`int_(0)^(pi)[cos x] dx, [ ]` denotes the greatest integer function , is equal to

Options

A. `(pi)/(2)`
B. 1
C. `(-1)`
D. `-(pi)/(2)`

Answer

To solve the integral \( I = \int_{0}^{\pi} [\cos x] \, dx \), where \([ \cdot ]\) denotes the greatest integer function, we will follow these steps:

### Step 1: Understand the behavior of \(\cos x\) on the interval \([0, \pi]\)

The function \(\cos x\) decreases from 1 to -1 as \(x\) goes from 0 to \(\pi\). Therefore, we can find the intervals where \([\cos x]\) takes specific integer values.

- At \(x = 0\), \(\cos(0) = 1\) so \([\cos(0)] = 1\).
- At \(x = \frac{\pi}{3}\), \(\cos\left(\frac{\pi}{3}\right) = \frac{1}{2\) so \([\cos\left(\frac{\pi}{3}\right)] = 0\).
- At \(x = \frac{\pi}{2}\), \(\cos\left(\frac{\pi}{2}\right) = 0\) so \([\cos\left(\frac{\pi}{2}\right)] = 0\).
- At \(x = \frac{2\pi}{3}\), \(\cos\left(\frac{2\pi}{3}\right) = -\frac{1}{2}\) so \([\cos\left(\frac{2\pi}{3}\right)] = -1\).
- At \(x = \pi\), \(\cos(\pi) = -1\) so \([\cos(\pi)] = -1\).

### Step 2: Determine the intervals for the greatest integer function

From the above analysis, we can conclude:
- For \(x \in [0, \frac{\pi}{3})\), \([\cos x] = 1\).
- For \(x \in [\frac{\pi}{3}, \frac{2\pi}{3})\), \([\cos x] = 0\).
- For \(x \in [\frac{2\pi}{3}, \pi]\), \([\cos x] = -1\).

### Step 3: Break down the integral into parts

Now we can express the integral \(I\) as the sum of integrals over these intervals:
\[ I = \int_{0}^{\frac{\pi}{3}} 1 \, dx + \int_{\frac{\pi}{3}}^{\frac{2\pi}{3}} 0 \, dx + \int_{\frac{2\pi}{3}}^{\pi} (-1) \, dx \]

### Step 4: Calculate each integral

1. **First integral**:
\[ \int_{0}^{\frac{\pi}{3}} 1 \, dx = \left[ x \right]_{0}^{\frac{\pi}{3}} = \frac{\pi}{3} - 0 = \frac{\pi}{3} \]

2. **Second integral**:
\[ \int_{\frac{\pi}{3}}^{\frac{2\pi}{3}} 0 \, dx = 0 \]

3. **Third integral**:
\[ \int_{\frac{2\pi}{3}}^{\pi} (-1) \, dx = -\left[ x \right]_{\frac{2\pi}{3}}^{\pi} = -\left( \pi - \frac{2\pi}{3} \right) = -\left( \frac{\pi}{3} \right) = -\frac{\pi}{3} \]

### Step 5: Combine the results

Now, we can combine the results of the integrals:
\[ I = \frac{\pi}{3} + 0 - \frac{\pi}{3} = 0 \]

### Final Answer

Thus, the value of the integral is:
\[ I = 0 \]