To solve the integral \( \int_{-4}^{4} e^{|x|} \{x\} \, dx \), where \( \{x\} \) denotes the fractional part of \( x \), we can use the property of definite integrals that states:
\[
\int_{-a}^{a} f(x) \, dx = \int_{0}^{a} (f(x) + f(-x)) \, dx
\]
### Step 1: Split the Integral
We can express the integral as:
\[
\int_{-4}^{4} e^{|x|} \{x\} \, dx = \int_{0}^{4} e^{x} \{x\} \, dx + \int_{-4}^{0} e^{-x} \{-x\} \, dx
\]
### Step 2: Simplify the Second Integral
For \( x < 0 \), we have \( |x| = -x \) and \( \{-x\} = 1 - \{x\} \). Therefore,
\[
\int_{-4}^{0} e^{-x} \{-x\} \, dx = \int_{-4}^{0} e^{-x} (1 - \{x\}) \, dx
\]
### Step 3: Combine the Integrals
Now we can combine the two parts:
\[
\int_{-4}^{4} e^{|x|} \{x\} \, dx = \int_{0}^{4} e^{x} \{x\} \, dx + \int_{-4}^{0} e^{-x} (1 - \{x\}) \, dx
\]
### Step 4: Evaluate the Integral from 0 to 4
For \( x \) in the range \( [0, 4] \), \( \{x\} = x - \lfloor x \rfloor \). Since \( x \) is between 0 and 4, we can break it down into intervals:
- 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 5: Evaluate Each Integral
1. **From 0 to 1**:
\[
\int_{0}^{1} e^{x} x \, dx
\]
Use integration by parts.
2. **From 1 to 2**:
\[
\int_{1}^{2} e^{x} (x - 1) \, dx
\]
Again, use integration by parts.
3. **From 2 to 3**:
\[
\int_{2}^{3} e^{x} (x - 2) \, dx
\]
Use integration by parts.
4. **From 3 to 4**:
\[
\int_{3}^{4} e^{x} (x - 3) \, dx
\]
Use integration by parts.
### Step 6: Combine Results
After evaluating all the integrals, combine the results from both parts of the integral.
### Final Step: Calculate the Result
The final result will yield a numerical value, which can be simplified to find the fractional part.
