calculus

Improper Integral

An improper integral has either an infinite limit or an integrand that is unbounded somewhere on the interval. Evaluated as a limit of proper integrals.

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An improper integral has at least one of:

  1. Infinite limit: ∫a∞f(x) dx\int_a^\infty f(x) \, dx or ∫−∞∞f(x) dx\int_{-\infty}^\infty f(x) \, dx.
  2. Unbounded integrand somewhere in [a,b][a, b] (vertical asymptote).

Both are evaluated as limits of proper integrals:

∫a∞f(x) dx=lim⁡b→∞∫abf(x) dx\int_a^\infty f(x) \, dx = \lim_{b \to \infty} \int_a^b f(x) \, dx

If finite, converges; otherwise diverges.

Famous examples:

  • ∫1∞1x2dx=1\int_1^\infty \frac{1}{x^2} dx = 1 ✓
  • ∫1∞1xdx=∞\int_1^\infty \frac{1}{x} dx = \infty ✗ (slower decay diverges)
  • ∫−∞∞e−x2dx=π\int_{-\infty}^\infty e^{-x^2} dx = \sqrt{\pi} — Gaussian integral.

Convergence tests (comparison, p-test) decide whether to bother integrating. Improper integrals appear in probability (PDF normalisation), Fourier transforms, and physics.