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To find the half-range cosine series for the function \( f(x) = 2x^2 + 3 \) on the interval \( (0, 1) \), we are asked to compute the Fourier cosine series, specifically for a half-range extension on this interval.

### Steps to Solve:

1. **Set up the Fourier Cosine Series**:  
   A half-range cosine series assumes that we are extending \( f(x) \) as an even function. This is typically used for functions defined on half the interval, say \( (0, L) \), where \( L = 1 \). For this case, the cosine series for \( f(x) \) over \( (0, 1) \) is given by:
   \[
   f(x) \sim \frac{a_0}{2} + \sum_{n=1}^{\infty} a_n \cos\left(\frac{n\pi x}{L}\right)
   \]
   where \( L = 1 \), and the coefficients \( a_0 \) and \( a_n \) are computed as:
   \[
   a_0 = \frac{2}{L} \int_0^L f(x) \, dx
   \]
   \[
   a_n = \frac{2}{L} \int_0^L f(x) \cos\left(\frac{n \pi x}{L}\right) \, dx
   \]

2. **Calculate the Coefficients**:

   - \( a_0 \):  
   \[
   a_0 = 2 \int_0^1 (2x^2 + 3) \, dx
   \]
   This is the average value of the function over the interval.

   - \( a_n \):  
   \[
   a_n = 2 \int_0^1 (2x^2 + 3) \cos(n \pi x) \, dx
   \]
   This will involve computing integrals of \( x^2 \cos(n\pi x) \) and \( \cos(n\pi x) \).

Let me now compute these integrals to find the Fourier coefficients \( a_0 \) and \( a_n \).The calculations for the Fourier cosine series yield the following results:

1. **Constant Term \( a_0 \)**:  
   \[
   a_0 = \frac{22}{3}
   \]

2. **Fourier Coefficients \( a_n \) for \( n \geq 1 \)**:  
   The general formula for \( a_n \) is given by:
   \[
   a_n = \frac{2(5\pi^2 n^2 \sin(\pi n) + 4\pi n \cos(\pi n) - 4\sin(\pi n))}{\pi^3 n^3}, \quad \text{for} \, n \neq 0
   \]
   When \( n = 0 \), \( a_n = \frac{22}{3} \).

Given that sine and cosine terms involving multiples of \( \pi \) behave in specific ways (e.g., \( \sin(\pi n) = 0 \) and \( \cos(\pi n) = (-1)^n \)), these terms simplify further for specific values of \( n \).

Would you like more details on the simplification for specific values of \( n \), or do you have any other questions?

Here are 5 relative questions you might find interesting:

1. How do the sine and cosine functions influence the values of the Fourier coefficients?
2. How would the Fourier sine series differ from the cosine series for this function?
3. What happens to the series representation as \( n \to \infty \)?
4. How is the half-range Fourier series extended to a full-range interval?
5. What are the physical interpretations of the Fourier coefficients in real-world applications?

**Tip**: Understanding the behavior of sine and cosine at multiples of \( \pi \) helps greatly in simplifying Fourier series.

Find the half-range cosine series for f(x) = 2x^2 + 3 in (0, 1).

Find the half-range Fourier cosine series for the function f(x) = 2x^2 + 3 on the interval (0, 1). Detailed steps are included for calculating the Fourier coefficients using integration, including key formulas and concepts.

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