Evaluate the following integrals:
`int_(0)^(2)[x]dx` where `[x]` is greatest integers function.

Answer

To evaluate the integral \( \int_{0}^{2} [x] \, dx \), where \([x]\) is the greatest integer function, we can break the integral into two parts based on the behavior of the greatest integer function over the interval from 0 to 2.

### Step-by-Step Solution:

1. **Understanding the Greatest Integer Function**:
The greatest integer function \([x]\) gives the largest integer less than or equal to \(x\).
- For \(0 \leq x < 1\), \([x] = 0\).
- For \(1 \leq x < 2\), \([x] = 1\).

2. **Breaking the Integral**:
We can split the integral at \(x = 1\):
\[ \int_{0}^{2} [x] \, dx = \int_{0}^{1} [x] \, dx + \int_{1}^{2} [x] \, dx \]

3. **Evaluating the First Integral**:
For the interval \(0 \leq x < 1\):
\[ \int_{0}^{1} [x] \, dx = \int_{0}^{1} 0 \, dx = 0 \]

4. **Evaluating the Second Integral**:
For the interval \(1 \leq x < 2\):
\[ \int_{1}^{2} [x] \, dx = \int_{1}^{2} 1 \, dx \]
This integral evaluates to:
\[ = [x]_{1}^{2} = 2 - 1 = 1 \]

5. **Combining the Results**:
Now, we combine the results from both integrals:
\[ \int_{0}^{2} [x] \, dx = 0 + 1 = 1 \]

### Final Answer:
Thus, the value of the integral \( \int_{0}^{2} [x] \, dx \) is \( \boxed{1} \).