If [x] is the greatest integer function not greater than x , then the value of ₀^2 x[x^2] ,dx is...

Step-by-step solution

Let I = ₀^2 x[x^2] ,dx Substitute x^2 = t 2x ,dx = dt x ,dx = dt 2 When x = 0 , t = 0 and when x = 2 , t = 4 I = 1 2 ₀^4 [t] ,dt I = 1 2 ( ₀^1 0 ,dt + ₁^2 1 ,dt + ₂^3 2 ,dt + ₃^4 3 ,dt ) I = 1 2 (0 + 1(2-1) + 2(3-2) + 3(4-3)) I = 1 2 (1 + 2 + 3) = 6 2 = 3 Answer: 3