Recurrence Relations (Reduction Formulas)

Recurrence relations, or reduction formulas, are part of the Further integration topic in NSW Year 12 Mathematics Extension 2 (NESA outcome ME2-12-04). This subtopic derives and uses recurrence relations that express an integral \(I_n\) in terms of an integral with a lower index, usually by applying integration by parts.

Students and teachers will cover how to derive a reduction formula and then use it repeatedly to evaluate a definite integral such as \(\displaystyle I_n = \int_1^2 x^n\sqrt{x-1}\,dx\) — a demanding, frequently examined skill in the HSC Extension 2 course.