To find the Fourier series of $f(x) = |\cos x|$ using integration by parts, we'll need to compute the Fourier coefficients. The expression provided,

$I = u \int v dx - u' \int_2 v dx + u'' \int_3 v dx$

is a generalized integration by parts formula, but for the Fourier series, we focus on computing the series as follows.

Fourier Series Overview

The Fourier series of a function $f(x)$ over $[-\pi, \pi]$ is given by:

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

Where:

• $a_n = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x) \cos(nx) dx$
• $b_n = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x) \sin(nx) dx$
• $a_0 = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x) dx$

Step 1: Symmetry of $f(x) = |\cos x|$

Since $f(x) = |\cos x|$ is an even function, all the sine terms $b_n$ will vanish. Thus, we only need to compute the cosine coefficients $a_n$ and $a_0$.

Step 2: Compute $a_0$

The constant term $a_0$ is given by:

$a_0 = \frac{1}{\pi} \int_{-\pi}^{\pi} |\cos x| dx$

Given the symmetry, we can simplify this to:

$a_0 = \frac{2}{\pi} \int_0^{\pi} \cos x \, dx$

This integral is straightforward:

$\int_0^{\pi} |\cos x| dx = \int_0^{\frac{\pi}{2}} \cos x \, dx + \int_{\frac{\pi}{2}}^{\pi} -\cos x \, dx = 2$

Thus,

$a_0 = \frac{4}{\pi}$

Step 3: Compute $a_n$

For $a_n$ (where $n \geq 1$), we compute:

$a_n = \frac{2}{\pi} \int_0^{\pi} |\cos x| \cos(nx) dx$

Given the symmetry of $|\cos x|$, we can split this into two intervals as before:

$a_n = \frac{2}{\pi} \left( \int_0^{\frac{\pi}{2}} \cos x \cos(nx) dx + \int_{\frac{\pi}{2}}^{\pi} -\cos x \cos(nx) dx \right)$

Using trigonometric identities and solving these integrals yields:

$a_n = \frac{4(1 - (-1)^n)}{\pi(1 - n^2)}$

Thus, $a_n$ is non-zero only for even $n$, and the Fourier series becomes:

$|\cos x| = \frac{2}{\pi} + \sum_{k=1}^{\infty} \frac{4}{\pi(1 - 4k^2)} \cos(2kx)$

This is the Fourier series for $f(x) = |\cos x|$.

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Would you like more details or clarifications on any of these steps?

Here are some related questions to explore further:

1. What are the differences between odd and even Fourier series?
2. How would you calculate the Fourier series for $f(x) = \sin^2(x)$?
3. What is the physical interpretation of the Fourier series coefficients?
4. Can the Fourier series be used to approximate non-periodic functions?
5. How do complex Fourier series differ from the real Fourier series?

Tip: When computing Fourier coefficients, always check the symmetry of the function (even/odd) to simplify the integrals.