okay so I was integrating sin^4(x)/4+sin^2(x)/2+sin(x) and then I took the derivative of a that and got something other than sin^4(x)/4+sin^2(x)/2+sin(x) because of the technique I used to integrate (and I'll outline that below).
So anyway I noticed that any sin^a(x) can be written as something like (1-cos(2x)+cos(4x)-cos(6x))/6. where 2, 4, and 6 and 1 are all some sort of multiple of a. So you essentially get sin^a(x) in a form without any exponents which is actually pretty amazing considering there is a giant obnoxious reduction formula for integration and its shit. So being able to convert sin^a(x) to something else would be way more convenient instead of the reduction formula, at least for me.
So now I'm here trying to use binomial theorem to expand sin^a(x) into some sort of polynomial in terms of (a1+a2cos(b1(x))+a3cos(b2)+....)/something
which will be fun
and yea here is the way I integrated sin^4(x)/4+sin^2(x)/2+sin(x). I'm hoping you can follow it not because I think you're stupid but because desmos only supports derivatives and doesn't like integral symbols so