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Let's break down the problem step by step:

### (c) Find the Fourier sine series of \( x^3 \).

The Fourier sine series for a function \( f(x) \) defined on the interval \( [0, L] \) is given by:

\[
f(x) = \sum_{n=1}^{\infty} b_n \sin\left(\frac{n\pi x}{L}\right)
\]
where the coefficients \( b_n \) are given by:

\[
b_n = \frac{2}{L} \int_0^L f(x) \sin\left(\frac{n\pi x}{L}\right) dx
\]

For the function \( f(x) = x^3 \), we need to compute:

\[
b_n = \frac{2}{L} \int_0^L x^3 \sin\left(\frac{n\pi x}{L}\right) dx
\]

This integral can be solved using integration by parts or by using a standard table of integrals.

### (d) Using the Fourier cosine series of \( f(x) \) to determine the Fourier sine series of \( x^3 \).

The Fourier cosine series for \( f(x) \) on \( [0, L] \) is of the form:

\[
f(x) = a_0 + \sum_{n=1}^{\infty} a_n \cos\left(\frac{n\pi x}{L}\right)
\]
where the coefficients \( a_n \) are given by:

\[
a_n = \frac{2}{L} \int_0^L f(x) \cos\left(\frac{n\pi x}{L}\right) dx
\]

To answer whether we can use the Fourier cosine series to determine the Fourier sine series, note that both series represent the same function but in different forms (sine vs cosine expansion). You can apply integration by parts to relate the cosine coefficients to the sine coefficients through the orthogonality properties of sine and cosine functions.

Now, let me know if you'd like me to solve the integrals or go deeper into any of these steps!

### Follow-up questions:
1. How do you compute the Fourier sine series step by step for a cubic function?
2. How is the Fourier sine series different from the Fourier cosine series?
3. What method do we use to compute integrals involving products of polynomials and sine functions?
4. Can you explain why the sine series applies to odd extensions of functions?
5. How do you check if the result in (d) matches the result in (c)?

### Tip:
When dealing with Fourier series, make sure to distinguish between sine and cosine series depending on whether the function exhibits odd or even symmetry.

c) Find the Fourier sine series of x^3. d) Can we use the Fourier cosine series of f(x) to determine the Fourier sine series of x^3?

Explore how to find the Fourier sine series of x^3 using integral formulas and understand the relationship between Fourier cosine and sine series. This tutorial covers integration by parts and Fourier series fundamentals for undergraduate students.

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