Improper integrals
An improper integral extends the definite integral to an infinite limit of integration or an integrand that is unbounded within the interval; it has a value only when the corresponding limit of ordinary integrals actually converges, not merely because the integrand shrinks.
Practise this conceptAn improper integral shows up whenever a definite integral's interval or integrand refuses to sit still: an infinite limit of integration, or an integrand that blows up somewhere inside the interval. Neither case can be handed straight to the fundamental theorem of calculus, because there is no ordinary area to compute over an infinite strip or through a vertical asymptote.
The fix is to fence off the trouble spot with a variable, integrate normally up to that fence, and then take a limit as the fence moves toward infinity or toward the singular point. If that limit exists and is finite, the improper integral converges to it; if the limit is infinite or does not exist, the integral diverges — and a divergent integral has no value at all, not a very large one.
The classic test case is ∫₁^∞ x⁻ᵖ dx: it converges exactly when p > 1, and diverges — including at the boundary p = 1 — otherwise. This single family is worth memorising, because comparing an unfamiliar integrand against x⁻ᵖ is usually the fastest way to decide convergence without computing anything.
Key ideas
Requires: The definite integral
Formulas
| Symbol | Name | Unit |
|---|---|---|
| exponent | 1 (dimensionless) | |
| improper integral | 1 (dimensionless) |
Common misconceptions
- Believes an integral with an infinite limit is automatically infinite, without checking whether the limit of the partial integrals actually converges.
- Believes that if the integrand tends to zero as x approaches infinity, the integral must converge — the mistake behind thinking ∫₁^∞ 1/x dx converges just because 1/x → 0.
- Treats an integral with an unbounded integrand near an endpoint (e.g. 1/√x near x = 0) exactly like an ordinary definite integral, without taking a limit as that endpoint is approached.
Resources
Part of these UT courses
This concept is mapped as covering part of the following course(s) in the University of Tartu curriculum.
- Higher mathematics IMTMM.00.340