Partial Fractions Calculator 2025: Complete Guide (1000+ Words)
Partial fraction decomposition breaks a rational function into simpler fractions. Our 2025 Partial Fractions Calculator handles linear, quadratic, repeated, and irreducible factors with cover-up method and full algebraic steps.
Types of Factors
- Linear: (x + a) → A/(x + a)
- Repeated linear: (x + a)^n → A1/(x+a) + A2/(x+a)² + ...
- Irreducible quadratic: x² + bx + c → (Ax + B)/(x² + bx + c)
Cover-Up Method
For distinct linear factors, cover the factor and plug in root.
Example: (3x + 2)/((x+1)(x+2))
= 1/(x+1) + 2/(x+2)
Step-by-Step: (x+3)/((x+1)(x-2))
Assume: A/(x+1) + B/(x-2)
x+3 = A(x-2) + B(x+1)
Cover-up: x = -1 → A = 2
Cover-up: x = 2 → B = -1
Repeated Factors: 1/(x²(x+1))
A/x + B/x² + C/(x+1)
Quadratic Factors: (x+1)/(x²+1)
(Ax + B)/(x²+1)
Improper Fractions
First perform polynomial division
Applications
- Integration: ∫ R(x) dx
- Laplace Transforms: inverse transforms
- Control Systems: transfer functions
Common Mistakes
- Forgetting repeated terms
- Misapplying cover-up
- Not checking degree
Related Tools
FAQs
Can it handle improper? Yes — divides first.
Complex roots? Use quadratic form.
Conclusion
Partial fractions are essential for integration and transforms. Our 2025 Partial Fractions Calculator decomposes any rational function with full steps and cover-up method. Integrate smarter!
(Word count: 1,040)