Partial Fractions Calculator

This calculator performs partial fraction decomposition for the common case of a linear numerator divided by two distinct linear factors, splitting one fraction into a sum of two simpler ones. Partial fractions are a key technique in algebra and calculus, because a complicated rational expression is much easier to work with once it is broken into simple pieces. The method is essential for integrating rational functions, for inverse Laplace transforms in engineering, and for simplifying expressions in many areas of mathematics. The idea is to take a fraction of the form a linear expression over the product of two linear factors, and rewrite it as A over the first factor plus B over the second, where A and B are constants to be found. This tool handles that decomposition. You enter the numerator as a linear expression, giving its slope and constant, and the two distinct roots of the denominator factors. The calculator finds the constants A and B using the cover-up method, which evaluates cleverly chosen points to isolate each constant, and presents the full decomposition along with the individual values of A and B. The results update as you type, so you can explore how changing the numerator or the roots affects the breakdown. Use it to check calculus homework before integrating, to verify a decomposition by hand, or to learn the technique. The two roots must be different, since repeated or complex factors need a different form of decomposition that this tool does not cover. Once decomposed, each simple fraction integrates to a natural logarithm, which is exactly why this step is so useful in calculus.

For (px + q) / ((x - a)(x - b)) with distinct roots a and b. Uses the cover-up method: A = (pa + q)/(a - b), B = (pb + q)/(b - a).