An improper integral with a single endpoint at infinity, like \(\int_a^\infty f(x)\,dx\) or \(\int_{-\infty}^b f(x)\,dx\), can be interpreted as a single limit of definite integrals, as long as \(f\) has no infinite singularities on the interval of integration (namely, \([a,\infty)\) or \((-\infty,b]\)). Specifically, \(\int_a^\infty f(x)\,dx\) is defined as \(\lim_{t\to\infty} \int_a^t f(x)\,dx\), while \(\int_{-\infty}^b f(x)\,dx\) is defined as \(\lim_{t\to-\infty} \int_t^b f(x)\,dx\).