AP Calculus BC glossary

Partial fractions

Also called: Partial fraction decomposition

Partial fraction decomposition rewrites a rational function as a sum of simpler fractions with linear denominators. Each piece then integrates to a natural logarithm, which is what makes otherwise impossible rational integrals routine.

On the AP exam the denominator always factors into distinct linear factors, so the setup is P(x)(x−a)(x−b)=Ax−a+Bx−b\frac{P(x)}{(x-a)(x-b)} = \frac{A}{x-a} + \frac{B}{x-b}. Clear the denominators and solve for the constants.

The fastest way to find the constants is to substitute the values that make each factor vanish, which knocks out all but one unknown at a time.

Check the degree first

Partial fractions require the numerator degree to be lower than the denominator degree. If it is not, do polynomial long division before decomposing.

Appears in: Unit 6: Integration and Accumulation