Number of integral values of `x` satisfying the inequality
`((e^(x)-pi^(x))log_(2)((x^(2)-5x+6)/(2))(x-8)^(31)(x-5)^(5))/((x+2)^(3)(x^(8)-x^(6)+x^(4)-x^(2)+1))ge0` is/are

Answer

`8`

Explanation

`:' x^(8) - x^(6) + x^(4) - x^(2) + 1 gt 0 AA xx in R`
Domin `x^(2) - 5x + 6 gt 0`
`(x - 3) ( x- 2) gt 0`
`x in (-oo, 2) cup (3, oo)`
Critical points `x^(x) - e^(x) = 0`
`log_(2) ((x^(2) - 5x + 6)/(2)) = 0`
`(x^(2) - 5x + 6)/(2) = 1`
`x^(2) + 5x + 4 = 0`
`(x - 4)(x - 1) = 0`
`x = 1, 4`
`((e^(x)-pi^(x))log_(2)'((x^(2)-5x+6)/(2))(x-8)^(31)(x-5)^(5))/((x+2)^(3)(x^(8)-x^(6)+x^(4)-x^(2)+1))`
`rArr((pi^(x)-e^(x))log_(2)'((x^(2)-5x+6)/(2))(x-8)^(31)(x-5)^(5))/((x+2)^(3))`

`x in (-2,]uu(1,2)uu(3,4]uu[5,8]`