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We are given the function \( f(x) = x \) in the interval \( (0, 2\pi) \) and asked to find the value of the Fourier coefficient \( b_n \), based on the multiple-choice options. 

### Fourier Coefficients:
For a periodic function \( f(x) \) with period \( 2\pi \), the Fourier series expansion is given by:

\[
f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty} \left( a_n \cos(nx) + b_n \sin(nx) \right)
\]

The \( b_n \) coefficient is computed as:

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

### Applying this to \( f(x) = x \):

1. \( f(x) = x \), so we need to compute:

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

### Integration by Parts:
To evaluate \( \int_0^{2\pi} x \sin(nx) \, dx \), we can use integration by parts. Let:

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

Now, applying the integration by parts formula \( \int u \, dv = uv - \int v \, du \):

\[
\int_0^{2\pi} x \sin(nx) \, dx = \left[ -\frac{x}{n} \cos(nx) \right]_0^{2\pi} + \frac{1}{n} \int_0^{2\pi} \cos(nx) \, dx
\]

Evaluating each term:

- \( \left[ -\frac{x}{n} \cos(nx) \right]_0^{2\pi} = 0 \) (since \( \cos(n \cdot 2\pi) = \cos(0) = 1 \))
- \( \int_0^{2\pi} \cos(nx) \, dx = 0 \) (since the integral of \( \cos(nx) \) over a full period is zero)

Thus:

\[
b_n = \frac{1}{\pi} \times \left( -\frac{2\pi}{n} \right) = -\frac{2}{n}
\]

### Answer:
The correct option is \( \boxed{b_n = -\frac{2}{n}} \), which corresponds to option (b).

Let me know if you'd like more details or have questions! 

Here are five related questions:
1. How would the calculation change if the function were \( f(x) = x^2 \)?
2. What is the significance of the Fourier coefficient \( a_n \)?
3. How does the Fourier series representation change for a function with period \( 4\pi \)?
4. Can we compute the Fourier coefficients for a discontinuous function like \( f(x) = |x| \)?
5. What is the physical interpretation of the \( b_n \) terms in the Fourier series?

**Tip**: Always double-check integration by parts setup when working with trigonometric functions in Fourier series problems!

If f(x) = x in (0, 2π), what is the value of b_n?

Solve for the Fourier coefficient b_n for the function f(x) = x in the interval (0, 2π) using integration by parts. Detailed step-by-step solution and multiple-choice options are provided.

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