Partial Fraction Calculator

Partial fraction decomposition is a method used to break down a rational function (a fraction with polynomials) into a sum of simpler fractions. It's commonly used in calculus to integrate rational functions, in algebra to solve equations, and in engineering for Laplace transforms.

You typically need partial fractions when:

• Integrating rational functions in calculus
• Solving differential equations using Laplace transforms
• Simplifying complex rational expressions
• Finding inverse Laplace transforms
• Analyzing control systems in engineering

There are four main cases:

• Case 1: Distinct linear factors → A/(ax+b) + B/(cx+d)
• Case 2: Repeated linear factors → A/(ax+b) + B/(ax+b)² + ...
• Case 3: Distinct quadratic factors → (Ax+B)/(ax²+bx+c)
• Case 4: Repeated quadratic factors → (Ax+B)/(ax²+bx+c) + (Cx+D)/(ax²+bx+c)²

To verify your decomposition, combine the partial fractions back into a single fraction by finding a common denominator. The result should equal the original rational function. Our calculator does this verification automatically and shows the steps.

If the degree of the numerator is greater than or equal to the degree of the denominator (improper fraction), you must first perform polynomial long division to separate the quotient and remainder. Then you apply partial fraction decomposition to the remainder term. Our calculator handles this automatically.

Partial fractions break complex rational functions into simpler terms that are easy to integrate. For example:

• ∫ A/(x-a) dx = A ln|x-a| + C
• ∫ A/(x-a)² dx = -A/(x-a) + C
• ∫ (Ax+B)/(x²+px+q) dx involves arctan or natural log

This makes integration much more manageable.

The Heaviside cover-up method is a shortcut for finding coefficients in partial fraction decomposition when factors are linear and distinct. You "cover up" one factor at a time in the original fraction and evaluate the remaining expression at the root of that factor to find its coefficient. It's much faster than solving systems of equations.