   Chapter 8.3, Problem 79E

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.) ∫ sin 2 x   d x = − sin n − 1 x cos x n + n − 1 n ∫ sin n − 2 x   d x

To determine

To prove: The reduction formula given as, sinnxdx=sinn1xcosxn+n1nsinn2xdx by making use of integration by parts.

Explanation

Given:

The provided expression is:

sinnxdx=sinn1xcosxn+n1nsinn2xdx

Formula used:

Integration by parts:

uvdx=uvdx(d(u)dxvdx)dx

Proof:

Consider the integral given as,

sinnxdx

Rewrite the integral as,

sinnxdx=sinn1xsinxdx

Apply the integral formula,

uvdx=uvdx(d(u)dxvdx)dx

Here, u=sinn1x , v=sinx

Hence,

sinnxdx=sinn1x(sinx)dx(d(sinn1x)dxsinxdx)dx

Therefore,

sinnxdx=sinn1x(cosx)

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