I have a hard time solving integrals of trig equations. Any advice? I'm working through a section of integrals of trig functions, and I always have the hardest time knowing off the bat the next step to make.  I know the techniques, but knowing which step to take is difficult.  There's a tree of possible solutions and I'm expected to know the correct route, and I don't.  I'm magically supposed to know that the half-angle formula is the correct approach, or whatever it is.
It's a real sticking point right now.  How do you approach these problems?  Any advice?
Btw, the question that really threw me was this:
$\int cos^4(2t)dt$
The solution required two shots of half-angle formula, and a substitution.  I didn't magically know that. @_@

I believe knowing how to do it is good enough. And actually not being able to (or to see the proper route to) evaluate the integral instantly isn't such a problem. You're not going to get the points for not solving it, but that's not important. When you see $\int \tan^2(x) \sec^4(x) \, \ dx$, you should obviously not be thinking about using the partial fraction method, say, to crack it. All it takes is a simple substitution and that's what important is if you can narrow down the potential ways to evaluate it and apply the one that works, you're good.
