`int_(a)^(b) [|x-a| + |x-b|]dx`=

Options

A. `((b-a)^(2))/(2)`
B. `(b^(2)-a^(2))/(2)`
C. `(a^(3)-b^(3))/(2)`
D. `(b-a)^(2)`

Answer

To solve the integral \( \int_{a}^{b} [|x-a| + |x-b|] \, dx \), we will analyze the expression inside the integral step by step.

### Step 1: Analyze the Absolute Values
The expression \( |x-a| + |x-b| \) needs to be evaluated based on the intervals defined by \( a \) and \( b \).

- For \( x < a \):
- \( |x-a| = a-x \)
- \( |x-b| = b-x \)
- Thus, \( |x-a| + |x-b| = (a-x) + (b-x) = a + b - 2x \)

- For \( a \leq x \leq b \):
- \( |x-a| = x-a \)
- \( |x-b| = b-x \)
- Thus, \( |x-a| + |x-b| = (x-a) + (b-x) = b - a \)

- For \( x > b \):
- \( |x-a| = x-a \)
- \( |x-b| = x-b \)
- Thus, \( |x-a| + |x-b| = (x-a) + (x-b) = 2x - (a+b) \)

Since the limits of integration are from \( a \) to \( b \), we only need to consider the case where \( a \leq x \leq b \).

### Step 2: Set Up the Integral
From the analysis above, we have:
\[ |x-a| + |x-b| = b - a \quad \text{for } x \in [a, b] \]
Now, we can set up the integral:
\[ \int_{a}^{b} [|x-a| + |x-b|] \, dx = \int_{a}^{b} (b - a) \, dx \]

### Step 3: Evaluate the Integral
The integral simplifies to:
\[ \int_{a}^{b} (b - a) \, dx = (b - a) \int_{a}^{b} 1 \, dx \]
The integral of 1 from \( a \) to \( b \) is simply:
\[ \int_{a}^{b} 1 \, dx = b - a \]
Thus, we have:
\[ \int_{a}^{b} (b - a) \, dx = (b - a)(b - a) = (b - a)^2 \]

### Final Answer
Therefore, the value of the integral is:
\[ \int_{a}^{b} [|x-a| + |x-b|] \, dx = (b - a)^2 \]

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