Question

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

Short answer

A reduction formula is a method in calculus for solving integrals by expressing them in a simpler or reduced form. One example concerns the sine function where: \(\int sin^n (x) dx = -\frac{1}{n} sin^{n-1}(x) cos(x) + \frac{n-1}{n} \int sin^{n-2}(x) dx\).

Step-by-step solution

Define Reduction Formula

A reduction formula is a method often used in calculus, particularly when integrating, where one expresses an integral recursively in terms of an integral with a lower degree.

Give an Example

Let's look at the standard reduction formula for the integral of sin(x) raised to the power of n. The reduction formula is given by:\[\int sin^n (x) dx = -\frac{1}{n} sin^{n-1}(x) cos(x) + \frac{n-1}{n} \int sin^{n-2}(x) dx\]This formula provides a method to repeatedly reduce the power of n by 2 until the integral becomes a standard, easier to solve integral.