Proving that an $\int_{-\infty}^\infty\int_k^{k+1}\sin(\exp(x))dxdk$ integral converges

Question

Well, how can I prove that the following integral converges:

$$\mathcal{I}:=\int_{-\infty}^\infty\int_\text{k}^{\text{k}+1}\sin\left(\exp\left(x\right)\right)\space\text{d}x\space\text{dk}\tag1$$

Answer 1

$$\mathcal{I}= \int_{-\infty}^{+\infty}\int_{e^k}^{e^{k+1}}\frac{\sin z}{z}\,dz\,dk=\int_{-\infty}^{+\infty}\left[\text{Si}(e^{k+1})-\text{Si}(e^k)\right]\,dk=\int_{0}^{+\infty}\frac{\text{Si}(eu)-\text{Si}(u)}{u}\,du$$ and we may invoke Frullani's integral to get $\mathcal{I}=\color{red}{\frac{\pi}{2}}$.