`int_1^4(x-0. 4)dx equals (where {x }is a fractional part of (x)` (a) 13 (b) 6.3 (c) 1.5 (d) 7.5

Options

A. `13`
B. `6.3`
C. `1.5`
D. `7.5`

Explanation

To solve the integral \( \int_1^4 (x - 0.4) \, dx \) where \( x \) is the fractional part of \( x \), we will follow these steps:

### Step 1: Change of Variable
Let \( t = x - 0.4 \). Then, the differential \( dx = dt \).

### Step 2: Change the Limits of Integration
When \( x = 1 \):
\[ t = 1 - 0.4 = 0.6 \]
When \( x = 4 \):
\[ t = 4 - 0.4 = 3.6 \]
Thus, the integral becomes:
\[ \int_{0.6}^{3.6} t \, dt \]

### Step 3: Break the Integral into Intervals
Next, we need to consider the fractional part of \( t \). The fractional part function \( \{t\} \) can be expressed piecewise over the intervals:
- From \( 0.6 \) to \( 1 \): \( \{t\} = t \)
- From \( 1 \) to \( 2 \): \( \{t\} = t - 1 \)
- From \( 2 \) to \( 3 \): \( \{t\} = t - 2 \)
- From \( 3 \) to \( 3.6 \): \( \{t\} = t - 3 \)

Thus, we can break the integral into four parts:
\[ \int_{0.6}^{1} t \, dt + \int_{1}^{2} (t - 1) \, dt + \int_{2}^{3} (t - 2) \, dt + \int_{3}^{3.6} (t - 3) \, dt \]

### Step 4: Evaluate Each Integral
1. **First Integral**:
\[ \int_{0.6}^{1} t \, dt = \left[ \frac{t^2}{2} \right]_{0.6}^{1} = \frac{1^2}{2} - \frac{0.6^2}{2} = \frac{1}{2} - \frac{0.36}{2} = \frac{1}{2} - 0.18 = 0.32 \]

2. **Second Integral**:
\[ \int_{1}^{2} (t - 1) \, dt = \left[ \frac{(t-1)^2}{2} \right]_{1}^{2} = \left[ \frac{(2-1)^2}{2} \right] = \frac{1^2}{2} = \frac{1}{2} = 0.5 \]

3. **Third Integral**:
\[ \int_{2}^{3} (t - 2) \, dt = \left[ \frac{(t-2)^2}{2} \right]_{2}^{3} = \left[ \frac{(3-2)^2}{2} \right] = \frac{1^2}{2} = \frac{1}{2} = 0.5 \]

4. **Fourth Integral**:
\[ \int_{3}^{3.6} (t - 3) \, dt = \left[ \frac{(t-3)^2}{2} \right]_{3}^{3.6} = \left[ \frac{(3.6-3)^2}{2} \right] = \frac{0.6^2}{2} = \frac{0.36}{2} = 0.18 \]

### Step 5: Sum the Results
Now, we add all the results from the integrals:
\[ 0.32 + 0.5 + 0.5 + 0.18 = 1.5 \]

### Final Answer
Thus, the value of the integral \( \int_1^4 (x - 0.4) \, dx \) is:
\[ \boxed{1.5} \]