The value of `int_-1^10 sgn (x -[x])dx` is equal to (where, [:] denotes the greatest integer function

Options

A. 9
B. 10
C. 11
D. 12

Answer

To solve the integral \( \int_{-1}^{10} \text{sgn}(x - [x]) \, dx \), where \([x]\) denotes the greatest integer function, we will break down the steps systematically.

### Step 1: Understanding the Sign Function
The function \(\text{sgn}(x - [x])\) gives us the sign of the fractional part of \(x\). The fractional part of \(x\) is defined as \(x - [x]\), which is always between 0 and 1 for any real number \(x\). Therefore, we can conclude:
- If \(x\) is an integer, then \(x - [x] = 0\) and \(\text{sgn}(x - [x]) = 0\).
- If \(x\) is not an integer, then \(x - [x] > 0\) and \(\text{sgn}(x - [x]) = 1\).

### Step 2: Identifying the Intervals
The integral from \(-1\) to \(10\) includes both integer and non-integer values. The integers in this range are \(-1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10\). The non-integer values will contribute \(1\) to the integral.

### Step 3: Counting the Integers
From \(-1\) to \(10\), the integers are:
- \(-1\)
- \(0\)
- \(1\)
- \(2\)
- \(3\)
- \(4\)
- \(5\)
- \(6\)
- \(7\)
- \(8\)
- \(9\)
- \(10\)

There are a total of \(12\) integers in this range.

### Step 4: Setting Up the Integral
The integral can be split into segments:
- From \(-1\) to \(0\): \(\text{sgn}(x - [x]) = 0\) at \(x = -1\) and \(x = 0\).
- From \(0\) to \(1\): \(\text{sgn}(x - [x]) = 1\).
- From \(1\) to \(2\): \(\text{sgn}(x - [x]) = 1\).
- Continuing this way until \(9\) to \(10\): \(\text{sgn}(x - [x]) = 1\).

### Step 5: Evaluating the Integral
The integral can be computed as follows:
- From \(-1\) to \(0\): contributes \(0\).
- From \(0\) to \(1\): contributes \(1\).
- From \(1\) to \(2\): contributes \(1\).
- From \(2\) to \(3\): contributes \(1\).
- From \(3\) to \(4\): contributes \(1\).
- From \(4\) to \(5\): contributes \(1\).
- From \(5\) to \(6\): contributes \(1\).
- From \(6\) to \(7\): contributes \(1\).
- From \(7\) to \(8\): contributes \(1\).
- From \(8\) to \(9\): contributes \(1\).
- From \(9\) to \(10\): contributes \(0\).

Thus, the total contribution from non-integer segments is \(10\).

### Step 6: Final Calculation
The total value of the integral is:
\[ 0 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 0 = 10 \]

### Conclusion
The value of the integral \( \int_{-1}^{10} \text{sgn}(x - [x]) \, dx \) is \(10\).