This problem involves a Fourier series and integral calculation. The function $f(x)$ is given as a Fourier series expansion:

$f(x) = a_0 + \sum_{n=1}^{\infty} a_n \cos(nx) + \sum_{n=1}^{\infty} b_n \sin(nx)$ where the series is valid for $x \in (-\pi, \pi)$.

We are asked to evaluate:

$\frac{1}{2\pi} \int_{-\pi}^{\pi} [f(x)]^2 \, dx$

This expression involves finding the energy (or the square of the function) over one period $(-\pi, \pi)$, which is a typical application in Fourier analysis.

Key steps:

1. Parse the Fourier series: The function $f(x)$ consists of:

   • A constant term $a_0$,
   • Cosine terms $a_n \cos(nx)$,
   • Sine terms $b_n \sin(nx)$.

2. Square the Fourier series: When squaring $f(x)$, we will have several terms:

   • $a_0^2$,
   • Cross terms involving products of $a_0$ with the cosine and sine terms,
   • Squares of cosine and sine terms (these will contribute significantly).

3. Simplification using orthogonality of trigonometric functions: The integrals of sine and cosine functions over $(-\pi, \pi)$ simplify greatly due to their orthogonality: $\int_{-\pi}^{\pi} \cos(nx) \cos(mx) \, dx = 0 \quad \text{for} \, n \neq m$ and similarly for sine functions. The non-zero contributions come only when $n = m$.

The correct expression is:

$\frac{1}{2\pi} \int_{-\pi}^{\pi} [f(x)]^2 \, dx = a_0^2 + \frac{1}{2} \sum_{n=1}^{\infty} (a_n^2 + b_n^2)$

Thus, the correct answer is:

$\boxed{(d)}$

Would you like a detailed explanation or any clarifications on this?

Here are 5 follow-up questions to expand your understanding:

1. What is the physical interpretation of the integral $\frac{1}{2\pi} \int_{-\pi}^{\pi} f(x)^2 dx$ in Fourier analysis?
2. How does the orthogonality of sine and cosine functions simplify the calculation of integrals in Fourier series?
3. What would change in the calculation if the interval for $x$ was different from $(-\pi, \pi)$?
4. How are the Fourier coefficients $a_n$ and $b_n$ related to the original function $f(x)$?
5. Can you describe how Parseval's theorem relates to this problem?

Tip:

In Fourier series problems, always utilize the orthogonality of trigonometric functions to simplify calculations when integrating products of sines and cosines.