Evaluate :
`int_(2)^(5)[|x-2|+|x-3|+|x-4|]dx`

Answer

To evaluate the integral

\[ I = \int_{2}^{5} \left( |x-2| + |x-3| + |x-4| \right) dx, \]

we will break it down into simpler parts by considering the behavior of the absolute value functions within the limits of integration.

### Step 1: Identify the points of interest
The expressions inside the absolute values change at \(x = 2\), \(x = 3\), and \(x = 4\). We will evaluate the integral in segments based on these points.

### Step 2: Break the integral into segments
We can break the integral into three parts:

1. From \(2\) to \(3\)
2. From \(3\) to \(4\)
3. From \(4\) to \(5\)

Thus, we can write:

\[ I = \int_{2}^{3} (|x-2| + |x-3| + |x-4|) dx + \int_{3}^{4} (|x-2| + |x-3| + |x-4|) dx + \int_{4}^{5} (|x-2| + |x-3| + |x-4|) dx \]

### Step 3: Evaluate each segment

#### Segment 1: \(x \in [2, 3]\)
In this interval:
- \( |x-2| = x-2 \)
- \( |x-3| = 3-x \)
- \( |x-4| = 4-x \)

Thus,

\[ \int_{2}^{3} (|x-2| + |x-3| + |x-4|) dx = \int_{2}^{3} \left( (x-2) + (3-x) + (4-x) \right) dx \]

Simplifying the integrand:

\[ = \int_{2}^{3} (5 - 2x) dx \]

Now, we calculate the integral:

\[ = \left[ 5x - x^2 \right]_{2}^{3} = \left( 5(3) - (3)^2 \right) - \left( 5(2) - (2)^2 \right) \]
\[ = (15 - 9) - (10 - 4) = 6 - 6 = 0 \]

#### Segment 2: \(x \in [3, 4]\)
In this interval:
- \( |x-2| = x-2 \)
- \( |x-3| = x-3 \)
- \( |x-4| = 4-x \)

Thus,

\[ \int_{3}^{4} (|x-2| + |x-3| + |x-4|) dx = \int_{3}^{4} \left( (x-2) + (x-3) + (4-x) \right) dx \]

Simplifying the integrand:

\[ = \int_{3}^{4} (x - 1) dx \]

Now, we calculate the integral:

\[ = \left[ \frac{x^2}{2} - x \right]_{3}^{4} = \left( \frac{(4)^2}{2} - 4 \right) - \left( \frac{(3)^2}{2} - 3 \right) \]
\[ = \left( 8 - 4 \right) - \left( \frac{9}{2} - 3 \right) = 4 - \left( \frac{9}{2} - \frac{6}{2} \right) = 4 - \frac{3}{2} = \frac{8}{2} - \frac{3}{2} = \frac{5}{2} \]

#### Segment 3: \(x \in [4, 5]\)
In this interval:
- \( |x-2| = x-2 \)
- \( |x-3| = x-3 \)
- \( |x-4| = x-4 \)

Thus,

\[ \int_{4}^{5} (|x-2| + |x-3| + |x-4|) dx = \int_{4}^{5} \left( (x-2) + (x-3) + (x-4) \right) dx \]

Simplifying the integrand:

\[ = \int_{4}^{5} (3x - 9) dx \]

Now, we calculate the integral:

\[ = \left[ \frac{3x^2}{2} - 9x \right]_{4}^{5} = \left( \frac{3(5)^2}{2} - 9(5) \right) - \left( \frac{3(4)^2}{2} - 9(4) \right) \]
\[ = \left( \frac{75}{2} - 45 \right) - \left( \frac{48}{2} - 36 \right) = \left( \frac{75}{2} - \frac{90}{2} \right) - \left( \frac{48}{2} - \frac{72}{2} \right) \]
\[ = \left( -\frac{15}{2} \right) - \left( -\frac{24}{2} \right) = -\frac{15}{2} + \frac{24}{2} = \frac{9}{2} \]

### Step 4: Combine the results
Now, we combine the results from all segments:

\[ I = 0 + \frac{5}{2} + \frac{9}{2} = \frac{14}{2} = 7 \]

### Final Answer
Thus, the value of the integral is:

\[ \boxed{7} \]