Let `f(x)=x-[x],` for ever real number `x ,w h e r e[x]` is the greatest integer less than or equal to `xdot` Then, evaluate `int_(-1)^1f(x)dx` .

Explanation

To evaluate the integral \( \int_{-1}^{1} f(x) \, dx \) where \( f(x) = x - [x] \) (the fractional part of \( x \)), we can follow these steps:

### Step 1: Understand the function \( f(x) \)
The function \( f(x) = x - [x] \) represents the fractional part of \( x \). This means that for any real number \( x \), \( f(x) \) will yield a value between 0 (inclusive) and 1 (exclusive).

### Step 2: Analyze the function over the interval \([-1, 1]\)
We need to evaluate the integral from \(-1\) to \(1\). The function \( f(x) \) behaves differently in different segments of the interval:
- For \( x \in [-1, 0) \): Here, \( [x] = -1 \), so \( f(x) = x - (-1) = x + 1 \).
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