For improper integrals of the form \(\int_0^\infty f(t)\,dt\), the author defines modes of convergence analogous to the various methods of summability for infinite series. Corresponding to Cesàro summability, for \(\alpha>-1\), \(f\) is \((C,\alpha)\) integrable if there is a number \(a\) such that \(\lim_{T\to+\infty}\int_0^T(1-\frac{t}{T})^\alpha f(t)\,dt=a\). Among the results contained in the present article are these: For \(\alpha >-1\) and \(h > 0\), if \(f\) is \((C,\alpha)\) integrable, with value \(a\), then \(f\) is \((C, \alpha+h)\) integrable with value \(a\); if \(\int_0^\infty f(t)\,dt\) converges to \(a\), then it converges to \(a\) in the sense of Cesàro, as well; if \(\int_0^\infty f(t)\,dt\) and \(\int_0^\infty g(t)\,dt\) converge to \(a\) and \(b\), and if \(h(t):=\int_0^t f(t-s)g(s)\,ds\) then \(\int_0^\infty h(t)\,dt\) converges \((C, 1)\) to \(ab\); and examples are given of divergent integrals that are Cesàro integrable. Corresponding to Abel summability, for \(f\) continuous on \([0,\infty)\), the author defines \(f\) to be A-integrable if \(\lim_{\lambda\to0+}\int_0^\infty e^{-\lambda t}f(t)\,dt\) exists. In a forthcoming note, the author promises to examine consequences of A-integrability as well as relations between the Cesàro and Abel notions of integrability.