Improper Integrals

An integral is improper if a limit is infinite, or if the integrand blows up somewhere in the interval.

Replace infinity with a limit

Integrate to a finite bound b, then take the limit as b grows without bound. The limit is the definition.

Converge or diverge

If the limit is finite the integral converges. If it is infinite or does not exist, it diverges.

A break inside

For a vertical asymptote at an endpoint, take the limit as the bound approaches it. A break inside the interval must be split at that point.

The hidden trap

Integrating straight through an interior asymptote without splitting gives a confident, wrong, finite answer.

The p-integral

The integral of 1 ÷ xᵖ from 1 to infinity converges when p > 1, mirroring the p-series exactly.

Integrating to infinity, or across a break

An improper integral has an infinite limit or an unbounded integrand. It is defined as a limit of proper integrals, and it exists only if that limit exists.

Write the limit explicitly

Replace the infinite limit with a variable, integrate, then take the limit. The exam expects the limit notation to appear; substituting infinity directly into an antiderivative is not accepted working.

Breaks inside the interval

If the integrand is unbounded at an interior point, split the integral there and take a limit from each side. Both parts must converge for the whole to converge.

The integral analogue of p-series

The integral of 1/xᵖ from 1 to infinity converges when p > 1 and diverges when p ≤ 1 — the same boundary as p-series, which is no coincidence: the integral test connects them.