`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 can follow these steps:

### Step 1: Understanding the Function
The function \(\cos x\) oscillates between -1 and 1. We need to determine the intervals where \([\cos x]\) takes specific integer values.

### Step 2: Finding the Range of \(\cos x\)
- For \(x = 0\), \(\cos(0) = 1\)
- For \(x = \frac{\pi}{2}\), \(\cos\left(\frac{\pi}{2}\right) = 0\)
- For \(x = \pi\), \(\cos(\pi) = -1\)

Thus, \(\cos x\) decreases from 1 to -1 as \(x\) goes from 0 to \(\pi\).

### Step 3: Identifying Intervals
- On the interval \([0, \frac{\pi}{2})\), \(\cos x\) is in the range \([0, 1)\), so \([\cos x] = 0\).
- At \(x = \frac{\pi}{2}\), \(\cos x = 0\), so \([\cos(\frac{\pi}{2})] = 0\).
- On the interval \((\frac{\pi}{2}, \pi]\), \(\cos x\) is in the range \((-1, 0)\), so \([\cos x] = -1\).

### Step 4: Setting Up the Integral
We can split the integral into two parts:
\[ I = \int_{0}^{\frac{\pi}{2}} [\cos x] \, dx + \int_{\frac{\pi}{2}}^{\pi} [\cos x] \, dx \]
Substituting the values we found:
\[ I = \int_{0}^{\frac{\pi}{2}} 0 \, dx + \int_{\frac{\pi}{2}}^{\pi} (-1) \, dx \]

### Step 5: Evaluating the Integrals
- The first integral:
\[ \int_{0}^{\frac{\pi}{2}} 0 \, dx = 0 \]
- The second integral:
\[ \int_{\frac{\pi}{2}}^{\pi} (-1) \, dx = -\left[x\right]_{\frac{\pi}{2}}^{\pi} = -\left(\pi - \frac{\pi}{2}\right) = -\frac{\pi}{2} \]

### Step 6: Combining the Results
Now, we combine both parts:
\[ I = 0 - \frac{\pi}{2} = -\frac{\pi}{2} \]

### Final Answer
Thus, the value of the integral is:
\[ I = -\frac{\pi}{2} \]