Integration

When using this integration rule, do I still divide by the derivative?

Not quite sure what you mean. The rule you have written (at the bottom) is correct. What do you mean by "divide by the derivative"?

Because when you are for example integrating sin2x you get -cos2x/2 because you are dividing by the derivative of the bracket (2x), so I’m asking if I have to divide ln(f(x)) by f’(x)?

The best thing you could do would be to rewrite the integrand as (1/2)(2 cos(2x))/(1 + sin(2x)), and then factorise the 1/2 oustide the integral sign. The integrand is then of the exact form f'(x)/f(x).

Yes, that makes sense so I’m guessing that when using that rule, you don’t divide by the bracket. Thanks

Although looking through this again now, if you mean "do I divide by 2?", then it is a yes. But I prefer to get an integrand of the exact form f'(x)/f(x) before integrating, it may take an extra line that doesn't do anything particularly clever, but it makes it very clear what you are doing.

Yes your way is better, thank you

you can use direct u-substitution $u = 1 + \sin(2x)$