Integration by Partial Fractions Calculator

The Integration by Partial Fractions Calculator simplifies rational-function integrals by decomposing a fraction into smaller terms that are easier to integrate. Use the tool to work through the calculation and review the result.

Definite
Indefinite

with respect to

Upper value

Lower value

The Integration by Partial Fractions Calculator simplifies rational-function integrals by decomposing a fraction into smaller terms that are easier to integrate.

When partial fractions apply

The method is used for a proper rational function whose denominator can be factored into suitable linear or irreducible quadratic factors. Repeated factors require corresponding repeated terms in the decomposition.

Worked example

Consider ∫ 1/[x(x+1)] dx. Decompose the fraction as 1/[x(x+1)] = 1/x − 1/(x+1). Integrating gives ln|x| − ln|x+1| + C, or equivalently ln|x/(x+1)| + C.

How the calculator helps

Factor the denominator when possible.
Set up the partial-fraction terms.
Solve for the unknown constants.
Integrate the simpler fractions and combine the result.

Frequently Asked Questions FAQ

Why use partial fractions before integrating?

A rational expression can often be split into simpler fractions whose antiderivatives are easier to find.

What is a simple partial-fractions example?

1/[x(x+1)] can be written as 1/x−1/(x+1), giving ln(abs(x)) − ln(abs(x+1)) + C after integration.

What should I check before decomposing the fraction?

Factor the denominator and confirm the rational function is proper; repeated or irreducible factors require the appropriate decomposition terms.

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