If [x] denotes the greatest integer function, then ₀^5 x^2[x] d x=

Step-by-step solution

aligned ₀^5 x^2[x] d x= & ₀^1 x^2[x] d x+ ₁^2 x^2[x] d x & + ₂^3 x^2[x] d x+ ₃^4 x^2[x]+ ₄^5 x^2[x] d x aligned aligned = ₀^1 x^2(0) d x+ ₁^2 x^2(1) d x & + ₂^3 x^2(2) d x & + ₃^4 x^2(3) d x+ ₄^5 x^2(4) d x aligned aligned & = [ x^3 3 ]_i^2+2 [ x^3 3 ]₂^3+3 [ x^3 3 ]₃^4+4 [ x^3 3 ]₄^5 & = 1 3 (8-1)+ 2 3 (27-8)+(64-27)+ 4 3 (125-64) & = 400 3 aligned