   Chapter 7.1, Problem 47E

Chapter
Section
Textbook Problem

# (a) Use the reduction formula in Example 6 to show that ∫ sin 2 x   d x = x 2 − sin 2 x 4 + C (b) Use part(a) and the reduction formula to evaluate ∫ sin 4 x   d x .

To determine

(a)

To show: expression sin2xdx and x2sin2x4+C are equivalent using reduction formula.

Explanation

Reduction formula is obtained by using the technique of integration by parts to solve the integral sinnxdx.

Formula used:

Reduction formula is as given below:

sinnxdx=1ncosxsinn1x+n1nsinn2xdx

Given:

The equation to prove, sin2xdx=x2sin2x4+C.

Calculation:

The integral sin2xdx can be solved by using reduction formula with n as 2. So, substitute n=2 in the reduction formula:

sin2xdx=12cosxsin21x+

To determine

(b)

To evaluate: the integral sin4xdx using reduction formula and result from part (a)

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