How to obtain the Inverse DT Fourier Transform formula in terms of f in hertz (from the formula in terms of $ \omega $)
Recall:
$ \, x(t)=\mathcal{F}^{-1}(\mathcal{X}(\omega))=\frac{1}{2\pi} \int_{-\infty}^{\infty}\mathcal{X}(\omega)e^{i\omega t} d \omega\, $
To obtain X(f), use the substitution
$ \omega= 2 \pi f $.
More specifically
$ \begin{align} x(t) &=\mathcal{F}^{-1}(\mathcal{X}(\omega)) \\ &=\mathcal{F}^{-1}(\mathcal{X}(2\pi f)) \\ &=\frac{1}{2\pi}\int_{-\infty}^{\infty}\mathcal{X}(2\pi f)e^{i2\pi ft} d2\pi f \\ &= \int_{-\infty}^{\infty}X(f)e^{i2\pi ft} df \end{align} $
