Improper Integrals

The central idea

An integral is improper when an interval is unbounded or the integrand is unbounded. It is defined by a limit, such as $\int_a^\infty f=\lim_{b\to\infty}\int_a^bf$. An interior singularity requires splitting and separate one-sided limits. The integral converges only if every required limit is finite.

Definitions, hypotheses, and notation

The $p$-integral tests differ at zero and infinity: $\int_1^\infty x^{-p}dx$ converges for $p>1$, while $\int_0^1x^{-p}dx$ converges for $p<1$. The same formula has opposite thresholds because different endpoints cause the singular behavior.

A Cauchy principal value may assign a symmetric cancellation to some divergent expressions, but it is not the ordinary improper integral. Introductory convergence requires each one-sided piece separately finite, preventing positive and negative infinities from being combined.

Conceptual meaning

Improper notation abbreviates a limiting process; infinity is not a bound that can be substituted into an antiderivative. Convergence means finite signed accumulation, while divergence means at least one limiting contribution fails to settle.

A dependable method and decision rule

1. Locate every infinite endpoint and singular point.
2. Split at each interior singularity before computing.
3. Replace each improper piece by its own limit with a finite variable endpoint.
4. Evaluate the proper integral first, then take the limit.
5. Declare convergence only when all pieces have finite limits.

Fully worked example

Common mistakes and why they fail