Integration using partial fractions
Decompose, then integrate term by term. Each fraction usually integrates to ln or arctan.
Integrals of rational functions become tractable after partial fractions. Each piece either integrates to ln|x − a| or to arctan(...) (for irreducible quadratic factors).
Worked examples
∫ (3x+1)/[(x+1)(x−2)] dx: decompose, then sum two ln integrals.
∫ 1/(x²+1) dx = arctan x + c.
∫ 1/[(x²+1)(x+1)] dx: decomposes into A/(x+1) + (Bx+C)/(x²+1).
Frequently asked questions
Repeated factor?
1/(x−a)² integrates to −1/(x−a). Different from 1/(x−a).
Irreducible quadratic on top with x?
Use the form (Bx + C)/(x² + bx + c). Bx integrates to ln; C-term integrates to arctan.