Let [t] denot the greatest integer less than or equal to t . Then the value of ₁^2|2 x-[3 x]| d x is

Step-by-step solution

aligned & ₁^2|2 x-[3 x]| d x= ₁^ 4 3 |2 x-3| d x+ _ 4 3 ^ 5 3 |2 x-4| d x+ _ 5 3 ^2|2 x-5| d x & = ₁^ 4 3 (3-2 x) d x+ _ 4 3 ^ 5 3 (4-2 x) d x+ _ 5 3 ^2(5-2 x) d x & = [3 x-x^2 ]₁^ 4 / 3 + [4 x-x^2 ]_ 4 / 3 ^ 5 / 3 + [5 x-x^2 ]_ 5 / 3 ^2 & =3[x]₁^ 4 / 3 +[4 x]_ 4 / 3 ^ 5 / 3 +5 [x^2 ]_ 5 / 3 ^2- [x^2 ]₁^2 & =(3+4+5) 1 3 - (2^2-1^2 ) & =4-3=1 aligned