Nonlinear integrals (NIs) are useful integration tools. It can get a set of virtual values by projecting original data onto a virtual space for classification purpose using NIs. The classical NIs implement projection along a line with respect to the features. But, in many cases, the linear projection cannot achieve good performance for classification or regression due to the limitation of the integrand. The linear function used for the integrand is just a special type of function with respect to the features. In this paper, we propose a nonlinear integrals with polynomial kernel (NIPK). A polynomial function with respect to the features is used as the integrand of NIs. It enables the projection to be along different types of curves to the virtual space so that the virtual values gotten by NIs can be better regularized and have higher separation power for classification. We use genetic algorithm to learn the fuzzy measures so that a larger solution space can be searched. To test the capability of the NIPK, we apply it to classification on several benchmark datasets and a bioinformatics project. Experiments show that there is evident improvement on performance for the NIPK compared to classical NIs.