Evaluating ∫₁∞ 1/x² dx

Real Analysis · Axiom Academy

Turn an infinite upper limit into a limit of definite integrals, then test for convergence. Evaluate the improper integral . Because the upper limit is infinite, decide whether it converges to a finite value — and if so, find that value. The shaded region sits under y = 1/x^2 from x = 1 stretching out toward . The region is infinitely long, yet its total area is finite — it converges to exactly 1 , because 1/x^2 decays fast enough for the tail to contribute almost nothing. Contrast: under y = 1/x the same tail does not shrink fast enough — diverges to . Whether the area is finite is entirely a question of how quickly the curve decays. Nice work — you evaluated an improper integral end to end and showed it converges. The moves that did the work: Improper integrals need a limit: when a bound is infinite, rewrite and work the ordinary definite integral inside. The limit decides convergence: as the term , so . A finite limit means the integral converges — here to 1 . Decay rate is everything: converges, but diverges ( ). The p -integral converges if and only if . That p -test is a cornerstone of Real Analysis — and it is why a region of infinite length can still enclose a finite, exact area.

This is the written version of the interactive lesson above. See the full Real Analysis course.