The general (composite) Newton–Cotes rules are studied for Hadamard finite-part integrals. We prove that the error of the *k*th-order Newton–Cotes rule is O$$\left({h}^{k}\right|\mathrm{ln}h\left|\right)$$ for odd *k* and O$$\left({h}^{k+1}\right|\mathrm{ln}h\left|\right)$$ for even *k* when the singular point coincides with an element junction point. Two modified Newton–Cotes rules are proposed to remove the factor ln *h* from the error bound. The convergence rate (accuracy) of even-order Newton–Cotes rules at element junction points is the same as the superconvergence rate at certain Gaussian points as presented in Wu & Lü (2005, *IMA J. Numer. Anal.*, **25**, 253–263) and Wu & Sun (2008, *Numer. Math.*, **109**, 143–165). Based on the analysis, a class of collocation-type methods are proposed for solving integral equations with Hadamard finite-part kernels. The accuracy of the collocation method is the same as the accuracy of the proposed even-order Newton–Cotes rules. Several numerical examples are provided to illustrate the theoretical analysis. |