Sorry for the unclear title. Normally, to find the coefficient of the Fourier series we do the integral $$a_n=\frac2T\int^T_0 f(t) \cos(n\omega t) \, dt$$ $$b_n=\frac2T\int^T_0 f(t) \sin(n\omega t) \, dt$$
However, if the function take $\omega t$ as its variable i.e $f(\omega t)$, how can we solve the integral
My lecturer replace $x=\omega t$ and do the integral
$$\int^T_0 f(x) \cos(nx) \, dx$$
because the function take $\omega t$ as its variable so the period of $\omega t$ is $T$. However, what I confuse is why it isn't like this
$$\int^T_0 f(x) \cos(n\omega x) \, dx = \int^T_0 f(\omega t) \cos(n\omega^2 t) \, dx$$