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The problem asks to calculate the Fourier coefficients \( b_n \) of the periodic function \( f(x) \) with a period of \( 2\pi \). The function is defined piecewise as:

\[
f(x) =
\begin{cases}
-5x - 4, & 0 \leq x \leq \pi \\
0, & \pi < x < 2\pi
\end{cases}
\]

### Fourier Series Setup

The Fourier series for a \( 2\pi \)-periodic function \( f(x) \) is given by:

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

The sine coefficients \( b_n \) are given by the formula:

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

Since \( f(x) \) is piecewise, we split the integral over the two regions:

1. From \( 0 \) to \( \pi \): \( f(x) = -5x - 4 \)
2. From \( \pi \) to \( 2\pi \): \( f(x) = 0 \)

Thus, the formula for \( b_n \) becomes:

\[
b_n = \frac{1}{\pi} \left( \int_0^{\pi} (-5x - 4) \sin(nx) \, dx + \int_{\pi}^{2\pi} 0 \cdot \sin(nx) \, dx \right)
\]

Since the second integral vanishes, we are left with:

\[
b_n = \frac{1}{\pi} \int_0^{\pi} (-5x - 4) \sin(nx) \, dx
\]

### Calculation of \( b_n \)

Now, let's break down the integral:

\[
b_n = \frac{1}{\pi} \left( -5 \int_0^{\pi} x \sin(nx) \, dx - 4 \int_0^{\pi} \sin(nx) \, dx \right)
\]

Each of these integrals can be solved using integration by parts and standard trigonometric integrals. I will now calculate the result of each term.The expression for the Fourier coefficients \( b_n \) is:

\[
b_n =
\begin{cases}
\frac{4n(\cos(\pi n) - 1) + 5\pi n \cos(\pi n) - 5 \sin(\pi n)}{\pi n^2}, & n \neq 0 \\
0, & n = 0
\end{cases}
\]

This is a piecewise result where the Fourier coefficient is zero when \( n = 0 \), and for \( n \neq 0 \), it follows the more complex expression involving trigonometric functions of \( n \).

Would you like further details on the derivation of this result, or clarification on any specific part?

Here are five follow-up questions related to this topic:
1. How do the sine and cosine terms behave in Fourier series for different functions?
2. What is the role of \( a_0 \) in a Fourier series, and how is it calculated?
3. How do Fourier coefficients relate to the shape and symmetry of the original function?
4. What is the significance of the periodicity assumption in Fourier series?
5. How would the Fourier coefficients change if the function were defined over a different interval?

**Tip:** Always pay attention to symmetry when solving Fourier series problems. Symmetric functions can lead to simplifications by eliminating either sine or cosine terms.

Berechnen Sie die Fourier-Koeffizienten b_n zu der 2π-periodischen Funktion f(x) gegeben durch: f(x) = { -5x - 4, 0 ≤ x ≤ π ; 0, π < x < 2π }

Learn how to calculate the Fourier coefficients b_n for a 2π-periodic piecewise function. This solution involves breaking down the function and solving using trigonometric integrals, focusing on sine terms. Suitable for undergraduate-level math students.

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