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 given problem, we need to evaluate the integral of the function \( f(x) \) defined as follows:

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

We need to compute the integral:

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

### Step 1: Determine the intervals and the function values

1. **Interval \( x \in \left[-\frac{3}{2}, -1\right) \)**:
- Here, \( \lfloor x \rfloor = -2 \) and \( |x| = -x \).
- Thus, \( f(x) = 3(-2) - \frac{5(-x)}{x} = -6 + 5 = -1 \).

2. **Interval \( x \in [-1, 0) \)**:
- Here, \( \lfloor x \rfloor = -1 \).
- Thus, \( f(x) = 3(-1) - \frac{5(-x)}{x} = -3 + 5 = 2 \).

3. **Interval \( x \in (0, 1) \)**:
- Here, \( \lfloor x \rfloor = 0 \).
- Thus, \( f(x) = 3(0) - \frac{5x}{x} = -5 \).

4. **Interval \( x \in [1, 2) \)**:
- Here, \( \lfloor x \rfloor = 1 \).
- Thus, \( f(x) = 3(1) - \frac{5x}{x} = 3 - 5 = -2 \).

### Step 2: Set up the integral

Now we can express the integral as a sum of integrals over the defined intervals:

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

### Step 3: Calculate each integral

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

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

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

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

### Step 4: 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:

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

### Final Answer

Thus, the value of the integral is:

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