The vaue of `int_(-1)^(2) (|x|)/(x)dx` is

Options

A. 0
B. 1
C. 3
D. none of these

Answer

To solve the integral \( \int_{-1}^{2} \frac{|x|}{x} \, dx \), we need to analyze the absolute value function and break the integral into appropriate intervals.

### Step-by-Step Solution:

1. **Identify the intervals based on the absolute value function**:
The function \( |x| \) behaves differently depending on whether \( x \) is negative or non-negative:
- For \( x < 0 \), \( |x| = -x \)
- For \( x \geq 0 \), \( |x| = x \)

Therefore, we will split the integral at \( x = 0 \):
\[ \int_{-1}^{2} \frac{|x|}{x} \, dx = \int_{-1}^{0} \frac{|x|}{x} \, dx + \int_{0}^{2} \frac{|x|}{x} \, dx \]

2. **Evaluate the first integral** \( \int_{-1}^{0} \frac{|x|}{x} \, dx \):
In this interval, \( |x| = -x \), so:
\[ \int_{-1}^{0} \frac{|x|}{x} \, dx = \int_{-1}^{0} \frac{-x}{x} \, dx = \int_{-1}^{0} -1 \, dx \]
This simplifies to:
\[ \int_{-1}^{0} -1 \, dx = -1 \cdot (0 - (-1)) = -1 \cdot 1 = -1 \]

3. **Evaluate the second integral** \( \int_{0}^{2} \frac{|x|}{x} \, dx \):
In this interval, \( |x| = x \), so:
\[ \int_{0}^{2} \frac{|x|}{x} \, dx = \int_{0}^{2} \frac{x}{x} \, dx = \int_{0}^{2} 1 \, dx \]
This simplifies to:
\[ \int_{0}^{2} 1 \, dx = 1 \cdot (2 - 0) = 2 \]

4. **Combine the results**:
Now we can combine the results of the two integrals:
\[ \int_{-1}^{2} \frac{|x|}{x} \, dx = \int_{-1}^{0} \frac{|x|}{x} \, dx + \int_{0}^{2} \frac{|x|}{x} \, dx = -1 + 2 = 1 \]

### Final Answer:
The value of the integral \( \int_{-1}^{2} \frac{|x|}{x} \, dx \) is \( 1 \).

---