₀^2[x] d x+ ₀^2|x-1| d x= (where [x] denotes the greatest integer function.)

Step-by-step solution

aligned & ₀^2[x] d x+ ₀^2[x-1] d x & = ₀^1 0 . d x+ ₁^2 1 . d x+ ₀^1(1-x) d x+ ₁^2(x-1) d x & =[0]₀^1+[x]₁^2+ [x- x^2 2 ]₀^1+ [ x^2 2 -x ]₁^2 & =0+1+ 1 2 + 1 2 =2 aligned