Evaluate: `int_(2)^(8)|x-5|dx`

Answer

To evaluate the integral \( \int_{2}^{8} |x-5| \, dx \), we need to analyze the expression inside the absolute value, \( |x-5| \).

### Step 1: Determine the points where the expression changes
The expression \( |x-5| \) changes at \( x = 5 \). We will split the integral at this point.

### Step 2: Break the integral into two parts
We can express the integral as:
\[ \int_{2}^{8} |x-5| \, dx = \int_{2}^{5} |x-5| \, dx + \int_{5}^{8} |x-5| \, dx \]

### Step 3: Evaluate \( |x-5| \) in each interval
- For \( x \in [2, 5] \), \( x - 5 < 0 \) so \( |x-5| = -(x-5) = 5 - x \).
- For \( x \in [5, 8] \), \( x - 5 \geq 0 \) so \( |x-5| = x - 5 \).

### Step 4: Rewrite the integral with the expressions
Now we can rewrite the integral:
\[ \int_{2}^{8} |x-5| \, dx = \int_{2}^{5} (5-x) \, dx + \int_{5}^{8} (x-5) \, dx \]

### Step 5: Calculate the first integral
Calculate \( \int_{2}^{5} (5-x) \, dx \):
\[ \int (5-x) \, dx = 5x - \frac{x^2}{2} \]
Evaluating from 2 to 5:
\[ \left[ 5(5) - \frac{5^2}{2} \right] - \left[ 5(2) - \frac{2^2}{2} \right] \]
Calculating:
\[ = \left[ 25 - \frac{25}{2} \right] - \left[ 10 - 2 \right] = \left[ 25 - 12.5 \right] - 8 = 12.5 - 8 = 4.5 \]

### Step 6: Calculate the second integral
Calculate \( \int_{5}^{8} (x-5) \, dx \):
\[ \int (x-5) \, dx = \frac{x^2}{2} - 5x \]
Evaluating from 5 to 8:
\[ \left[ \frac{8^2}{2} - 5(8) \right] - \left[ \frac{5^2}{2} - 5(5) \right] \]
Calculating:
\[ = \left[ \frac{64}{2} - 40 \right] - \left[ \frac{25}{2} - 25 \right] = \left[ 32 - 40 \right] - \left[ 12.5 - 25 \right] = -8 - (-12.5) = -8 + 12.5 = 4.5 \]

### Step 7: Combine the results
Now, we add the results of both integrals:
\[ \int_{2}^{8} |x-5| \, dx = 4.5 + 4.5 = 9 \]

### Final Answer
Thus, the value of the integral \( \int_{2}^{8} |x-5| \, dx \) is \( \boxed{9} \).

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