Difference between the greatest and least values opf the function `f (x) = int _(0)^(x) (cos ^(2) t + cos t +2) ` dt in the interval `[0, 2pi]` is `K pi, `then K is equal to:

Options

A. 1
B. 3
C. 5
D. None of these

Answer

To find the difference between the greatest and least values of the function \( f(x) = \int_{0}^{x} (\cos^2 t + \cos t + 2) \, dt \) in the interval \([0, 2\pi]\), we will follow these steps:

### Step 1: Differentiate the function
Using the Fundamental Theorem of Calculus, we differentiate \( f(x) \):
\[ f'(x) = \cos^2 x + \cos x + 2 \]

### Step 2: Analyze the derivative
We need to analyze \( f'(x) \):
\[ f'(x) = \cos^2 x + \cos x + 2 \]
Since \( \cos^2 x \) is always non-negative and \( \cos x \) ranges from -1 to 1, we can see that:
\[ f'(x) \geq 0 \quad \text{for all } x \]
This means \( f(x) \) is a non-decreasing function on the interval \([0, 2\pi]\).

### Step 3: Evaluate \( f(x) \) at the endpoints
Next, we evaluate \( f(x) \) at the endpoints of the interval:
1. **At \( x = 0 \)**:
\[ f(0) = \int_{0}^{0} (\cos^2 t + \cos t + 2) \, dt = 0 \]

2. **At \( x = 2\pi \)**:
\[ f(2\pi) = \int_{0}^{2\pi} (\cos^2 t + \cos t + 2) \, dt \]

### Step 4: Calculate \( f(2\pi) \)
To compute \( f(2\pi) \), we can split the integral:
\[ f(2\pi) = \int_{0}^{2\pi} \cos^2 t \, dt + \int_{0}^{2\pi} \cos t \, dt + \int_{0}^{2\pi} 2 \, dt \]

- **Calculate \( \int_{0}^{2\pi} \cos^2 t \, dt \)**:
Using the identity \( \cos^2 t = \frac{1 + \cos 2t}{2} \):
\[ \int_{0}^{2\pi} \cos^2 t \, dt = \int_{0}^{2\pi} \frac{1 + \cos 2t}{2} \, dt = \frac{1}{2} \left[ t + \frac{1}{2} \sin 2t \right]_{0}^{2\pi} = \frac{1}{2} \left[ 2\pi + 0 - (0 + 0) \right] = \pi \]

- **Calculate \( \int_{0}^{2\pi} \cos t \, dt \)**:
\[ \int_{0}^{2\pi} \cos t \, dt = [\sin t]_{0}^{2\pi} = \sin(2\pi) - \sin(0) = 0 \]

- **Calculate \( \int_{0}^{2\pi} 2 \, dt \)**:
\[ \int_{0}^{2\pi} 2 \, dt = 2 \cdot (2\pi - 0) = 4\pi \]

### Step 5: Combine the results
Now, we can combine these results:
\[ f(2\pi) = \pi + 0 + 4\pi = 5\pi \]

### Step 6: Find the difference
The greatest value of \( f(x) \) is \( f(2\pi) = 5\pi \) and the least value is \( f(0) = 0 \). Therefore, the difference is:
\[ \text{Difference} = f(2\pi) - f(0) = 5\pi - 0 = 5\pi \]

### Step 7: Express in terms of \( K \)
We are given that the difference is \( K\pi \). Thus, we have:
\[ K\pi = 5\pi \implies K = 5 \]

### Final Answer
Thus, the value of \( K \) is:
\[ \boxed{5} \]