The value of ∫₀² x² [x] dx, where [x] is greatest integer function, is:

The value of ∫₀² x² [x] dx, where [x] is greatest integer function, is:

• A. 1/3
• B. 8/3
• C. 3
• D. 7/3

Answer: B) 8/3

Explanation: ∫₀² x² [x] dx = ∫₀¹ x²·0 dx + ∫₁² x²·1 dx = 0 + [x³/3]₁² = 8/3 − 1/3 = 7/3. We check: x in [0,1): [x]=0. x in [1,2): [x]=1. At x=2, [2]=2 but point measure zero. So ∫ = 0 + ∫₁² x² dx = [x³/3]₁² = 8/3 − 1/3 = 7/3. Option 3 is 7/3. So correct index 3.