Sum of integers satisfying ` sqrt(log_(2)x-1)-1//2 log_(2)(x^(3))+2 gt 0` is ________.

Answer

5

Explanation

`sqrt(log_(2) x-1)-3/2 log_(2) x+2 gt 0 (xgt 0)`
` rArr sqrt(log_(2)x-1)-3/2 (log_(2) x-1)+1/2 gt 0` ...(i)
Let ` sqrt(log_(2)x-1)=tge 0`, we have ….(ii)
` log_(2) x-1 ge 0`
Then from Eq. (i), we have ` t-3/2 t^(2) + 1/2 gt 0`
` 3t^(2) - 2t -1 lt 0 `
` rArr 1//3 lt t lt 1` ...(iii)
Form Eqs. (ii) and (iii), we have ` 0 le t lt 1`.
` 0 le sqrt(log_(2)x-1) lt 1`
` 0 le log_(2) x - 1 lt 1`
` 1 le log_(2) x lt 2`
` 2 le x lt 4`
Hence, the integral values are 2 and 3, and their sum is 5.