Evaluate the definite integrals`int_1^4[|x-1|+|x-2|+|x-3|]dx`

Explanation

To evaluate the definite integral \( I = \int_1^4 [|x-1| + |x-2| + |x-3|] \, dx \), we need to analyze the absolute value expressions within the integral. The critical points where the expressions inside the absolute values change sign are at \( x = 1, 2, \) and \( 3 \). We will split the integral at these points.

### Step 1: Identify the intervals
The integral can be split into three intervals based on the critical points:
1. From \( 1 \) to \( 2 \)
2. From \( 2 \) to \( 3 \)
3. From \( 3 \) to \( 4 \)

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