What is `int_(2)^(8) |x-5|dx` equal to ?

Options

A. `2`
B. `3`
C. `4`
D. `9`

Explanation

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

### Step 1: Identify the points where the expression inside the absolute value changes sign.
The expression \( |x-5| \) changes sign at \( x = 5 \). Thus, we will evaluate the integral from \( 2 \) to \( 5 \) and from \( 5 \) to \( 8 \).

### Step 2: Rewrite the integral by splitting it at the point where the absolute value changes.
\[ \int_{2}^{8} |x-5| \, dx = \int_{2}^{5} |x-5| \, dx + \int_{5}^{8} |x-5| \, dx \]

### Step 3: Determine the expression for \( |x-5| \) in each interval.
- For \( x \) in the interval \( [2, 5] \), \( x-5 \) is negative, so \( |x-5| = -(x-5) = 5-x \).
- For \( x \) in the interval \( [5, 8] \), \( x-5 \) is positive, so \( |x-5| = x-5 \).

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

### Step 5: Evaluate the first integral \( \int_{2}^{5} (5-x) \, dx \).
\[ \int (5-x) \, dx = 5x - \frac{x^2}{2} \]
Now, evaluate 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 this gives:
\[ \left( 25 - \frac{25}{2} \right) - \left( 10 - 2 \right) = \left( 25 - 12.5 \right) - 8 = 12.5 - 8 = 4.5 \]

### Step 6: Evaluate the second integral \( \int_{5}^{8} (x-5) \, dx \).
\[ \int (x-5) \, dx = \frac{x^2}{2} - 5x \]
Now, evaluate 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 this gives:
\[ \left( \frac{64}{2} - 40 \right) - \left( \frac{25}{2} - 25 \right) = \left( 32 - 40 \right) - \left( 12.5 - 25 \right) = -8 - (-12.5) = -8 + 12.5 = 4.5 \]

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

Thus, the final answer is:
\[ \int_{2}^{8} |x-5| \, dx = 9 \]