Partial Fractions Calculator to Decompose Ration...
Decompose any rational function into partial fractions. Essential for calculus integration, inverse Laplace transforms, and solving differential equations.
What is Partial Fraction Decomposition?
Partial fraction decomposition splits a complex rational function into simpler fractions that are easier to integrate or invert. It is a standard technique in calculus (for integration), control theory (inverse Laplace transforms), and signal processing. The denominator's roots determine the form of the decomposition.
📊 Partial Fractions Calculator
Free calculator for instant results.
📐 Formula
P(x)/Q(x) = A/(ax+b) + B/(cx+d) + ...
For each distinct linear factor (ax+b) in denominator: add term A/(ax+b). For repeated factor (ax+b)²: add A/(ax+b)+B/(ax+b)². Quadratic factors: add (Ax+B)/(x²+bx+c).
📝 Worked Example
Decompose 5x+3 / [(x+1)(x+2)]:
= A/(x+1) + B/(x+2)
5x+3 = A(x+2)+B(x+1)
x=−1: −2=A(1) → A=−2
x=−2: −7=B(−1) → B=7
Answer: −2/(x+1) + 7/(x+2)
📝 How to Use
❓ FAQ
When can partial fractions be applied?
Only when degree of numerator < degree of denominator. If not, do polynomial long division first to get a proper fraction remainder.
How are partial fractions used in integration?
∫ P(x)/Q(x) dx = ∫ [A/(x+a) + B/(x+b)] dx = A ln|x+a| + B ln|x+b| + C. Each simple term is directly integrable.
Veer Kumavat
Founder & AuthorVeer is a 14-year-old student from Nashik, Maharashtra, who built SciFi Calculators to help students worldwide master STEM subjects. He is passionate about making complex science and math problems accessible through intuitive digital tools.