What do you mean by d[x]? Is that the "greatest integer function" or "floor function", ? If so that is a "Stieljes" integral, not a "Riemann" integral. changes value only at integer values so the Stieljes integral is just the sum of the function being integrated at -1, 0, 1, 2, and 3. Since here the function is just "x", that would be -1+ 0+ 1+ 2+ 3= 5.

I have no idea why you think you should justadda new integral to it! That certainly has nothing to do with "integration by parts".