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

Options

A. `(-5pi)/3`
B. `-pi`
C. `(5pi)/3`
D. `-2pi`

Answer

B

Explanation

`I=int_(pi)^(2pi)[2sin x]dx`

From the graph in the figure
`I=int_(pi//6)^(5pi//6)1 dx+int_(pi)^(7pi//6)-1dx+int_(7pi//6)^(11pi//6)-2 dx+int_(11pi//6)^(2pi)-1 dx`
`=((5pi)/6-(pi)/6)+(-(7pi)/6+pi)+2(-(11pi)/6+(7pi)/6)`
`+(-2pi+(11pi)/6)`
`=(2pi)/3-(pi)/6-(8pi)/6-(pi)/6=-pi`