Improper Integrals: Limits, Convergence, and Principal Values

All flashcards

Improper integral

An integral defined using limits when its interval is unbounded or its integrand is unbounded or discontinuous.

First-kind improper integral

An integral made improper by at least one infinite endpoint.

Second-kind improper integral

An integral over a bounded interval with a singularity or relevant discontinuity.

Mixed improper integral

An integral with multiple sources of impropriety, such as an infinite endpoint and a singularity.

Cauchy principal value

A value defined by a prescribed limiting procedure that may exist even when the ordinary improper integral diverges.

Quiz questions

1. For $\operatorname{p.v.}\int_{-1}^{1}\frac{1}{x}\,dx=0$, what can be concluded about the ordinary improper integral?Answer: It diverges because its two one-sided integrals do not converge separately.
2. Which condition defines absolute convergence for an improper integral of a real-valued function?Answer: The improper integral of the absolute value is finite.
3. Which statement describes the improper integral $\int_0^\infty \frac{\sin x}{x}\,dx$?Answer: It converges as an improper Riemann integral but not absolutely.
4. How is ordinary convergence of $\int_{-\infty}^{\infty} f(x)\,dx$ defined?Answer: Both one-sided improper integrals, split at a finite point, must converge separately.
5. What is the value of $\int_1^\infty \frac{1}{x^2}\,dx$?Answer: $1$