Let `f:R to R` be a function defined by f(x) `{:{([x]",",xle 2),(0",",x gt 2):}`where [x] is the greatest integer less than or equal to x.If `I=int_(-1)^(2) (xf(x^(2)))/(2+f(x+1))dx`, then the value, is

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)`