Types of Improper Integrals

Real Analysis · Axiom Academy

LESSON Types of Improper Integrals Extending the integral to infinite intervals and unbounded integrands — every improper integral is really a limit. 1. Type 1 — An Infinite Interval A Type 1 improper integral has an endpoint at infinity. We never "reach" ; instead we integrate up to a finite cutoff b and let it run off to the right: Integrate to a finite b , then take the limit Take on . The region runs forever to the right, yet — watch the running total — its area approaches a finite limit : 2. Type 1 — When the Limit Is Infinite Same setup, gentler decay. Take on . It still goes to 0 , but not fast enough : the cutoff antiderivative has no finite limit. As the right edge runs off, the running area keeps climbing — it never levels off. The integral diverges : 3. Type 2 — An Unbounded Integrand The other way to be improper: the interval is finite, but the integrand has a vertical asymptote inside it. Here blows up at x = 0 . We pull the bad endpoint in to t and take a one-sided limit: Start just to the right of 0 , then let The region grows taller without bound as — but it also gets razor-thin, and the area still approaches a finite limit: 4. Type 2 — When the Spike Wins Push the singularity harder. For on (0, 1] the antiderivative is , which dives to as . The accumulated area has no finite limit: As the left edge approaches the asymptote, the running area climbs without bound — the integral diverges :

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