Introduction:
The Fourier sine transform is used to transform a function from the time domain to the frequency domain. In this question, we are asked to find the Fourier sine transform of xe^-x^2/2.

Step 1: Rewrite the function in terms of an odd function:
Since we are looking for the Fourier sine transform, we need to rewrite the function as an odd function. We can do this by multiplying the function by the sine function, i.e., xe^-x^2/2 * sin(x).

Step 2: Use integration by parts:
We can use integration by parts to evaluate the Fourier sine transform. Let u = xe^-x^2/2 and dv = sin(x) dx. Then du/dx = e^-x^2/2 - x^2e^-x^2/2 and v = -cos(x). Using the formula for integration by parts, we get:

∫xe^-x^2/2 * sin(x) dx = -xe^-x^2/2 * cos(x) + ∫(e^-x^2/2 - x^2e^-x^2/2) * cos(x) dx

Step 3: Evaluate the integral:
We can evaluate the integral using integration by parts again. Let u = e^-x^2/2 - x^2e^-x^2/2 and dv = cos(x) dx. Then du/dx = -xe^-x^2 and v = sin(x). Using the formula for integration by parts, we get:

∫(e^-x^2/2 - x^2e^-x^2/2) * cos(x) dx = (e^-x^2/2 - x^2e^-x^2/2) * sin(x) + ∫xe^-x^2 * sin(x) dx

Step 4: Substitute the original function:
Substituting the original function back into the equation, we get:

∫xe^-x^2/2 * sin(x) dx = -xe^-x^2/2 * cos(x) + (e^-x^2/2 - x^2e^-x^2/2) * sin(x) + ∫xe^-x^2 * sin(x) dx

Step 5: Simplify the equation:
We can simplify the equation by combining like terms:

∫xe^-x^2/2 * sin(x) dx = (e^-x^2/2) * sin(x) - xe^-x^2/2 * cos(x) - ∫(e^-x^2/2) * cos(x) dx

Step 6: Evaluate the integral:
We can evaluate the integral using integration by parts again. Let u = e^-x^2/2 and dv = cos(x) dx. Then du/dx = -xe^-x^2 and v = sin(x). Using the formula for integration by parts, we get:

∫(e^-x^2/2) * cos(x) dx = e^-x^2/2 * sin(x) + ∫xe^-x^2 * sin(x) dx

Step 7: Substitute the integral back into the equation:

Fourrier sine transform