how do you evaluate integral of [lxl] from (0,2) , a step function.

would you just do it as if it were a absolute value and then do two different integrals?

To evaluate the integral of the step function f(x) = [x], also known as the greatest integer function, from 0 to 2, we can divide the integral into two separate regions. In each region, the function is constant, taking on only integral values. Here's how we can proceed:

1. Splitting the integral: Since the greatest integer function has a step at every integer, we can split the integral at each integer value in the interval (0, 2). In this case, we have a single integer value in the interval, namely 1. Therefore, we will split the integral into two separate integrals: one from 0 to 1 and the other from 1 to 2.

2. Evaluating each integral separately:
- Integral from 0 to 1: In this region, the greatest integer function is equal to 0. Thus, the integral becomes ∫[x] dx = ∫0 dx = x | from 0 to 1 = 1 - 0 = 1.
- Integral from 1 to 2: Here, the greatest integer function is equal to 1. So the integral becomes ∫[x] dx = ∫1 dx = x | from 1 to 2 = 2 - 1 = 1.

3. Adding the results: Since the two integrals are evaluated separately, we can add their individual values to get the overall result. In this case, the integral of [x] from 0 to 2 is equal to the sum of the two integrals we found: 1 + 1 = 2.

Therefore, the integral of [x] from 0 to 2, as a step function, is 2.