition expresses a rational function P(x)/Q(x) as a sum of simpler fractions. Requirements: degree of P < degree of Q (if not, divide first). Factor Q(x) into linear and irreducible quadratic factors. Set up equation with unknown coefficients, solve for coefficients.
Linear factors: (x-a) gives A/(x-a). Repeated linear: (x-a)ⁿ gives A₁/(x-a) + A₂/(x-a)² + ... + Aₙ/(x-a)ⁿ. Irreducible quadratic: ax²+bx+c gives (Ax+B)/(ax²+bx+c). Each factor type has specific decomposition pattern.
Partial fractions are crucial for: integrating rational functions, solving differential equations, inverse Laplace transforms, control system analysis, and electrical circuit analysis. Transforms difficult integrals into sums of easier integrals that can be solved using basic techniques.
