the value of `int_(0)^([x]) dx` (where , [.] denotes the greatest integer function)

Options

A. [x]
B. `([x])/(2)`
C. `x[x]`
D. None of these

Answer

To solve the integral \( I = \int_{0}^{[x]} dx \), where \([x]\) denotes the greatest integer function (also known as the floor function), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Greatest Integer Function**: The greatest integer function \([x]\) gives the largest integer less than or equal to \(x\). For example, if \(x = 3.7\), then \([x] = 3\); if \(x = 5\), then \([x] = 5\).

2. **Set Up the Integral**: We need to evaluate the integral \( I = \int_{0}^{[x]} dx \). This means we are integrating from 0 to the greatest integer less than or equal to \(x\).

3. **Evaluate the Integral**: The integral of \(dx\) is simply \(x\). Therefore, we can evaluate the definite integral:
\[ I = \left[ x \right]_{0}^{[x]} = [x] - 0 = [x] \]

4. **Final Result**: Thus, the value of the integral is:
\[ I = [x] \]

### Conclusion:
The value of the integral \( \int_{0}^{[x]} dx \) is equal to the greatest integer function of \(x\), denoted as \([x]\).