Let [ ] be the greatest integer function. If = ₀⁶⁴ (x^ 1 / 3 - [x^ 1 / 3 ] ) d x , then 1 ₀^ ( ² ⁶ + ⁶ ) d is equal to _ _ _ _ .

Step-by-step solution

₀⁶⁴ x^ 1 3 , dx = 3 4 [ x^ 4 3 ]₀⁶⁴ = 192 & ₀⁶⁴ [x^ 1/3 ] , dx = ₀¹ [x^ 1/3 ] , dx + ₁⁸ [x^ 1/3 ] , dx + ₈²⁷ [x^ 1/3 ] , dx + ₂₇⁶⁴ [x^ 1/3 ] , dx = 156 So = 192 - 156 = 36 Now E = 1 ₀^ 36 ^2 ^6 + ^6 , d = 36 ₀^ ^2 ^6 + ^6 , d E = 36 2 ₀^ /2 ^2 ^6 + ^6 , d Let J = ₀^ /2 ^2 ^6 + ^6 , d ....(1) Applying King J = ₀^ /2 ^2 ^6 + ^6 , d .....(2) Now 2J = ₀^ /2 1 ^6 + ^6 , d (add (1) & (2)) = ₀^ /2 ^6 ^6 + 1 , d = ₀^ (1 + ^2) ^4 - ^2 + 1 , d = ₀^ 1 + 1 ^2 ^2 - 1 + 1 ^2 , d = J = 2 E = 36 2 J = 36