This paper takes a new approach to goodness-of-fit testing, based on Kolmogorov-Smirnov and Cramr-von Mises type functionals of the suitably integrated empirical process. A key limiting process is the integrated Brownian bridge. The new test statistics are compared with their classical "non-integrated" counterparts with respect to local Bahadur efficiency in case of shift alternatives. Whereas the integrated Kolmogorov-Smirnov test is locally Bahadur optimal for the logistic distribution, an integrated   1n-statistic turns out to be locally Bahadur optimal for the "root-logistic distribution".