I disagree with your teacher and you--though I agree with you in part
Note if you differentiate your result you get part of the answer but not all
See attachment
As part of a larger problem I came across 1/8 the integral of cos^3(2x).
I simplified this to 1/16 the integral of 2(1-sin^2(2x))cosx and then solved with u substitution.
u=sin(2x)
du=2cos(2x)
This came out to [sin^3(2x)]/48.
My teacher tells me that the answer is [sin^3(2x)]/6. I don't see how this is possible. Who is right?