---
tag: 012RYET5E2
kind: concept
name: Reduction formula
msc: [26A36]
uses: [08Y1BBQTK9]
sources:
  - url: https://en.wikipedia.org/wiki/Integration_by_reduction_formulae
    locator: How to find the reduction formula
same_as:
  wikidata: Q3823543
  wikipedia: Integration by reduction formulae
---

## Summary

A reduction formula expresses an integral that depends on an integer $n$ through the same integral with a smaller $n$. Repeated use of the formula reduces the integral to a simple case. [[Integration by parts|08Y1BBQTK9]] usually produces the formula.

## Remarks

For $I_n = \int_0^{\pi/2} \sin^n x \, dx$, integration by parts gives $I_n = \frac{n - 1}{n} I_{n-2}$ for $n \ge 2$.
With $I_0 = \pi/2$ and $I_1 = 1$, this gives all $I_n$.
These integrals lead to the [[Wallis product|0XCYA4YERS]].

## Explanations

### L1 human

A reduction formula is a recursion for a family of integrals.
Each step lowers a power, until the integral is easy.