https://file.aiquickdraw.com/mathbot/file/1726927704796u9poxwkd.jpg

Let's go through each part of the question:

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### (i) **Formula used for \( f(x) \)**

The problem asks for the formula of the half-range sine series for the given function \( f(x) = \pi x - x^2 \) over \( (0, \pi) \).

The **half-range sine series** only includes sine terms, which means that \( f(x) \) is represented as:

\[
f(x) = \sum_{n=1}^{\infty} b_n \sin(nx)
\]

So, the correct option is **(d)**.

---

### (ii) **Formula used for \( b_n \)**

For the sine series, the coefficients \( b_n \) are given by:

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

Therefore, the correct option is **(b)**.

---

### (iii) **Value of \( b_n \)**

To determine the value of \( b_n \), we use the formula:

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

This integral evaluates to \( \frac{4}{n^3} [1 + (-1)^n] \). Therefore, the correct option is **(b)**.

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### Final Answers:
- (i) Formula used for \( f(x) \): **(d)**
- (ii) Formula used for \( b_n \): **(b)**
- (iii) Value of \( b_n \): **(b)**

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Would you like a more detailed explanation for any of these parts?

### Related Questions:
1. What is the full Fourier series representation of \( f(x) \)?
2. How is the half-range cosine series derived?
3. What is the general form for the coefficients \( a_n \) in a cosine series?
4. Can we compute \( b_n \) for a more complex function over \( (0, \pi) \)?
5. How do you determine if a function can be represented by a half-range sine series?

**Tip:** Always pay attention to the range of the function when selecting a half-range sine or cosine series!

For half-range sine series of the function f(x) = πx - x² in (0, π)

Explore the half-range sine series for the function f(x) = πx - x² over the interval (0, π). Learn how to compute the Fourier sine coefficients and use integration techniques to determine the value of b_n. This is suitable for undergraduate students studying Fourier series and integral calculus.

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