Improper integrals over unbounded intervals

Let $f$ be Riemann integrable on every compact interval $[a,R]$ with $R>a$. The improper integral over the right-unbounded interval is $\int_a^\infty f(x)\,dx:=\lim_{R\to\infty}\int_a^R f(x)\,dx,$ provided this limit exists as a finite real number. In that case the improper integral converges; otherwise it diverges. In particular, a limit of $+\infty$ or $-\infty$ is divergence, not convergence to an extended-real value.

If $f$ is Riemann integrable on every $[R,b]$ with $R<b$, define $\int_{-\infty}^b f(x)\,dx:=\lim_{R\to-\infty}\int_R^b f(x)\,dx$ under the same finite-limit convention. All finite integrals use the oriented convention.