Evaluate: `int_0^(2pi)[sinx]dx ,w h e r e[dot]` denotes the greatest integer function.

Explanation

The graph of the function `y=[sin x]` is as shown in the following figure.

Since for `xepsilon(0,pi)-{(pi)/2},[sinx]=0, int_(0)^(pi)[sin x]dx=0`
For `xepsilon(pi,2pi),[sin x]=-1`
`:.int_(pi)^(2pi)[sin x] dx=-(pixx1)=-pi`