If [*] denots the greatest integer function then the value of the integeral `underset(-pi//2)overset(pi//2)int([(x)/(pi)]+0.5) dx`, is

Options

A. `pi`
B. `(pi)/(2)`
C. 0
D. `-(pi)/(2)`

Answer

To solve the integral

\[ \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \left( \left[ \frac{x}{\pi} \right] + 0.5 \right) dx, \]

we will follow these steps:

### Step 1: Identify the function
Let \( f(x) = \left[ \frac{x}{\pi} \right] + 0.5 \).

### Step 2: Determine the behavior of the function
The greatest integer function \( \left[ \frac{x}{\pi} \right] \) will take different values depending on the interval of \( x \):
- For \( x \) in \( \left[-\frac{\pi}{2}, 0\right) \), \( \frac{x}{\pi} \) is in \( \left[-\frac{1}{2}, 0\right) \), so \( \left[ \frac{x}{\pi} \right] = -1 \).
- For \( x = 0 \), \( \left[ \frac{x}{\pi} \right] = 0 \).
- For \( x \) in \( \left(0, \frac{\pi}{2}\right) \), \( \frac{x}{\pi} \) is in \( \left(0, \frac{1}{2}\right) \), so \( \left[ \frac{x}{\pi} \right] = 0 \).

Thus, we can express \( f(x) \) as:
- \( f(x) = -1 + 0.5 = -0.5 \) for \( x \in \left[-\frac{\pi}{2}, 0\right) \)
- \( f(x) = 0 + 0.5 = 0.5 \) for \( x \in \left(0, \frac{\pi}{2}\right) \)

### Step 3: Split the integral
Now, we can split the integral into two parts:

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

### Step 4: Evaluate each part
1. For \( x \in \left[-\frac{\pi}{2}, 0\right) \):
\[ \int_{-\frac{\pi}{2}}^{0} f(x) \, dx = \int_{-\frac{\pi}{2}}^{0} (-0.5) \, dx = -0.5 \cdot \left(0 - \left(-\frac{\pi}{2}\right)\right) = -0.5 \cdot \frac{\pi}{2} = -\frac{\pi}{4} \]

2. For \( x \in \left(0, \frac{\pi}{2}\right) \):
\[ \int_{0}^{\frac{\pi}{2}} f(x) \, dx = \int_{0}^{\frac{\pi}{2}} (0.5) \, dx = 0.5 \cdot \left(\frac{\pi}{2} - 0\right) = 0.5 \cdot \frac{\pi}{2} = \frac{\pi}{4} \]

### Step 5: Combine the results
Now, we combine the results of both integrals:

\[ \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} f(x) \, dx = -\frac{\pi}{4} + \frac{\pi}{4} = 0 \]

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

\[ \boxed{0}. \]