`f(x)={(3[x]-(5|x|)/(x)",",xne0),(2",",x=0):}`. Then `int_(-3//2)^(2)f(x)dx=`
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Answer

To solve the integral \( \int_{-\frac{3}{2}}^{2} f(x) \, dx \) where

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

we will break the integral into pieces based on the behavior of the function \( f(x) \) in different intervals.

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

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

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

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

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

### Step 2: Set up the integral

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

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

### Step 3: Calculate each integral

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

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

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

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

### Step 4: Combine the results

Now, we combine all the results from the integrals:

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

Calculating this gives:

\[ = -\frac{1}{2} + 2 - 5 - 2 = -\frac{1}{2} + 0 - 5 = -\frac{1}{2} - 5 = -\frac{1}{2} - \frac{10}{2} = -\frac{11}{2} \]

Thus, the final answer is:

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