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Improper Integral

Definition

An improper integral is an integral over an unbounded interval or one whose integrand becomes unbounded near an endpoint or an interior point. It is defined by limits of ordinary definite integrals: limits at infinity for infinite endpoints, and one-sided limits at finite singularities. It converges if every required limit exists as a finite real number; otherwise, it diverges.

Interpretation

Improper integrals extend definite integrals to unbounded intervals and unbounded integrands, where the Riemann sums of a definite integral are not available. The idea is to integrate over a range where everything is ordinary and then let that range grow: the improper integral is the limit, if it exists, of these ordinary integrals.

The curve 1/x squared from 1 onwards, with the area from 1 to b shaded dark and the remaining tail beyond b shaded light; together they have area 1.

∫1bdxx2=1−1b\displaystyle\int_1^b\frac{\mathrm{d}x}{x^2}=1-\frac{1}{b} is the dark area. Letting b→∞b\to\infty adds the light tail: ∫1∞dxx2=lim⁡b→∞(1−1b)=1\displaystyle\int_1^{\infty}\frac{\mathrm{d}x}{x^2}=\lim_{b\to\infty}\Bigl(1-\frac{1}{b}\Bigr)=1. An infinitely long region can therefore have a finite area, provided the integrand decays fast enough.

An infinite integration boundary is handled by a limit at infinity of truncated integrals. A singularity at a finite boundary is handled by a limit at a point from the allowed side. When there are several problem points, the integral is split so that each piece has only one, and every piece must converge separately.

Two slips come from skipping this step. Applying the antiderivative across an interior singularity gives ∫−11dxx2=[−1x]−11=−2\displaystyle\int_{-1}^{1}\frac{\mathrm{d}x}{x^2}=\Bigl[-\frac{1}{x}\Bigr]_{-1}^{1}=-2, a negative value for a positive integrand; in fact both halves diverge. And ∫−∞∞x dx\displaystyle\int_{-\infty}^{\infty}x\,\mathrm{d}x diverges even though ∫−RRx dx=0\displaystyle\int_{-R}^{R}x\,\mathrm{d}x=0 for every RR: the two tails must converge on their own, not only cancel in a symmetric limit.

The gamma function is defined by an improper integral, and improper integrals return in Probability Theory, where probabilities for quantities with an unbounded range are computed this way.

Key Results

  • Improper integrals are defined via limits of definite integrals.
  • Convergence depends on the existence of the defining limit.

Examples

  • ∫1∞1x2 dx\displaystyle\int_1^\infty \frac{1}{x^2}\,\mathrm{d}x converges.
  • ∫1∞1x dx\displaystyle\int_1^\infty \frac{1}{x}\,\mathrm{d}x diverges.

Prerequisites

Sources

Symbols: infinity, integral

Last updated: 27 October 2026 11:52