Question

Let I denote the integer part of 4×∫₀¹.⁵ x⋅⌊x²⌋dx. What is the value of I?

Answer 1

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.