The value of `int_(-2)^1 [x[1+cos((pix)/2)]+1] dx,` where [.] denotes greatest integer function is

Options

A. `1`
B. `1//2`
C. `2`
D. none of these

Explanation

To solve the integral \( \int_{-2}^{1} \left[ x \left( 1 + \cos\left(\frac{\pi x}{2}\right) \right) + 1 \right] \, dx \), where \([.]\) denotes the greatest integer function, we will break it down step by step.

### Step 1: Analyze the function
The integrand is \( [x(1 + \cos(\frac{\pi x}{2})) + 1] \). We need to find the values of \( x \) in the interval \([-2, 1]\) and determine the greatest integer function for those values.

### Step 2: Determine the range of \( x \)
The interval we are integrating over is from \(-2\) to \(1\). We will evaluate the function \( 1 + \cos\left(\frac{\pi x}{2}\right) \) within this interval.

### Step 3: Evaluate \( 1 + \cos\left(\frac{\pi x}{2}\right) \)
- At \( x = -2 \):
\[ \cos\left(\frac{\pi \cdot -2}{2}\right) = \cos(-\pi) = -1 \implies 1 + \cos(-\pi) = 0 \]
- At \( x = -1 \):
\[ \cos\left(\frac{\pi \cdot -1}{2}\right) = \cos(-\frac{\pi}{2}) = 0 \implies 1 + \cos(-\frac{\pi}{2}) = 1 \]
- At \( x = 0 \):
\[ \cos\left(\frac{\pi \cdot 0}{2}\right) = \cos(0) = 1 \implies 1 + \cos(0) = 2 \]
- At \( x = 1 \):
\[ \cos\left(\frac{\pi \cdot 1}{2}\right) = \cos(\frac{\pi}{2}) = 0 \implies 1 + \cos(\frac{\pi}{2}) = 1 \]

### Step 4: Determine the greatest integer function values
Now we can evaluate \( [x(1 + \cos(\frac{\pi x}{2})) + 1] \) at key points:
- For \( x \in [-2, -1) \):
\[ [x(0) + 1] = [1] = 1 \]
- For \( x \in [-1, 0) \):
\[ [x(1) + 1] = [x + 1] \text{ where } x \in [-1, 0) \implies [0, 1) \text{ which gives } [0] = 0 \]
- For \( x \in [0, 1] \):
\[ [x(2) + 1] = [2x + 1] \text{ where } x \in [0, 1] \implies [1, 3) \text{ which gives } [1] = 1 \]

### Step 5: Set up the integral
Now we can break the integral into segments based on the greatest integer values:
\[ \int_{-2}^{-1} 1 \, dx + \int_{-1}^{0} 0 \, dx + \int_{0}^{1} 1 \, dx \]

### Step 6: Calculate each integral
1. \( \int_{-2}^{-1} 1 \, dx = [x]_{-2}^{-1} = -1 - (-2) = 1 \)
2. \( \int_{-1}^{0} 0 \, dx = 0 \)
3. \( \int_{0}^{1} 1 \, dx = [x]_{0}^{1} = 1 - 0 = 1 \)

### Step 7: Combine the results
Adding these results together:
\[ 1 + 0 + 1 = 2 \]

### Final Answer
The value of the integral is \( \boxed{2} \).

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