Decomposing (2x + 3)/((x - 1)(x + 2))

Calculus 2 · Axiom Academy

Decompose a rational function with distinct linear factors, then integrate it term by term. Evaluate the integral by first decomposing the integrand into partial fractions. Nice work — you decomposed a rational function with distinct linear factors and integrated it. Remember: Identify the form: for distinct linear factors (x - a)(x - b) , use . Clear denominators: multiply both sides by the common denominator to get a polynomial equation. Strategic substitution (cover-up): pick x values that zero out a factor — x = 1 kills B , x = -2 kills A . Integration is then easy: each partial fraction integrates to a natural log, . Check the degree: this method needs the numerator degree to be less than the denominator degree. Partial fractions turn a hard rational integral into a sum of logarithms — a workhorse technique throughout Calculus 2.

This is the written version of the interactive lesson above. See the full Calculus 2 course.