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To solve the problem of finding \( b_1 \) for the given function \( f(x) = x \cos(x) \) in the interval \( (-\pi, \pi) \), this looks like a problem involving Fourier series.

### Steps to Find \( b_1 \):
1. **Fourier Series for \( f(x) \):** 
   The Fourier series expansion of a function \( f(x) \) over \( (-\pi, \pi) \) is given by:
   \[
   f(x) = a_0 + \sum_{n=1}^{\infty} \left( a_n \cos(nx) + b_n \sin(nx) \right)
   \]
   where
   \[
   b_n = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x) \sin(nx) \, dx
   \]

2. **For \( b_1 \):**
   We need to compute the coefficient \( b_1 \), where:
   \[
   b_1 = \frac{1}{\pi} \int_{-\pi}^{\pi} x \cos(x) \sin(x) \, dx
   \]
   Using the identity \( \cos(x) \sin(x) = \frac{1}{2} \sin(2x) \), the integral simplifies:
   \[
   b_1 = \frac{1}{\pi} \int_{-\pi}^{\pi} x \cdot \frac{1}{2} \sin(2x) \, dx
   \]

3. **Simplification:**
   The integral is odd since the integrand \( x \sin(2x) \) is an odd function (odd function times another function). The integral of an odd function over a symmetric interval \( (-\pi, \pi) \) is zero:
   \[
   b_1 = 0
   \]

Thus, the correct value of \( b_1 \) is indeed **0**.

Would you like a deeper explanation or have any questions? 

Here are five related questions:
1. What are the conditions for a function to have a Fourier series expansion?
2. How do you find the \( a_n \) coefficients in a Fourier series?
3. Why do integrals of odd functions over symmetric intervals yield zero?
4. How can you extend the Fourier series definition to non-periodic functions?
5. What are the practical applications of Fourier series in signal processing?

**Tip:** Whenever you deal with Fourier series, always check if the function is odd or even to simplify integrals.

If f(x) = x cos(x) in (-π, π), then find the Fourier series coefficient b₁.

Learn how to find the Fourier series coefficient b₁ for the function f(x) = x cos(x) over the interval (-π, π). This detailed solution uses integral calculus and the properties of even and odd functions to simplify the process. Suitable for college-level students studying Fourier series.

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