Question

To integrate 1-4cos(2x)

Answer 1

Is it 2sin(2x)?

Answer 2

to integrate 1 you get x

Answer 3

and you are missing a negative in front of the 2sin(2x)

Answer 4

So the answer will be x+2sin(2x)?

Answer 5

The derivative of sin x is cos x, not -cos x.
So the antiderivative of cos x will not change signs to -sin x (just going the opposite direction).

I think you just need to flip the sign on 2sin(2x) there. And I suppose don't forget the constant +C if this is an indefinite integral!