The integral is positive, its integer part is the integral itself, so I = 1/6.
We first evaluate the definite integral:
∫₀¹.⁵ x⋅⌊x²⌋dx = ∫₀¹.⁵ x⋅⌊x⌋dx
Because ⌊x²⌋ = ⌊x⌋ for all 0 ≤ x ≤ 1/√2, we can rewrite the integral as:
∫₀¹.⁵ x⋅⌊x⌋dx = ∫₀¹.⁵ x²dx = x³/3 |₀¹.⁵ = (1/3)/8 = 1/24
Therefore, 4 times the value of the integral is 4 * (1/24) = 1/6.
Since the integral is positive, its integer part is the integral itself, so I = 1/6.