Use Parsevals identity to find the length
$$|f| = \sqrt{\frac{1}{\pi} \int_{-\pi}^{\pi}[f(x)]^{2}dx}$$
for $f(x) = 2 \cos(14x) + 4 \cos(11x) + 2 \sin(27x) - \cos(19x) + 5 \cos(140x)$.
So I'm a bit confused as to which side of the equation to work on-- as trying to square $f(x)$ is not something that seems overly fun. However, I am aware that the function can be reduced and rewritten as a Fourier series, but I also don't think that's entirely correct because it is such a complex function to begin with. As someone who is quite new to linear algebra and therefore does not necessarily have a whole grasp on this, can somebody help me and walk me through how to do this?
In fact, $f(x)$ as given to you is already in Fourier series form, so no need to reduce and rewrite. A real-valued Fourier series on $[-\pi,\pi]$ is a sum of terms of the form $\sin(nx)$ and $\cos(nx)$, and all your terms are already in that form. So $f$ is just a Fourier series where all but $5$ of the coefficients are zero.