Improper Integrals over Infinite Intervals

An integral such as $\int_a^\infty f(x)\,dx$ is defined by replacing infinity with a finite boundary $b$, evaluating $\int_a^b f(x)\,dx$, and then taking the limit as $b\to\infty$. The integral converges only if that limit exists and is finite. We never substitute infinity into an antiderivative as though it were an oversized real number.

Convergence depends on the long-run balance between interval length and function decay. A positive function may approach zero and still have an infinite total area if it decays too slowly. The classic $p$-integrals show the threshold clearly. This distinction becomes important in probability, physics, and series theory, where an infinite domain can still carry a finite total mass or energy.