Kernel machines are widely considered to be powerful tools in various fields of information science. By using a kernel, an unknown target is represented by a function that belongs to a reproducing kernel Hilbert space (RKHS) corresponding to the kernel. The application area is widened by enlarging the RKHS such that it includes a wide class of functions. In this study, we demonstrate a method to perform this by using parameter integration of a parameterized kernel. Some numerical experiments show that the unresolved problem of finding a good parameter can be neglected.