Find the value of `int_(0)^(4)[x]dx`, where `[.]` represents the gretest integer function.

Explanation

To find the value of the integral \( \int_{0}^{4} [x] \, dx \), where \([x]\) represents the greatest integer function (also known as the floor function), we will break down the integral into segments where the value of \([x]\) is constant.

### Step-by-step solution:

1. **Identify the intervals**: The greatest integer function \([x]\) takes constant integer values in the intervals:
- From \(0\) to \(1\), \([x] = 0\)
- From \(1\) to \(2\), \([x] = 1\)
- From \(2\) to \(3\), \([x] = 2\)
- From \(3\) to \(4\), \([x] = 3\)

2. **Break the integral into segments**:
\[ \int_{0}^{4} [x] \, dx = \int_{0}^{1} [x] \, dx + \int_{1}^{2} [x] \, dx + \int_{2}^{3} [x] \, dx + \int_{3}^{4} [x] \, dx \]

3. **Evaluate each segment**:
- For \( \int_{0}^{1} [x] \, dx \):
\[ \int_{0}^{1} 0 \, dx = 0 \]
- For \( \int_{1}^{2} [x] \, dx \):
\[ \int_{1}^{2} 1 \, dx = 1 \cdot (2 - 1) = 1 \]
- For \( \int_{2}^{3} [x] \, dx \):
\[ \int_{2}^{3} 2 \, dx = 2 \cdot (3 - 2) = 2 \]
- For \( \int_{3}^{4} [x] \, dx \):
\[ \int_{3}^{4} 3 \, dx = 3 \cdot (4 - 3) = 3 \]

4. **Sum the results**:
\[ \int_{0}^{4} [x] \, dx = 0 + 1 + 2 + 3 = 6 \]

### Final answer:
The value of \( \int_{0}^{4} [x] \, dx \) is \( 6 \).

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