What is the integral of sin²(kθ)cos²(kθ) over the interval from 0 to 2π?

[bigplanet401]

Homework Statement

What is

<br /> \int_0^{2 \pi} \; d\theta \sin^2 k\theta \cos^2 k\theta \; ?<br />

Homework Equations

Orthogonality of sines and cosines?

The Attempt at a Solution

I tried substitution and didn't get anywhere. Yeah, that's about it.

 

[Dick]

Use the trig identity sin(2x)=2*sin(x)*cos(x) for a start.

 

[bigplanet401]

Whoa, I completely missed that. Using that identity, the integral becomes

<br /> \begin{align*}<br /> &amp; \int_0^{2\pi} d\theta \; \frac{1}{4} \sin^2 2k\theta\\<br /> &amp;= \int_0^{2\pi} d\theta \; \frac{1}{8} (1 - \sin 4 k \theta )\\<br /> &amp;= \frac{\pi}{4} \, ,<br /> \end{align*}<br />
right?

The answer just seems too simple--like there should be some k's around or something.

 

[Dick]

You mean 1-cos(4k*theta), I hope. If k is an integer then there are no k's left around. If it isn't there are.