If [x] denotes the greatest integer less than or equal to x , then the value of the integral ₀^2 x^2[x] , dx is equal to

Step-by-step solution

The given integral is I = ₀^2 x^2[x] , dx . Using the property of the greatest integer function, we split the integral at integer points within the limits of integration: I = ₀^1 x^2[x] , dx + ₁^2 x^2[x] , dx For x [0, 1) , [x] = 0 . For x [1, 2) , [x] = 1 . Substituting these values into the integrals: I = ₀^1 x^2(0) , dx + ₁^2 x^2(1) , dx I = 0 + [ x^3 3 ]₁^2 I = 2^3 3 - 1^3 3 I = 8 3 - 1 3 = 7 3 Answer: 7 3