Homework Help: How to integrate this

1. Feb 3, 2013

phospho

$$\displaystyle\int sin^22tcos^2t\ dt$$

This was part (b) to a question, the previous part of the question was to integrate $\displaystyle\int sin^22tcost\ dt$ which I managed to do by expressing $sin^22t$ as $4(sin^2t - sin^4t)$

I tried a similar method for the integrand above, but didn't really go far with it. I'm not really sure what I'm trying to spot here, and I usually struggle with these integrals.

2. Feb 3, 2013

hikaru1221

You can try with $\int sin^2(2t)dt - \int sin^2(2t)sin^2(t)dt$, plus some trigonometric manipulations.

P.S.: Silly me, perhaps you only need some trigonometric manipulations.

3. Feb 3, 2013

thanks

4. Feb 3, 2013

vela

Staff Emeritus
Instead of changing the first factor, I'd use the identity $\cos^2 t = \frac{1+\cos 2t}{2}$ to get everything in terms of $2t$. It should be pretty straightforward after that.

Your original approach would also work. You should check your textbook on how to handle integrals of the form
$$\int \cos^n x \sin^m x\,dx.$$ They're tedious, but there's a recipe to follow that depends on the evenness and oddness of $m$ and $n$.

5. Feb 3, 2013

haruspex

After following vela's advice, you may also find it useful to know cos(3x) = 4 cos3(x) - 3 cos(x)