Let `f: RvecR` be a function defined by `f(x)={[x],(xlt=2) \ \ (0,x >2)` where `[x]` is the greatest integer less than or equal to `xdot` If `I=int_(-1)^2(xf(x^2))/(2+f(x+1))dx , then \ the \ value \ of \ (4I-1)i s`

Options

A. `(1)/(4)`
B. `(1)/(2)`
C. 8
D. `-(1)/(4)`

Answer

A

Explanation

We have, f(x)`{:{([x]",",xle 2),(0",",x gt 2):}`
`:.I=overset(2)underset(-1)int (xf(x^(2)))/(2+f(x+1))dx`
`rArr I=overset(0)underset(-1)int (x xx0)/(2xx0)dxoverset(1)underset(0)int (x xx0)/(2+1)dx+overset(sqrt(2))underset(1)int (x xx1)/(2+0)dxoverset(2)underset(sqrt(2))int (x xx0)/(2+f(x+1))dx`
`rArr I=(1)/(2)[(x^(2))/(2)]_(1)^(sqrt(2))=(1)/(2)(1-(1)/(2))=(1)/(4)`