Problem: Evaluate the integral
Solution: This trigonometric integral looks simple. There is an extra sine and an extra cosine in the numerator, so we can make a simple trig substitution. The denominator makes it clear which one:
We obtained an integral of a rational function, which is theoretically an easy integral. In real life it may get more difficult, since now we have to factor the denominator. It is a fourth degree polynomial, which means that we do not know formulas for its roots. Usually we then try to plug in some nice numbers to guess some roots. Try it, but it is pointless, this baby does not have any integrer roots, not even rational roots. For most people this means the end of the road. However, this polynomial is very simple, so surely somebody has been messing around with it before, so now it is time to rack one's memory, perhaps browse the Internet, and sure enough, it can be factored.
What now? We should decompose this expression into partial fractions, which is less ugly than it seems, and then try to integrate them. The first step is completing squares in those irreducible quadratic expressions there, which sounds less and less appealing. If you feel strong, you can try it, a brief sketch of the calculations is here.
For now we will leave it as an emergency back-up plan in case we will not come up with something different. This hoped-for alternative solution will not follow from any standard procedure, it would have to be an individual trick taylored to this problem. The best bet lies with substitution. What are the possibilities?
There are not that many candidates, essentially a substitution for sine, for sine squared and for sine to power four. The second one looks very interesting, since the derivative of sine squared is esentially in the numerator, a clear sign that we might be onto something. We try it.
We got lucky, the problem is solved.