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To find the Fourier sine transform of \( f(x) = e^{-|x|} \), we follow these steps:

### Step 1: Define the Fourier Sine Transform
The Fourier sine transform \( F_s(k) \) of a function \( f(x) \) is defined as:
\[ F_s(k) = \sqrt{\frac{2}{\pi}} \int_0^\infty f(x) \sin(kx) \, dx \]

Given \( f(x) = e^{-|x|} \), we need to split the function \( f(x) \) into two parts since it is defined piecewise:
\[ f(x) = \begin{cases} 
e^{-x}, & x \geq 0 \\
e^x, & x < 0 
\end{cases} \]

### Step 2: Simplify the Integral
Because the Fourier sine transform is defined for \( x \geq 0 \), we only consider the part of \( f(x) \) where \( x \geq 0 \):
\[ f(x) = e^{-x} \]

### Step 3: Set Up the Integral
The Fourier sine transform of \( f(x) = e^{-x} \) is then:
\[ F_s(k) = \sqrt{\frac{2}{\pi}} \int_0^\infty e^{-x} \sin(kx) \, dx \]

### Step 4: Evaluate the Integral
To evaluate the integral \( \int_0^\infty e^{-x} \sin(kx) \, dx \), we use integration by parts or a known integral result. This integral can be solved using the formula:
\[ \int_0^\infty e^{-ax} \sin(bx) \, dx = \frac{b}{a^2 + b^2} \]
where \( a = 1 \) and \( b = k \).

### Step 5: Apply the Formula
Using the above formula:
\[ \int_0^\infty e^{-x} \sin(kx) \, dx = \frac{k}{1^2 + k^2} = \frac{k}{1 + k^2} \]

### Step 6: Multiply by the Prefactor
Now, multiply by the prefactor \( \sqrt{\frac{2}{\pi}} \):
\[ F_s(k) = \sqrt{\frac{2}{\pi}} \cdot \frac{k}{1 + k^2} \]

### Final Result
The Fourier sine transform of \( f(x) = e^{-|x|} \) is:
\[ F_s(k) = \sqrt{\frac{2}{\pi}} \cdot \frac{k}{1 + k^2} \]

### Summary
- We defined the Fourier sine transform.
- We considered the function \( f(x) \) for \( x \geq 0 \).
- We set up and evaluated the integral using a known formula.
- We multiplied by the necessary prefactor to obtain the final result.

Would you like any more details or explanations on specific parts of this solution?

### Relative Questions
1. What is the definition of the Fourier sine transform?
2. How do you handle piecewise functions in Fourier transforms?
3. What is the known integral result for \( \int_0^\infty e^{-ax} \sin(bx) \, dx \)?
4. Can you show the steps for integration by parts used in evaluating the integral?
5. How does the prefactor \( \sqrt{\frac{2}{\pi}} \) affect the Fourier transform?
6. What is the Fourier cosine transform, and how does it differ from the sine transform?
7. Why do we only consider \( x \geq 0 \) in the sine transform?
8. How would the result change if we considered the Fourier transform instead of the sine transform?

### Tip
When dealing with Fourier transforms, always check the conditions of the integral and simplify the function within the given domain for accurate results.

Learn how to compute the Fourier sine transform of the function f(x) = e^{-|x|}. Detailed step-by-step solution with explanations, including the Fourier sine transform formula and integration techniques. Suitable for advanced mathematics enthusiasts and students.

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