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

Options

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

Answer

To find the value of the integral \(\int_{-1}^{2} \frac{|x|}{x} \, dx\), we can follow these steps:

### Step 1: Analyze the function \(\frac{|x|}{x}\)
The function \(\frac{|x|}{x}\) behaves differently depending on the sign of \(x\):
- For \(x < 0\), \(|x| = -x\), so \(\frac{|x|}{x} = \frac{-x}{x} = -1\).
- For \(x > 0\), \(|x| = x\), so \(\frac{|x|}{x} = \frac{x}{x} = 1\).

### Step 2: Split the integral at the point where the function changes
Since the function changes at \(x = 0\), we can split the integral into two parts:
\[ \int_{-1}^{2} \frac{|x|}{x} \, dx = \int_{-1}^{0} \frac{|x|}{x} \, dx + \int_{0}^{2} \frac{|x|}{x} \, dx \]

### Step 3: Evaluate each part of the integral
1. **For the interval \([-1, 0]\)**:
\[ \int_{-1}^{0} \frac{|x|}{x} \, dx = \int_{-1}^{0} -1 \, dx = -\int_{-1}^{0} 1 \, dx \]
Evaluating this integral:
\[ -\left[x\right]_{-1}^{0} = -\left(0 - (-1)\right) = -1 \]

2. **For the interval \([0, 2]\)**:
\[ \int_{0}^{2} \frac{|x|}{x} \, dx = \int_{0}^{2} 1 \, dx = \left[x\right]_{0}^{2} = 2 - 0 = 2 \]

### Step 4: Combine the results
Now, we can combine the results from both intervals:
\[ \int_{-1}^{2} \frac{|x|}{x} \, dx = -1 + 2 = 1 \]

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

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