` int _(a)^(b) (|x|)/x dx , a lt 0 lt b,` is equal to

Options

A. `|b|-|a|`
B. `|b|+|a|`
C. `|a-b|`
D. None of these

Explanation

To solve the integral \( \int_{a}^{b} \frac{|x|}{x} \, dx \) where \( a < 0 < b \), we will break the integral into two parts based on the definition of the absolute value function.

### Step-by-Step Solution:

1. **Understanding the Absolute Value Function**:
- The absolute value function \( |x| \) is defined as:
\[ |x| = \begin{cases} x & \text{if } x \geq 0 \\ -x & \text{if } x < 0 \end{cases} \]

2. **Breaking the Integral**:
- Since \( a < 0 \) and \( b > 0 \), we can split the integral at 0:
\[ \int_{a}^{b} \frac{|x|}{x} \, dx = \int_{a}^{0} \frac{|x|}{x} \, dx + \int_{0}^{b} \frac{|x|}{x} \, dx \]

3. **Evaluating the First Integral** (\( \int_{a}^{0} \frac{|x|}{x} \, dx \)):
- For \( x < 0 \), \( |x| = -x \), so:
\[ \frac{|x|}{x} = \frac{-x}{x} = -1 \]
- Therefore:
\[ \int_{a}^{0} \frac{|x|}{x} \, dx = \int_{a}^{0} -1 \, dx = -\int_{a}^{0} 1 \, dx \]
- This evaluates to:
\[ -[x]_{a}^{0} = -[0 - a] = -(-a) = a \]

4. **Evaluating the Second Integral** (\( \int_{0}^{b} \frac{|x|}{x} \, dx \)):
- For \( x \geq 0 \), \( |x| = x \), so:
\[ \frac{|x|}{x} = \frac{x}{x} = 1 \]
- Therefore:
\[ \int_{0}^{b} \frac{|x|}{x} \, dx = \int_{0}^{b} 1 \, dx = [x]_{0}^{b} = b - 0 = b \]

5. **Combining the Results**:
- Now, we combine the results of both integrals:
\[ \int_{a}^{b} \frac{|x|}{x} \, dx = a + b \]

6. **Final Result**:
- The value of the integral \( \int_{a}^{b} \frac{|x|}{x} \, dx \) is:
\[ a + b \]