Improper Integrals

Infinite intervals and unbounded integrands, handled by limits.

The idea

An improper integral extends the definite integral to two situations the original construction cannot handle. The interval may have no right-hand end, so there is no finite stretch to slice; or the integrand may grow without bound somewhere on the interval, so a slice there has no finite height.

We define both by the same limit: integrate the part that is finite, then move the edge. For an interval with no end, stop at a movable endpoint $t$, integrate over the finite piece, and push $t$ outward:

$\int_{a}^{\infty} f(x)\,dx = \lim_{t \to \infty} \int_{a}^{t} f(x)\,dx.$

For an integrand that blows up at an endpoint, trim the interval to a movable $t$ short of that endpoint and slide $t$ toward it.

If the limit is a finite number, the integral converges to that number. If the limit is infinite or does not exist, the integral diverges and has no value.

An endless or unbounded region may or may not enclose a finite area; whether the integrand shrinks fast enough is exactly what the limit decides.

Ways to work on it

• Walkthrough. Define an improper integral as a limit and test convergence.
• Practice. Evaluate a convergent integral over an infinite interval.
• Hardest. Handle an unbounded integrand and decide whether the integral converges.

Not sure where to start? Take the ten-question placement test.