Why is this limit 0?

["TCL"]

Let  p(n)= int_0^pi  | sin(n+ 0.5)x| /  sin (0.5 x)  dx.
I am trying to prove that
lim    p(n)/ ln n   = 0.

I am able to prove  lim sup p(n)/ ln n  is finite, but of course that is not 
good enough.