# Integrate me

1. Mar 8, 2009

### bigplanet401

1. The problem statement, all variables and given/known data
What is

$$\int_0^{2 \pi} \; d\theta \sin^2 k\theta \cos^2 k\theta \; ?$$

2. Relevant equations
Orthogonality of sines and cosines?

3. The attempt at a solution
I tried substitution and didn't get anywhere. Yeah, that's about it.

2. Mar 8, 2009

### Dick

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

3. Mar 8, 2009

### bigplanet401

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

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

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

4. Mar 8, 2009

### 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.