Chapter 8.3, Problem 80E

### Calculus of a Single Variable

11th Edition
Ron Larson + 1 other
ISBN: 9781337275361

Chapter
Section

### Calculus of a Single Variable

11th Edition
Ron Larson + 1 other
ISBN: 9781337275361
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:

âˆ«cosnxdx=cosnâˆ’1xsinxn+nâˆ’1nâˆ«cosnâˆ’2xdx

Formula used:

Integration by parts:

âˆ«uvdx=uâˆ«vdxâˆ’âˆ«(d(u)dxâˆ«vdx)dx

Proof:

Consider the integral given as,

âˆ«cosnxdx

Rewrite the given integral as,

âˆ«cosnxdx=âˆ«cosnâˆ’1xcosxdx

Apply the integral formula,

âˆ«uvdx=uâˆ«vdxâˆ’âˆ«(d(u)dxâˆ«vdx)dx

Here,Â u=cosnâˆ’1xÂ ,Â v=cosx

Hence,

âˆ«cosnxdx=cosnâˆ’1xâˆ«(cosx)dxâˆ’âˆ«(d(cosnâˆ’1x)dxâˆ«cosxdx)dx

Therefore,

âˆ«cosnxdx=cosnâˆ’1x(sinx)

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