Partial Fraction Decomposition
Decompose rational expressions into partial fractions for easier integration and simplification.
Input Rational Expression
Use polynomial numerators and products of rational linear factors, including repeated factors (degree at most 12).
Theory & Formula
Partial Fraction Decomposition
Partial fraction decomposition is a method to break down complex rational expressions into simpler fractions. This technique is particularly useful for integration.
Method
For linear factors: Split the fraction into sum of simpler fractions
\(\frac{P(x)}{(x-a)(x-b)} = \frac{A}{x-a} + \frac{B}{x-b}, \quad a \ne b, \quad \deg P < 2\)Example
\(\frac{5x + 7}{(x+1)(x+2)} = \frac{A}{x+1} + \frac{B}{x+2}\)\(5x + 7 = A(x+2) + B(x+1)\)\(A = 2, B = 3\)\(\frac{5x + 7}{(x+1)(x+2)} = \frac{2}{x+1} + \frac{3}{x+2}\)