The complex form $$A(z,t) = \int_{\omega_1}^{\omega_2} A ( \omega ) e^{j (\omega t - kz) } d\omega$$ is the most common because it is compact (easy to write and read). As usual in physics, one writes complex expressions and when one wants the "real" thing, one takes the real part of the expression.
However, if you really want to have an expression without complex numbers, you can write your wavepacket under the form $$B(z,t) = \int_{\omega_1}^{\omega_2} \alpha( \omega ) \cos(\omega t - kz) + \beta( \omega ) \sin(\omega t - kz)\,d\omega$$$$B(z,t) = \int_{\omega_1}^{\omega_2} \alpha( \omega ) \cos(\omega t - kz) + \beta( \omega ) \sin(\omega t - kz)\,d\omega,$$ with $0 \leq \omega_1 \leq \omega_2$, bothboth $\alpha( \omega ) $ and $\beta( \omega )$ being real numbers (and possibly $0 \leq \omega_1 \leq \omega_2$ if you like to consider only positive frequencies, which is all you need in the real world). This is for a packet travelling towards positive $z$. Change $\omega t - kz$ into $\omega t + kz$ for a packet travelling towards negative $z$.
Note that for $$\alpha( \omega ) = \Re(A( \omega )),\ \beta( \omega )=-\Im(A( \omega )),$$ you have exactly $$B(z,t)=\Re(A(z,t)).$$
Indeed, using this definition of $\alpha$ and $\beta$, that is: $$A(\omega)=\alpha(\omega)-j\beta(\omega),$$ and the identity $\Re((\alpha-j\beta)(C+jS))=\Re(\alpha C+\beta S+j(\alpha S-\beta C))=\alpha C+\beta S$ where $C$ and $S$ stand for $\cos$ and $\sin$, you have: $$\Re(A(z,t))=\Re \int_{\omega_1}^{\omega_2} (\alpha( \omega ) - j \beta( \omega ))\; e^{j (\omega t - kz) } d\omega=\\ \int_{\omega_1}^{\omega_2} \Re ((\alpha( \omega ) - j \beta( \omega )) (\cos (\omega t - kz) +j\sin (\omega t - kz))) \;d\omega=\\ \int_{\omega_1}^{\omega_2} \alpha( \omega ) \cos (\omega t - kz) + \beta( \omega )\sin (\omega t - kz) \;d\omega=B(z,t).\\$$