   Chapter 8.3, Problem 80E

Chapter
Section
Textbook Problem

# Verifying a Reduction Formula In Exercises 79-82, use integration by parts to verify the reduction formula. (A reduction formula reduces a given integral to the sum of a function and a simpler integral.) ∫ cos 2 x   d x = cos n − 1 x sin x n + n − 1 n ∫ cos n − 2 x   d x

To determine

To prove: The reduction formula given as, cosnxdx=cosn1xsinxn+n1ncosn2xdx using integration by parts.

Explanation

Given:

The provided expression is:

cosnxdx=cosn1xsinxn+n1ncosn2xdx

Formula used:

Integration by parts:

uvdx=uvdx(d(u)dxvdx)dx

Proof:

Consider the integral given as,

cosnxdx

Rewrite the given integral as,

cosnxdx=cosn1xcosxdx

Apply the integral formula,

uvdx=uvdx(d(u)dxvdx)dx

Here, u=cosn1x , v=cosx

Hence,

cosnxdx=cosn1x(cosx)dx(d(cosn1x)dxcosxdx)dx

Therefore,

cosnxdx=cosn1x(sinx

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