# Why is this integral wrong?

I don't understand... When I take a definite integral from a to b of 4cos(x)sin(x), u-substitution tells me that the answer should be 2sin^2(x) evaluated from b - it's evaluation at a. But my first reflex when seeing this integral was to use the identity sin(2x) = 2cos(x)sin(x) to get 4cos(x)sin(x) = 2sin(2x) and therefore integrate on that. Doing so gives me an evaluation with similar boundaries of -cos(2x), so I thought -cos(2x) = 2sin^2(x). But I know this to be wrong from the double angle identity of cos(2x) = 1-2sin^2(x) => -cos(2x) = 2sin^2(x)-1. Where did the 1 go?

$$-\cos2x$$ and $$2\sin^2x$$ have the same derivative, so they differ by a constant. That constant is $$1$$.