If `n in N`, then `int_(-n)^(n) (-1)^([x]) dx` equals

Options

A. 2n
B. n
C. `n^(2)`
D. 0

Answer

To solve the integral \( I = \int_{-n}^{n} (-1)^{[x]} \, dx \), where \( n \) is a natural number, we can break it down step by step.

### Step 1: Understand the greatest integer function
The greatest integer function, denoted as \([x]\), returns the largest integer less than or equal to \(x\). For example:
- If \(x \in [-1, 0)\), then \([x] = -1\).
- If \(x \in [0, 1)\), then \([x] = 0\).
- If \(x \in [1, 2)\), then \([x] = 1\), and so on.

### Step 2: Break the integral into parts
We can break the integral from \(-n\) to \(n\) into two parts:
\[ I = \int_{-n}^{0} (-1)^{[x]} \, dx + \int_{0}^{n} (-1)^{[x]} \, dx \]

### Step 3: Evaluate the integral from \(-n\) to \(0\)
For \(x \in [-n, 0)\):
- The greatest integer function \([x] = -1\).
Thus,
\[ \int_{-n}^{0} (-1)^{[x]} \, dx = \int_{-n}^{0} (-1)^{-1} \, dx = \int_{-n}^{0} -1 \, dx \]
Calculating this integral:
\[ = -\left[ x \right]_{-n}^{0} = -\left( 0 - (-n) \right) = -n \]

### Step 4: Evaluate the integral from \(0\) to \(n\)
For \(x \in [0, n)\):
- The greatest integer function \([x] = 0\).
Thus,
\[ \int_{0}^{n} (-1)^{[x]} \, dx = \int_{0}^{n} (-1)^{0} \, dx = \int_{0}^{n} 1 \, dx \]
Calculating this integral:
\[ = \left[ x \right]_{0}^{n} = n - 0 = n \]

### Step 5: Combine the results
Now, we can combine the results from both parts:
\[ I = (-n) + n = 0 \]

### Conclusion
Thus, the value of the integral \( I = \int_{-n}^{n} (-1)^{[x]} \, dx \) is:
\[ \boxed{0} \]