`int_(2)^(8) |x-5| dx=?`

Answer

To solve the integral \( \int_{2}^{8} |x-5| \, dx \), we will break it down into two parts based on the definition of the absolute value function.

### Step 1: Determine the points where the expression inside the absolute value changes sign.
The expression \( |x-5| \) changes at \( x = 5 \). Therefore, we will split the integral at this point.

### Step 2: Rewrite the integral.
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 the first integral \( \int_{2}^{5} |x-5| \, dx \).
For \( x \) in the interval \([2, 5]\), \( x-5 \) is negative, so:
\[ |x-5| = -(x-5) = 5-x \]
Thus, we have:
\[ \int_{2}^{5} |x-5| \, dx = \int_{2}^{5} (5-x) \, dx \]

### Step 4: Calculate the first integral.
Now we compute:
\[ \int (5-x) \, dx = 5x - \frac{x^2}{2} + C \]
Evaluating from 2 to 5:
\[ \left[ 5x - \frac{x^2}{2} \right]_{2}^{5} = \left( 5(5) - \frac{5^2}{2} \right) - \left( 5(2) - \frac{2^2}{2} \right) \]
Calculating:
\[ = \left( 25 - \frac{25}{2} \right) - \left( 10 - 2 \right) = \left( 25 - 12.5 \right) - 8 = 12.5 - 8 = 4.5 \]

### Step 5: Evaluate the second integral \( \int_{5}^{8} |x-5| \, dx \).
For \( x \) in the interval \([5, 8]\), \( x-5 \) is non-negative, so:
\[ |x-5| = x-5 \]
Thus, we have:
\[ \int_{5}^{8} |x-5| \, dx = \int_{5}^{8} (x-5) \, dx \]

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

### Step 7: Combine the results.
Now we combine the two parts:
\[ \int_{2}^{8} |x-5| \, dx = \int_{2}^{5} |x-5| \, dx + \int_{5}^{8} |x-5| \, dx = 4.5 + 4.5 = 9 \]

### Final Answer:
\[ \int_{2}^{8} |x-5| \, dx = 9 \]