Evaluate the following integral: `int_1^4{|x-1|+|x-2|+|x-4|}dx`

Explanation

To evaluate the integral \( I = \int_1^4 (|x-1| + |x-2| + |x-4|) \, dx \), we first need to analyze the absolute value expressions over the interval from 1 to 4.

### Step 1: Identify the critical points
The critical points where the expressions inside the absolute values change sign are \( x = 1 \), \( x = 2 \), and \( x = 4 \). We will break the integral into segments based on these points.

### Step 2: Break the integral into segments
We will evaluate the integral in three segments:
1. From \( 1 \) to \( 2 \)
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