   Chapter 7.2, Problem 59E

Chapter
Section
Textbook Problem

# Use a graph of the integrated to guess the value of the integral. Then use the methods of this section to prove that your guess is correct. ∫ 0 2 π cos 3 x   d x

To determine

To evaluate: the value of integral from the graph of the integrand and then perform the integration to verify.

Explanation

Trigonometric integral of the form sinmxcosnxdx can be solved using strategies depending on whether m and n are odd or even.

Formula used:

When power of cosine in the integral is odd, save one cosine factor and use the identity cos2x=1sin2x to rewrite other terms in sine function form:

sinmxcos2k+1xdx=sinmx(1sin2x)kcosxdx

Then, use the substitution u=sinx

Given:

The integral, 02πcos3xdx.

Calculation:

The graph of integrand from x=0 to 2π is as shown below:

Notice from the graph that the total area above x-axis under the graph seems to be equal to the area under the graph below the x-axis. Hence, the integral from 0 to 2π will be zero. Now, perform the integration and confirm that it evaluates to zero

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