If [x] is the greatest integer function not greater than x , then ₀^8[x] d x is equal to

Step-by-step solution

Let I= ₀^8[x] d x aligned & I= ₀^1[x] d x+ ₁^2[x] d x+ ₂^3[x] d x+ ₃^4[x] d x+ ₄^5[x] d x & + ₅^6[x] d x+ ₆^7[x] d x+ ₇^8[x] d x & = ₀^1 0 d x+ ₁^2 1 d x+ ₂^3 2 d x+ ₃^4 3 d x+ ₄^5 4 d x & + ₅^6 5 d x+ ₆^7 6 d x+ ₇^8 7 d x & =0+1[x]₁^2+2[x]₂^3+3[x]₃^4+4[x]₄^5 & +5[x]₅^6+6[x]₆^7+7[x]₇^8 & =0+1(2-1)+2(3-2)+3(4-3) & +4(5-4)+5(6-5)+6(7-6)+7(8-7) & =1+2+3+4+5+6+7 & = 7 (7+1) 2 = 7 8 2 =7 4=28 . & aligned