Given a signal: $$x(t) = -5\cos{(93.6\sqrt{2}\pi t - 2\pi/3)} + 3 \cos{(135.2\sqrt{2}\pi t + \pi/13)}$$
I am confused about the right process to take or if there is an easier way than following all the integration. I know I need the fundamental frequency, I need to find the coefficients, but am not sure how to proceed.
Ignore what the comment by Dan Boschen says.
First, determine whether the sum of sinusoids is indeed a periodic signal. Keep in mind that while each sinusoid in a sum is a periodic signal, the sum need not be: for example, $\cos(t) + \sin(\pi t)$ is not a periodic signal even though both $\cos(t)$ and $\sin(\pi t)$ individually are periodic signals. In this instance, you should be able to determine that your given signal is indeed periodic, and perhaps even be able to determine the fundamental frequency.
When you have done all that, there is no need to do any integrations etc. Simply expand the sinusoids using $$\cos(A\pm B) = \cos(A)\cos(B) \mp \sin(A)\sin(B)$$ to get that your signal is of the form $$\alpha \cos(93.6\sqrt{2}\pi t) + \beta \sin(93.6\sqrt{2}\pi t) + \gamma\cos(135.2\sqrt{2}\pi t) + \delta\sin(135.2\sqrt{2}\pi t)$$ which reveals to you the values of the only four nonzero coefficients of the Fourier series (the $a_0 + \sum_{n=1}^\infty a_n \cos(n\omega_0 t) + \sum_{n=1}^\infty b_n \sin(n\omega_0 t)$ version) of your given signal, but not their identities. If you had managed to determine the fundamental frequency $\omega_0$, then you can express the signal as $$a_n \cos(n\omega_0t) + b_n\sin(n\omega_0t) + a_m \cos(m\omega_0t) + b_m\sin(m\omega_0t)$$ where $a_n=\alpha, b_n = \beta, a_m = \gamma, b_m = \delta$, and of course, all the other $a_i's$ and $b_i's$ have value $0$.
