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Chapter 8: Problem 64

Describe what is meant by a reduction formula. Give an example.

Short Answer

Expert verified
A reduction formula expresses an integral or sum of 'n' terms in terms of an 'n-1' terms problem. An example is the integral \( \int x^n dx \), which can be recursively defined as \( \int x^n dx = \frac{1}{n}x^n - \frac{n-1}{n} \int x^{n-1} dx\).

Step by step solution

01

Definition of a Reduction Formula

A reduction formula can be defined as a formula that expresses an integral or summation of 'n' terms in terms of an integral or summation of 'n-1' terms. It reduces the problem of computing an integral or summation of 'n' terms to a easier problem of 'n-1' terms.
02

Example of a Reduction Formula

For instance, let's choose an integral, \( \int x^n dx \), a very typical example of a reduction formula. You could solve it by using the power rule for integration which states that \( \int x^n dx = \frac{1}{n+1}x^{n+1} + c \) when n 鈮� -1. But it can also be written in a recursive way: \( \int x^n dx = \frac{1}{n}x^n - \frac{n-1}{n} \int x^{n-1} dx\). This is a reduction formula because it expresses the problem of integrating \(x^n\) in terms of a simpler problem, integrating \(x^{n-1}\).

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