If [.] denotes the greatest integer function and
`f(x)={:{(3[x]-(5|x|)/(x)","x ne 0),(" "2","x= 0):}`
then `int_(-3//2)^(2) f(x)dx` is equal to

Options

A. `-(11)/(2)`
B. `-(7)/(2)`
C. -6
D. `-(17)/(2)`

Answer

To solve the integral \( \int_{-\frac{3}{2}}^{2} f(x) \, dx \) where the function \( f(x) \) is defined as follows:

\[ f(x) = \begin{cases} 3[x] - \frac{5|x|}{x} & \text{if } x \neq 0 \\ 2 & \text{if } x = 0 \end{cases} \]

we will evaluate the integral by breaking it into segments based on the behavior of the greatest integer function \([x]\).

### Step 1: Identify the intervals for integration
The integral will be split into the following intervals:
1. From \( -\frac{3}{2} \) to \( -1 \)
2. From \( -1 \) to \( 0 \)
3. From \( 0 \) to \( 1 \)
4. From \( 1 \) to \( 2 \)

### Step 2: Evaluate \( f(x) \) in each interval

1. **Interval \( \left[-\frac{3}{2}, -1\right) \)**:
- Here, \( [x] = -2 \) (since \(-2 < x < -1\)).
- Thus, \( f(x) = 3(-2) - \frac{5(-x)}{x} = -6 + 5 = -1 \).

2. **Interval \( [-1, 0) \)**:
- Here, \( [x] = -1 \) (since \(-1 \leq x < 0\)).
- Thus, \( f(x) = 3(-1) - \frac{5(-x)}{x} = -3 + 5 = 2 \).

3. **Interval \( [0, 1) \)**:
- Here, \( [x] = 0 \) (since \(0 \leq x < 1\)).
- Thus, \( f(x) = 3(0) - \frac{5x}{x} = -5 \).

4. **Interval \( [1, 2] \)**:
- Here, \( [x] = 1 \) (since \(1 \leq x < 2\)).
- Thus, \( f(x) = 3(1) - \frac{5x}{x} = 3 - 5 = -2 \).

### Step 3: Set up the integral
Now we can set up the integral as follows:

\[ \int_{-\frac{3}{2}}^{2} f(x) \, dx = \int_{-\frac{3}{2}}^{-1} (-1) \, dx + \int_{-1}^{0} (2) \, dx + \int_{0}^{1} (-5) \, dx + \int_{1}^{2} (-2) \, dx \]

### Step 4: Evaluate each integral

1. **For \( \int_{-\frac{3}{2}}^{-1} (-1) \, dx \)**:
\[ = -1 \cdot \left(-1 + \frac{3}{2}\right) = -1 \cdot \left(\frac{1}{2}\right) = -\frac{1}{2} \]

2. **For \( \int_{-1}^{0} (2) \, dx \)**:
\[ = 2 \cdot (0 - (-1)) = 2 \cdot 1 = 2 \]

3. **For \( \int_{0}^{1} (-5) \, dx \)**:
\[ = -5 \cdot (1 - 0) = -5 \]

4. **For \( \int_{1}^{2} (-2) \, dx \)**:
\[ = -2 \cdot (2 - 1) = -2 \]

### Step 5: Combine the results
Now, we combine all the results:

\[ \int_{-\frac{3}{2}}^{2} f(x) \, dx = -\frac{1}{2} + 2 - 5 - 2 \]

Calculating this gives:

\[ = -\frac{1}{2} + 2 - 7 = -\frac{1}{2} - 5 = -\frac{11}{2} \]

### Final Answer
Thus, the value of the integral is:

\[ \int_{-\frac{3}{2}}^{2} f(x) \, dx = -\frac{11}{2} \]