Given f(x) where
`={(x|x|,"for" xle -1),([x+1]+[1-x],"for"-1lt x lt 1 ","),(-x|x|,"for" xge1):}` [.] denotes the greatest integer function. If `I= int_(-2)^(2) f ( x) dx`,then |3I| =

Options

A. `8
B. `-4//3`
C. `8//3`
D. `4//3`

Answer

To solve the problem, we need to evaluate the integral \( I = \int_{-2}^{2} f(x) \, dx \) where the function \( f(x) \) is defined piecewise. Let's break down the solution step by step.

### Step 1: Define the function \( f(x) \)

The function \( f(x) \) is given as:
- \( f(x) = x |x| \) for \( x \leq -1 \)
- \( f(x) = [x + 1] + [1 - x] \) for \( -1 < x < 1 \)
- \( f(x) = -x |x| \) for \( x \geq 1 \)

Here, \([.]\) denotes the greatest integer function.

### Step 2: Break the integral into parts

We can split the integral into three parts based on the definition of \( f(x) \):

\[ I = \int_{-2}^{-1} f(x) \, dx + \int_{-1}^{1} f(x) \, dx + \int_{1}^{2} f(x) \, dx \]

### Step 3: Evaluate each part of the integral

1. **For \( x \in [-2, -1] \)**:
\[ f(x) = x |x| = x(-x) = -x^2 \]
So,
\[ \int_{-2}^{-1} f(x) \, dx = \int_{-2}^{-1} -x^2 \, dx \]
Evaluating this integral:
\[ = -\left[ \frac{x^3}{3} \right]_{-2}^{-1} = -\left( \frac{(-1)^3}{3} - \frac{(-2)^3}{3} \right) = -\left( -\frac{1}{3} + \frac{8}{3} \right) = -\left( \frac{7}{3} \right) = \frac{7}{3} \]

2. **For \( x \in [-1, 1] \)**:
We need to evaluate \( f(x) = [x + 1] + [1 - x] \).
- For \( -1 < x < 0 \): \( [x + 1] = 0 \) and \( [1 - x] = 1 \) (since \( 1 - x > 1 \)).
- For \( 0 < x < 1 \): \( [x + 1] = 1 \) and \( [1 - x] = 1 \).

Therefore, we can compute:
\[ \int_{-1}^{0} f(x) \, dx = \int_{-1}^{0} 1 \, dx = [x]_{-1}^{0} = 0 - (-1) = 1 \]
\[ \int_{0}^{1} f(x) \, dx = \int_{0}^{1} 2 \, dx = [2x]_{0}^{1} = 2 - 0 = 2 \]
Thus,
\[ \int_{-1}^{1} f(x) \, dx = 1 + 2 = 3 \]

3. **For \( x \in [1, 2] \)**:
\[ f(x) = -x |x| = -x^2 \]
So,
\[ \int_{1}^{2} f(x) \, dx = \int_{1}^{2} -x^2 \, dx = -\left[ \frac{x^3}{3} \right]_{1}^{2} = -\left( \frac{8}{3} - \frac{1}{3} \right) = -\left( \frac{7}{3} \right) = \frac{7}{3} \]

### Step 4: Combine the results

Now we can combine the results of the three integrals:
\[ I = \frac{7}{3} + 3 + \frac{7}{3} = \frac{7}{3} + \frac{9}{3} + \frac{7}{3} = \frac{23}{3} \]

### Step 5: Calculate \( |3I| \)

Now, we need to find \( |3I| \):
\[ |3I| = |3 \cdot \frac{23}{3}| = |23| = 23 \]

### Final Answer

Thus, the final answer is:
\[ \boxed{23} \]