Evaluate: `int_0^(100)(x-[x]dx(w h e r e[dot]` represents the greatest integer function).

Explanation

To evaluate the integral

\[ \int_0^{100} \left( x - [x] \right) \, dx \]

where \([x]\) represents the greatest integer function (also known as the floor function), we can follow these steps:

### Step 1: Rewrite the integrand
The expression \( x - [x] \) represents the fractional part of \( x \), denoted as \(\{x\}\). Thus, we can rewrite the integral as:

\[ \int_0^{100} \{x\} \, dx \]

### Step 2: Understand the periodicity
The fractional part function \(\{x\}\) is periodic with a period of 1. This means that the behavior of \(\{x\}\) from 0 to 1 will repeat for each integer interval. Therefore, we can express the integral from 0 to 100 as:

\[ \int_0^{100} \{x\} \, dx = \int_0^{1} \{x\} \, dx \times 100 \]

### Step 3: Evaluate the integral from 0 to 1
Now, we need to evaluate the integral:

\[ \int_0^{1} \{x\} \, dx \]

In the interval from 0 to 1, \(\{x\} = x\). Thus, we can rewrite the integral as:

\[ \int_0^{1} x \, dx \]

### Step 4: Calculate the integral
The integral of \(x\) from 0 to 1 is calculated as follows:

\[ \int_0^{1} x \, dx = \left[ \frac{x^2}{2} \right]_0^1 = \frac{1^2}{2} - \frac{0^2}{2} = \frac{1}{2} \]

### Step 5: Multiply by the number of periods
Since we have established that the integral from 0 to 100 can be expressed as 100 times the integral from 0 to 1, we have:

\[ \int_0^{100} \{x\} \, dx = 100 \times \frac{1}{2} = 50 \]

### Final Answer
Thus, the value of the integral is:

\[ \int_0^{100} \left( x - [x] \right) \, dx = 50 \]

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