Firstly, it's important to realise that integration is not the same as differentiation where you can differentiate almost anything using the same set of methods. You will come across some expressions that seem simple but they may be very hard to integrate or it may not even be possible.
In C4 you'll meet a variety of methods in order to tackle integrals but there are some
exceptions which you'll need to get used to - there aren't many.
∫sin2x dx and
∫cos2x dx are exceptions.
So far you have met the 'reverse chain rule' rule. That only works when the stuff in the brackets is linear. Looking at
sin2x, the first thing you could do is write it as
(sinx)2. This cannot be integrated using reverse chain rule since the stuff in the brackets
sinx is not a linear function.
So you need another method. But as I said before, there isn't a standard method that you can use; instead you need to use the trig identity and go from there as you've seen. You just need to practice this for the integrals of
sin2x and
cos2x.
At this point you really need to be familiar with the derivatives and integrals of
sinx and
cosx. Set aside some time to learn these off-by-heart. It shouldn't take too long.