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We are given the function \( f(x) = 2x \) in the interval \( (0, 2\pi) \), and we need to determine the Fourier sine series coefficient \( b_n \).

### Formula for \( b_n \):

The Fourier sine series coefficient \( b_n \) is given by the formula:

\[
b_n = \frac{1}{\pi} \int_0^{2\pi} f(x) \sin(nx) \, dx
\]

Here, \( f(x) = 2x \), and we will apply this in the interval \( (0, 2\pi) \).

### Step 1: Set up the integral

We need to compute:

\[
b_n = \frac{1}{\pi} \int_0^{2\pi} 2x \sin(nx) \, dx
\]

### Step 2: Integration by parts

To solve the integral \( \int 2x \sin(nx) \, dx \), we can use integration by parts. Let:

- \( u = 2x \), so \( du = 2 \, dx \)
- \( dv = \sin(nx) \, dx \), so \( v = -\frac{1}{n} \cos(nx) \)

Now applying integration by parts:

\[
\int 2x \sin(nx) \, dx = -\frac{2x}{n} \cos(nx) \Big|_0^{2\pi} + \frac{2}{n} \int_0^{2\pi} \cos(nx) \, dx
\]

### Step 3: Evaluate the boundary terms

First, let's evaluate the term \( -\frac{2x}{n} \cos(nx) \Big|_0^{2\pi} \):

At \( x = 2\pi \):
\[
-\frac{2(2\pi)}{n} \cos(n(2\pi)) = -\frac{4\pi}{n} \times 1 = -\frac{4\pi}{n}
\]

At \( x = 0 \):
\[
-\frac{2(0)}{n} \cos(n(0)) = 0
\]

So the boundary term is \( -\frac{4\pi}{n} \).

### Step 4: Evaluate the integral of \( \cos(nx) \)

Now, evaluate \( \frac{2}{n} \int_0^{2\pi} \cos(nx) \, dx \). We know that the integral of \( \cos(nx) \) over one full period \( (0, 2\pi) \) is zero:

\[
\int_0^{2\pi} \cos(nx) \, dx = 0
\]

### Step 5: Combine the results

So, the integral simplifies to:

\[
b_n = \frac{1}{\pi} \left( -\frac{4\pi}{n} \right) = -\frac{4}{n}
\]

### Conclusion:

The correct answer is:

\[
\boxed{d) \, b_n = -\frac{4}{n}}
\]

Would you like further details on the integration process or the Fourier series expansion?

Here are five related questions for further exploration:

1. How would you compute the Fourier cosine coefficients \( a_n \) for the same function \( f(x) = 2x \)?
2. What would change if the function were defined on a different interval, like \( (-\pi, \pi) \)?
3. Can you generalize the solution for \( f(x) = Cx \), where \( C \) is a constant?
4. What is the physical interpretation of the Fourier sine series in applications such as signal processing?
5. How would the Fourier series change if \( f(x) \) was an even or odd function?

**Tip:** When applying integration by parts, always carefully choose the terms for \( u \) and \( dv \) to simplify the calculation.

If f(x) = 2x in (0, 2π), then what is bn?

Solve for the Fourier sine series coefficient bn for the function f(x) = 2x over the interval (0, 2π) using integration by parts. Includes step-by-step breakdown and formula application.

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