If [x] denotes the greatest integer function then `int_(0.5)^(4.5) [x] dx+ int_(-1)^(1) |x| dx` is equal to

Options

A. 9
B. 8
C. 7
D. 6

Answer

To solve the given problem, we need to evaluate the following expression:

\[ I = \int_{0.5}^{4.5} [x] \, dx + \int_{-1}^{1} |x| \, dx \]

where \([x]\) denotes the greatest integer function.

### Step 1: Evaluate the first integral \(I_1 = \int_{0.5}^{4.5} [x] \, dx\)

We will break this integral into parts based on the intervals where the greatest integer function \([x]\) remains constant:

1. From \(0.5\) to \(1\): \([x] = 0\)
2. From \(1\) to \(2\): \([x] = 1\)
3. From \(2\) to \(3\): \([x] = 2\)
4. From \(3\) to \(4\): \([x] = 3\)
5. From \(4\) to \(4.5\): \([x] = 4\)

Thus, we can write:

\[ I_1 = \int_{0.5}^{1} 0 \, dx + \int_{1}^{2} 1 \, dx + \int_{2}^{3} 2 \, dx + \int_{3}^{4} 3 \, dx + \int_{4}^{4.5} 4 \, dx \]

Calculating each part:

- \(\int_{0.5}^{1} 0 \, dx = 0\)
- \(\int_{1}^{2} 1 \, dx = [x]_{1}^{2} = 2 - 1 = 1\)
- \(\int_{2}^{3} 2 \, dx = 2[x]_{2}^{3} = 2(3 - 2) = 2\)
- \(\int_{3}^{4} 3 \, dx = 3[x]_{3}^{4} = 3(4 - 3) = 3\)
- \(\int_{4}^{4.5} 4 \, dx = 4[x]_{4}^{4.5} = 4(4.5 - 4) = 2\)

Now, summing these values:

\[ I_1 = 0 + 1 + 2 + 3 + 2 = 8 \]

### Step 2: Evaluate the second integral \(I_2 = \int_{-1}^{1} |x| \, dx\)

We will break this integral into two parts based on the definition of \(|x|\):

1. From \(-1\) to \(0\): \(|x| = -x\)
2. From \(0\) to \(1\): \(|x| = x\)

Thus, we can write:

\[ I_2 = \int_{-1}^{0} -x \, dx + \int_{0}^{1} x \, dx \]

Calculating each part:

- \(\int_{-1}^{0} -x \, dx = -\left[\frac{x^2}{2}\right]_{-1}^{0} = -\left(0 - \frac{(-1)^2}{2}\right) = \frac{1}{2}\)
- \(\int_{0}^{1} x \, dx = \left[\frac{x^2}{2}\right]_{0}^{1} = \frac{1^2}{2} - 0 = \frac{1}{2}\)

Now, summing these values:

\[ I_2 = \frac{1}{2} + \frac{1}{2} = 1 \]

### Step 3: Combine the results

Now, we can combine both integrals to find \(I\):

\[ I = I_1 + I_2 = 8 + 1 = 9 \]

### Final Answer

Thus, the final answer is:

\[ \boxed{9} \]