`int_(pi//2)^(3pi//2) [2cos x]dx`is equal to

Options

A. `(5pi)/(3)`
B. `-(5pi)/(3)`
C. `-pi`
D. `-2pi`

Answer

To solve the integral \( I = \int_{\frac{\pi}{2}}^{\frac{3\pi}{2}} \lfloor 2 \cos x \rfloor \, dx \), we will follow these steps:

### Step 1: Understand the Function
The function \( \lfloor 2 \cos x \rfloor \) represents the greatest integer less than or equal to \( 2 \cos x \). We need to analyze the behavior of \( \cos x \) in the interval \( \left[\frac{\pi}{2}, \frac{3\pi}{2}\right] \).

### Step 2: Evaluate \( \cos x \) in the Interval
In the interval \( \left[\frac{\pi}{2}, \frac{3\pi}{2}\right] \):
- At \( x = \frac{\pi}{2} \), \( \cos\left(\frac{\pi}{2}\right) = 0 \)
- At \( x = \frac{3\pi}{2} \), \( \cos\left(\frac{3\pi}{2}\right) = 0 \)
- The maximum value of \( \cos x \) in this interval is \( -1 \) (which occurs at \( x = \pi \)).

Thus, \( 2 \cos x \) ranges from \( 0 \) to \( -2 \) in this interval.

### Step 3: Determine the Values of \( \lfloor 2 \cos x \rfloor \)
- For \( 2 \cos x \) in \( \left(0, -1\right) \), \( \lfloor 2 \cos x \rfloor = -1 \).
- For \( 2 \cos x \) in \( \left[-2, -1\right) \), \( \lfloor 2 \cos x \rfloor = -2 \).

### Step 4: Find the Points of Change
To find the points where \( 2 \cos x \) crosses the integer boundaries:
- Set \( 2 \cos x = -1 \):
\[ \cos x = -\frac{1}{2} \]
This occurs at \( x = \frac{2\pi}{3} \) and \( x = \frac{4\pi}{3} \).

### Step 5: Break the Integral into Intervals
Now we can break the integral into three parts:
1. From \( \frac{\pi}{2} \) to \( \frac{2\pi}{3} \): Here, \( \lfloor 2 \cos x \rfloor = -1 \).
2. From \( \frac{2\pi}{3} \) to \( \frac{4\pi}{3} \): Here, \( \lfloor 2 \cos x \rfloor = -2 \).
3. From \( \frac{4\pi}{3} \) to \( \frac{3\pi}{2} \): Here, \( \lfloor 2 \cos x \rfloor = -1 \).

### Step 6: Write the Integral
Thus, we can write:
\[ I = \int_{\frac{\pi}{2}}^{\frac{2\pi}{3}} (-1) \, dx + \int_{\frac{2\pi}{3}}^{\frac{4\pi}{3}} (-2) \, dx + \int_{\frac{4\pi}{3}}^{\frac{3\pi}{2}} (-1) \, dx \]

### Step 7: Evaluate Each Integral
1. For \( \int_{\frac{\pi}{2}}^{\frac{2\pi}{3}} (-1) \, dx \):
\[ = -\left(\frac{2\pi}{3} - \frac{\pi}{2}\right) = -\left(\frac{4\pi}{6} - \frac{3\pi}{6}\right) = -\frac{\pi}{6} \]

2. For \( \int_{\frac{2\pi}{3}}^{\frac{4\pi}{3}} (-2) \, dx \):
\[ = -2\left(\frac{4\pi}{3} - \frac{2\pi}{3}\right) = -2\left(\frac{2\pi}{3}\right) = -\frac{4\pi}{3} \]

3. For \( \int_{\frac{4\pi}{3}}^{\frac{3\pi}{2}} (-1) \, dx \):
\[ = -\left(\frac{3\pi}{2} - \frac{4\pi}{3}\right) = -\left(\frac{9\pi}{6} - \frac{8\pi}{6}\right) = -\frac{\pi}{6} \]

### Step 8: Combine the Results
Now, combine all parts:
\[ I = -\frac{\pi}{6} - \frac{4\pi}{3} - \frac{\pi}{6} \]
\[ = -\frac{\pi}{6} - \frac{8\pi}{6} - \frac{\pi}{6} = -\frac{10\pi}{6} = -\frac{5\pi}{3} \]

### Final Answer
Thus, the value of the integral is:
\[ \boxed{-\frac{5\pi}{3}} \]