Evaluate ∫02π [sin⁡x]dx, where [.] denotes  the greatest integer function.

y=sin⁡x is a non-monotonic function in [0,2π] .  Hence, draw the graph of f(x)=[sin⁡x] .  From the above graph we have, ∫02π [sin⁡x]dx= Algebraic area of the shaded region  =(π)(−1) (l×b) =−π