I found an old lecture on YouTube given by Walter Rudin (1990, in Wisconsin), and towards the beginning he mentions that if $f(x)$ were not integrable, on some interval, it would be obvious that it doesn't have a Fourier Series on that interval, since if you looked at the Fourier coefficient given by
$$ \frac{1}{\pi}\int_0^\pi f(x) \cos(nx) dx$$
it wouldn't be meaningful, given that $f$ is not integrable.
I was a bit confused because I'm wondering whether the product $f(x)\cos(nx)$ could be integrable, and so one could indeed compute the coefficients.
So my question is: if $f(x)$ is not integrable, how do we know that the product $f(x)\cos(nx)$ is not integrable?