We propose a new approach in numerical computations of integral equations of the first kind. Some important integral equations of the first kind arise in the analysis of inverse problems, but they are often typically ill-posed so that numerical construction of their solutions is hard. To overcome the difficulty, two devices play essential roles in our proposal: the use of multiple-precision arithmetic and that of high-accurate numerical integration formula. The latter is a generalization of the spectral collocation method, and we mainly discuss it in this paper. We also show some numerical examples.