Integration by Partial Fractions Calculator

The Integration by Partial Fractions Calculator is designed for rational functions, where one polynomial is divided by another. When the fraction can be decomposed into simpler fractions, each part may be integrated using familiar logarithmic or power rules.

When Partial Fractions Apply

The method is generally used after ensuring that the rational expression is proper, meaning the numerator degree is lower than the denominator degree. If it is improper, polynomial division is normally performed first.

Basic Decomposition

For a denominator with distinct linear factors, a fraction such as P(x)/[(x−a)(x−b)] can be written as A/(x−a) + B/(x−b). Repeated or irreducible quadratic factors require different decomposition forms.

Example

Consider 1/[x(x+1)]. It can be decomposed as 1/x − 1/(x+1). Integrating gives ln|x| − ln|x+1| + C. The decomposition turns one rational expression into two standard logarithmic integrals.

Using the Calculator

Enter the rational function in the supported format. Check that factors and powers are grouped correctly. The calculator may show the decomposition and the resulting integral, which makes it easier to compare the algebra with a manual solution.

Domain Restrictions

The original denominator determines where the function is undefined. Simplifying or combining logarithms in the final expression should not be used to ignore those restrictions. Keep the excluded values from the original integrand in mind.