The value of the integral `int_(0)^(3) (x^(2)+1)d[x]` is, where[*] is the greatest integer function

Options

A. 12
B. 15
C. 17
D. 19

Explanation

To solve the integral \( I = \int_{0}^{3} (x^2 + 1) \, d[\![x]\!] \), where \([\![x]\!]\) is the greatest integer function, we will follow these steps:

### Step 1: Understand the greatest integer function
The greatest integer function \([\![x]\!]\) gives the largest integer less than or equal to \(x\). For the interval from 0 to 3, we can break it down as follows:
- From \(0\) to \(1\), \([\![x]\!] = 0\)
- From \(1\) to \(2\), \([\![x]\!] = 1\)
- From \(2\) to \(3\), \([\![x]\!] = 2\)

### Step 2: Break the integral into segments
We can express the integral as:
\[ I = \int_{0}^{1} (x^2 + 1) \, d[\![x]\!] + \int_{1}^{2} (x^2 + 1) \, d[\![x]\!] + \int_{2}^{3} (x^2 + 1) \, d[\![x]\!] \]

### Step 3: Evaluate each segment
Since \([\![x]\!]\) is constant in each segment, the differential \(d[\![x]\!]\) will be zero except at the points where it jumps (at \(x = 1\) and \(x = 2\)). Therefore, we can evaluate the contributions at these points.

1. **At \(x = 1\)**:
\[ \int_{0}^{1} (x^2 + 1) \, d[\![x]\!] = (1^2 + 1) \cdot [\![1]\!] = 2 \cdot 1 = 2 \]

2. **At \(x = 2\)**:
\[ \int_{1}^{2} (x^2 + 1) \, d[\![x]\!] = (2^2 + 1) \cdot [\![2]\!] = 5 \cdot 1 = 5 \]

3. **At \(x = 3\)**:
\[ \int_{2}^{3} (x^2 + 1) \, d[\![x]\!] = (3^2 + 1) \cdot [\![3]\!] = 10 \cdot 1 = 10 \]

### Step 4: Combine the results
Now, we can sum the contributions from each of the points:
\[ I = 2 + 5 + 10 = 17 \]

### Final Answer
Thus, the value of the integral is:
\[ \boxed{17} \]