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

1.	Sum of integers satisfying ` sqrt(log_(2)x-1)-1//2 log_(2)(x^(3))+2 gt 0` is ________.
Answer» Correct Answer - 5 `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.

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