We have used the idea of Gaussian integration around the classical solution for the action to argue that the first-order result for the path integral is \(\exp \left\{\frac{i}{\hbar} S_{\mathrm{cl}}\left[x_{\mathrm{cl}}(t)\right]\right\}\). However, is Gaussian integration, and more generally the path integral itself, well defined for this particular exponential? How would you expect to modify the exponential in order to make sense of the Gaussian integration?