What is `int_2^ s |x-5|dx` equal to

Options

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

Answer

To solve the integral \( \int_2^8 |x - 5| \, dx \), we need to analyze the absolute value function \( |x - 5| \).

### Step 1: Identify the points where the expression inside the absolute value changes sign.
The expression \( x - 5 \) is zero when \( x = 5 \). Therefore, we will evaluate the integral in two parts: from \( 2 \) to \( 5 \) and from \( 5 \) to \( 8 \).

### 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: Determine the form of \( |x - 5| \) in each interval.
- For \( x \) in the interval \( [2, 5] \), \( x - 5 < 0 \), so \( |x - 5| = -(x - 5) = 5 - x \).
- For \( x \) in the interval \( [5, 8] \), \( x - 5 \geq 0 \), so \( |x - 5| = x - 5 \).

### Step 4: Write the integrals with the correct 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: Evaluate the first integral \( \int_2^5 (5 - x) \, dx \).
\[ \int (5 - x) \, dx = 5x - \frac{x^2}{2} + C \]
Now, we evaluate it 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 each term:
\[ = \left( 25 - \frac{25}{2} \right) - \left( 10 - 2 \right) = \left( 25 - 12.5 \right) - 8 = 12.5 - 8 = 4.5 \]

### Step 6: Evaluate the second integral \( \int_5^8 (x - 5) \, dx \).
\[ \int (x - 5) \, dx = \frac{x^2}{2} - 5x + C \]
Now, we evaluate it 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 each term:
\[ = \left( \frac{64}{2} - 40 \right) - \left( \frac{25}{2} - 25 \right) = (32 - 40) - \left( 12.5 - 25 \right) = -8 - (-12.5) = -8 + 12.5 = 4.5 \]

### Step 7: Combine the results from both integrals.
Now we add the results of the two integrals:
\[ \int_2^8 |x - 5| \, dx = 4.5 + 4.5 = 9 \]

### Final Answer:
Thus, the value of \( \int_2^8 |x - 5| \, dx \) is \( 9 \).

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