`int_(3pi//2)^(5pi//3) [2cos x]dx=`

Options

A. `(5pi)/(3)`
B. `(4pi)/(3)`
C. `(2pi)/(3)`
D. none of these

Answer

To solve the integral \( \int_{\frac{3\pi}{2}}^{\frac{5\pi}{3}} [2 \cos x] \, dx \), where \([ \cdot ]\) denotes the greatest integer function, we will follow these steps:

### Step 1: Determine the values of \(2 \cos x\) at the limits of integration.
- At \(x = \frac{3\pi}{2}\):
\[ \cos\left(\frac{3\pi}{2}\right) = 0 \implies 2 \cos\left(\frac{3\pi}{2}\right) = 2 \cdot 0 = 0 \]

- At \(x = \frac{5\pi}{3}\):
\[ \cos\left(\frac{5\pi}{3}\right) = \cos\left(2\pi - \frac{\pi}{3}\right) = \cos\left(-\frac{\pi}{3}\right) = \frac{1}{2} \implies 2 \cos\left(\frac{5\pi}{3}\right) = 2 \cdot \frac{1}{2} = 1 \]

### Step 2: Analyze the behavior of \(2 \cos x\) in the interval \(\left[\frac{3\pi}{2}, \frac{5\pi}{3}\right]\).
- The function \(2 \cos x\) varies from \(0\) to \(1\) as \(x\) moves from \(\frac{3\pi}{2}\) to \(\frac{5\pi}{3}\).

### Step 3: Determine the greatest integer value of \(2 \cos x\) in the interval.
- Since \(2 \cos x\) starts at \(0\) and reaches \(1\), the greatest integer function \([2 \cos x]\) will take the value \(0\) throughout the entire interval \(\left[\frac{3\pi}{2}, \frac{5\pi}{3}\right)\).

### Step 4: Set up the integral.
- Since \([2 \cos x] = 0\) for all \(x\) in \(\left[\frac{3\pi}{2}, \frac{5\pi}{3}\right)\), we can write:
\[ \int_{\frac{3\pi}{2}}^{\frac{5\pi}{3}} [2 \cos x] \, dx = \int_{\frac{3\pi}{2}}^{\frac{5\pi}{3}} 0 \, dx \]

### Step 5: Evaluate the integral.
- The integral of \(0\) over any interval is \(0\):
\[ \int_{\frac{3\pi}{2}}^{\frac{5\pi}{3}} 0 \, dx = 0 \]

### Final Answer:
\[ \int_{\frac{3\pi}{2}}^{\frac{5\pi}{3}} [2 \cos x] \, dx = 0 \]

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