The value of `int_(pi)^(2pi)[2sinx]dx` is equal to (where `[.]` represents the greatest integer function)

Options

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

Answer

To solve the integral \( \int_{\pi}^{2\pi} 2\sin x \, dx \) and find its value using the greatest integer function, we will follow these steps:

### Step 1: Evaluate the Integral
We start by evaluating the integral \( \int 2\sin x \, dx \).

\[ \int 2\sin x \, dx = -2\cos x + C \]

### Step 2: Set Up the Definite Integral
Now we need to evaluate the definite integral from \( \pi \) to \( 2\pi \):

\[ \int_{\pi}^{2\pi} 2\sin x \, dx = \left[-2\cos x\right]_{\pi}^{2\pi} \]

### Step 3: Calculate the Values at the Bounds
Next, we calculate the values of \( -2\cos x \) at the bounds \( \pi \) and \( 2\pi \):

1. At \( x = 2\pi \):
\[ -2\cos(2\pi) = -2(1) = -2 \]

2. At \( x = \pi \):
\[ -2\cos(\pi) = -2(-1) = 2 \]

### Step 4: Subtract the Values
Now, we subtract the value at the lower limit from the value at the upper limit:

\[ \int_{\pi}^{2\pi} 2\sin x \, dx = -2 - 2 = -4 \]

### Step 5: Apply the Greatest Integer Function
Finally, we apply the greatest integer function (denoted by \( [.] \)) to the result:

\[ \left[-4\right] = -4 \]

### Final Answer
Thus, the value of \( \int_{\pi}^{2\pi} 2\sin x \, dx \) is equal to \( -4 \).

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