How to obtain the CT Fourier transform formula in terms of f in hertz (from the formula in terms of $ \omega $)
Recall:
$ \mathcal{X}(\omega )=\mathcal{F}(x(t))=\int_{-\infty}^{\infty} x(t) e^{-i2\pi ft} dt $
To obtain X(f), use the substitution
$ \omega= 2 \pi f $.
More specifically
$ \begin{align} X(f) &=\mathcal{X}(2\pi f)\\ &=\int_{-\infty}^{\infty} x(t) e^{-i2\pi ft} dt \end{align} $
