The value of the integral `int_(-4)^(4)e^(|x|){x}dx` is equal to (where `{.}` denotes the fractional part function)

Options

A. `e^(4)`
B. `e^(4)+1`
C. `(e^(4)-1)`
D. `e^(2)`

Answer

To solve the integral \( I = \int_{-4}^{4} e^{|x|} \{x\} \, dx \), where \( \{x\} \) denotes the fractional part of \( x \), we can follow these steps:

### Step 1: Understand the Function
The fractional part function \( \{x\} \) is defined as \( \{x\} = x - \lfloor x \rfloor \), where \( \lfloor x \rfloor \) is the greatest integer less than or equal to \( x \).

### Step 2: Split the Integral
Since the integral is from \(-4\) to \(4\) and involves \( e^{|x|} \), we can use the property of even functions. The function \( e^{|x|} \) is even, and thus:
\[ I = \int_{-4}^{4} e^{|x|} \{x\} \, dx = 2 \int_{0}^{4} e^{x} \{x\} \, dx \]

### Step 3: Evaluate the Integral from 0 to 4
Now we need to evaluate \( \int_{0}^{4} e^{x} \{x\} \, dx \). We can break this integral into intervals where \( \{x\} \) behaves differently:
- From \(0\) to \(1\): \( \{x\} = x \)
- From \(1\) to \(2\): \( \{x\} = x - 1 \)
- From \(2\) to \(3\): \( \{x\} = x - 2 \)
- From \(3\) to \(4\): \( \{x\} = x - 3 \)

Thus, we can write:
\[ \int_{0}^{4} e^{x} \{x\} \, dx = \int_{0}^{1} e^{x} x \, dx + \int_{1}^{2} e^{x} (x - 1) \, dx + \int_{2}^{3} e^{x} (x - 2) \, dx + \int_{3}^{4} e^{x} (x - 3) \, dx \]

### Step 4: Compute Each Integral
1. **For \( \int_{0}^{1} e^{x} x \, dx \)**:
Using integration by parts, let \( u = x \) and \( dv = e^{x} dx \):
\[ du = dx, \quad v = e^{x} \]
\[ \int e^{x} x \, dx = x e^{x} - \int e^{x} \, dx = x e^{x} - e^{x} + C \]
Evaluating from \(0\) to \(1\):
\[ \left[ x e^{x} - e^{x} \right]_{0}^{1} = \left[ 1 e^{1} - e^{1} \right] - \left[ 0 - 1 \right] = e - e + 1 = 1 \]

2. **For \( \int_{1}^{2} e^{x} (x - 1) \, dx \)**:
\[ = \int_{1}^{2} e^{x} x \, dx - \int_{1}^{2} e^{x} \, dx \]
The first part can be computed similarly as above, and the second part evaluates to \( e^{2} - e \).

3. **For \( \int_{2}^{3} e^{x} (x - 2) \, dx \)**:
Similar to above, we compute this integral.

4. **For \( \int_{3}^{4} e^{x} (x - 3) \, dx \)**:
Again, compute this integral.

### Step 5: Combine the Results
After computing all the integrals, sum them up and multiply by \(2\) to get the final value of \(I\).

### Step 6: Find the Fractional Part
Finally, take the fractional part of the result.