   Chapter 4.5, Problem 80E

Chapter
Section
Textbook Problem

Even and Odd Functions In Exercises 79 and 80, write the integral as the sum of the integral of an odd function and the integral of an even function. Use this simplification to evaluate the integral. ∫ − π / 2 π / 2 ( sin 4 x + cos 4 x )   d x

To determine

To calculate: The value of the provided definite integral by the use of the properties of even and odd functions.

Explanation

Given:

The integral is π2π2(sin4x+cos4x)dx.

Formula used:

The integration of xn is:

xndx=xn+1n+1

If f(x) is an even function. Then,

aaf(x)dx=20af(x)dx

And if f(x) is an odd function. Then,

aaf(x)dx=0

Calculation:

Consider the provided integral,

π2π2(sin4x+cos4x)dx

It can be divided into odd and even.

So,

π2π2(sin4x)dx=14[cos(4x)]π2π2=14[cos(4π2)cos(4π2)]=0

So, it is an odd function

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