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

Explanation

To evaluate the definite integral \( \int_1^4 [|x-1| + |x-2| + |x-3|] \, dx \), we will first analyze the absolute value expressions and break the integral into intervals where the expressions are linear.

### Step 1: Identify the critical points
The critical points for the absolute values are at \( x = 1 \), \( x = 2 \), and \( x = 3 \). These points will help us determine the intervals for integration.

### Step 2: Break the integral into intervals
We will evaluate the integral over the intervals \( [1, 2] \), \( [2, 3] \), and \( [3, 4] \):
\[
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