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Calculus I / Techniques of Integration

Improper Integrals

The definition of the definite integral ∫abf(x) dx\int_a^b f(x)\,dx assumes a bounded interval and a bounded integrand. Both assumptions fail in practice: the perpetuity formula requires integrating over [0,∞)[0, \infty); the log-normal density has an integrand that diverges at 00; the Black-Scholes derivation integrates the Gaussian density over (−∞,∞)(-\infty, \infty). An improper integral extends the definite integral to these settings via a limit: integrate over a bounded sub-domain, then let the boundary approach the problematic limit. When the limit exists, the integral converges; when it does not, it diverges. The pp-test provides a clean criterion for power-law integrands and is the comparison benchmark for harder integrands.
Learning objectives. By the end of this note you should be able to:
  • evaluate Type I improper integrals (infinite limits) as limits of definite integrals;
  • evaluate Type II improper integrals (vertical asymptotes) and identify whether the blow-up is integrable;
  • apply the comparison test to determine convergence or divergence without explicit evaluation;
  • apply the pp-test for ∫1∞x−p dx\int_1^\infty x^{-p}\,dx and ∫01x−p dx\int_0^1 x^{-p}\,dx.

In this lesson

  1. Type I: Infinite limits of integration
  2. Type II: Vertical asymptotes
  3. The comparison test

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