Evaluate the integral. ∫0 π Cos 6θ dθ

Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter6: The Trigonometric Functions
Section6.2: Trigonometric Functions Of Angles
Problem 34E
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Evaluate the integral.

0 π Cos 6θ dθ

Expert Solution
Step 1

Consider the following integral:

0πcos6θdθ

Consider the following equation:

I=cos6θdθ

Consider the following formula:

cosnxdx=n-1ncosn-2xdx+cosn-1xsinxn

Step 2

I=cos6θdθ=cos6-1xsinx6+6-16cos6-2xdx=cos5xsinx6+56cos4xdx=cos5xsinx6+56cos4-1xsinx4+4-14cos4-2xdx=cos5xsinx6+5cos3xsinx24+58cos2xdx

I=cos5xsinx6+5cos3xsinx24+58cos2-1xsinx2+2-12cos2-2xdx=cos5xsinx6+5cos3xsinx24+5cosxsinx16+516dx=cos5xsinx6+5cos3xsinx24+5cosxsinx16+5x16+C

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