The value of `int_(0)^(|x|){x}` dx (where `[*]` and `{*}` denotes greatest integer and fraction part function respectively) is

Options

A. `1/2 [x]`
B. `2[x]`
C. `(1)/(2[x])`
D. `[x]`

Answer

To solve the integral \( \int_{0}^{|x|} \{x\} \, dx \), where \(\{x\}\) denotes the fractional part of \(x\) and \([x]\) denotes the greatest integer part of \(x\), we can follow these steps:

### Step 1: Understanding the fractional part function
The fractional part of \(x\) can be expressed as:
\[ \{x\} = x - [x] \]
where \([x]\) is the greatest integer less than or equal to \(x\).

### Step 2: Setting up the integral
We can rewrite the integral using the definition of the fractional part:
\[ \int_{0}^{|x|} \{x\} \, dx = \int_{0}^{|x|} (x - [x]) \, dx \]

### Step 3: Splitting the integral
This integral can be split into two parts:
\[ \int_{0}^{|x|} (x - [x]) \, dx = \int_{0}^{|x|} x \, dx - \int_{0}^{|x|} [x] \, dx \]

### Step 4: Calculating the first integral
The first integral is straightforward:
\[ \int_{0}^{|x|} x \, dx = \left[ \frac{x^2}{2} \right]_{0}^{|x|} = \frac{|x|^2}{2} \]

### Step 5: Calculating the second integral
To compute the second integral, we need to consider how the greatest integer function behaves over the interval \([0, |x|]\). If \(n\) is the largest integer such that \(n \leq |x|\), then we can break the integral into segments:
\[ \int_{0}^{|x|} [x] \, dx = \sum_{k=0}^{n-1} \int_{k}^{k+1} k \, dx + \int_{n}^{|x|} n \, dx \]
Calculating each part:
1. For \(k = 0\) to \(n-1\):
\[ \int_{k}^{k+1} k \, dx = k \cdot (1) = k \]
So, the sum becomes:
\[ \sum_{k=0}^{n-1} k = \frac{(n-1)n}{2} \]
2. For the last part:
\[ \int_{n}^{|x|} n \, dx = n \cdot (|x| - n) \]

Combining these gives:
\[ \int_{0}^{|x|} [x] \, dx = \frac{(n-1)n}{2} + n(|x| - n) \]

### Step 6: Putting it all together
Now, substituting back into our integral:
\[ \int_{0}^{|x|} \{x\} \, dx = \frac{|x|^2}{2} - \left( \frac{(n-1)n}{2} + n(|x| - n) \right) \]

### Step 7: Final expression
The final expression for the integral becomes:
\[ \int_{0}^{|x|} \{x\} \, dx = \frac{|x|^2}{2} - \left( \frac{(n-1)n}{2} + n|x| - n^2 \right) \]