The image you've uploaded shows the calculation of Fourier series coefficients $a_n$ for a function $F(t)$. Here’s a step-by-step outline of what is happening and what the final expression represents:

1. Expression for $a_0$: The first coefficient $a_0$ is calculated as: $a_0 = \frac{4}{T} \int_0^{T/2} \left( \frac{4c}{T} t - c \right) \, dt = 0$ This indicates that the function has no constant (DC) component.

2. General $a_n$ formula: For $n \geq 1$, the general formula for $a_n$ is: $a_n = \frac{4}{T} \int_0^{T/2} \left( \frac{4c}{T} t - c \right) \cos(n\omega t) \, dt$ which simplifies using integration by parts: $a_n = \frac{16c}{T} \int_0^{T/2} t \cos(n \omega t) \, dt - \frac{4c}{T} \int_0^{T/2} \cos(n \omega t) \, dt$

3. Solving integrals: After solving the integrals, you get: [