If [*] denots the greatest integer function then the value of the integeral `int_(-pi//2)^(pi//2)([(x)/(pi)]+0.5) dx`, is

Options

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

Answer

C

Explanation

We have,
`I=underset(-pi//2)overset(pi//2)int([(x)/(pi)]+0.5) dx`
`rArrI=underset(-1//2)overset(1//2)int([t]+0.5)dt,"where" x =pit`
`rArrI=pi{underset(-1//2)overset(0)int (-10.5)dt+underset(0)overset(1//2)int (0+0.5)dt}`
`rArr I=pi {-(1)/(4)+(1)/(4)}=0`