If for a real number y,[y] is the greatest integer less than or equal to y, then the value of the integral `int_(pi//2)^(3pi//2)[2 sin x]dx`, is

Options

A. `-pi`
B. 0
C. `-pi//2`
D. `pi//2`

Answer

C

Explanation

We have,
`overset(3pi//2)underset(pi//2)int[2sinx]dx= overset(5pi//6)underset(pi//2)int[2 sinx]dx overset(x)underset(5pi//6)int[2sin x]dx+overset(7pi//6)underset(pi)int [2 sinx]dx+overset(3pi//2)underset(pi) int [2 sin x]dx+overset(3pi//2)underset(7pi//6)int [2 sin x]dx`
`rArr overset(3pi//2)underset(pi//3)int [2 sin x]dx= overset(5pi//6)underset(pi//2)int 1dx+overset(pi)underset(5pi//6)int 0dx+overset(7pi//6)underset(pi)int -1dx+overset(3pi//2)underset(7pi//6)int -2dx`
`rArroverset(3pi//2)underset(pi//2)int[2 sin x]dx=((5pi)/(6)-(pi)/(2))-((7pi)/(6)-pi)-2((3pi)/(2)-(pi)/(6))=(pi)/(2)`