Concept / Calculus
Improper integrals are limits, not substitutions.
When an integrand is unbounded at an endpoint or the interval is infinite, the integral is defined by a limiting process. The notation is compact; the limit is the actual claim.
Endpoint definition
If f is integrable on every interval [ε, b] with ε greater than a, define the improper integral at the left endpoint by:
The improper integral converges only when that limit exists as a finite number. Writing the integral symbol does not guarantee convergence, and substituting the singular endpoint into an antiderivative is not valid.
The endpoint power test
| Exponent | Function at zero | Improper area |
|---|---|---|
| 0 < p < 1 | Unbounded | Finite: 1/(1-p) |
| p = 1 | Unbounded | Diverges logarithmically |
| p > 1 | Unbounded | Diverges as a power law |
Comparison is often more useful than exact integration
Near a singular endpoint, the leading-order behavior can determine convergence. If two positive functions are comparable near the endpoint, a known power-law model can often establish whether the unknown integral converges. The comparison must apply in a sufficiently small one-sided neighborhood and must preserve the inequality direction needed by the theorem.
Use the same model interactively
The Boundary Observatory computes the truncated integral from ε to 1. Lower ε to inspect how the partial area stabilizes for p below 1 and continues growing for p at or above 1.