Improper Integrals

Improper integrals extend the Riemann integral to unbounded intervals or unbounded functions. They are defined as limits of proper integrals. Convergence tests (comparison, limit comparison, absolute convergence) determine whether an improper integral converges or diverges. Improper integrals appear throughout analysis, probability, and physics.

Definitions

Definition6.1Improper integral (Type I: infinite interval)

If $f$ is integrable on $[a, R]$ for all $R > a$ , the improper integral is

$\int_a^\infty f(x) \, dx = \lim_{R \to \infty} \int_a^R f(x) \, dx,$

provided the limit exists.

Definition6.2Improper integral (Type II: unbounded function)

If $f$ is integrable on $[a+\epsilon, b]$ for all $\epsilon > 0$ but unbounded near $a$ , the improper integral is

$\int_a^b f(x) \, dx = \lim_{\epsilon \to 0^+} \int_{a+\epsilon}^b f(x) \, dx.$

Example∫₁^∞ 1/x² dx converges

$\int_1^\infty \frac{1}{x^2} \, dx = \lim_{R \to \infty} \int_1^R \frac{1}{x^2} \, dx = \lim_{R \to \infty} \left[-\frac{1}{x}\right]_1^R = \lim_{R \to \infty} \left(1 - \frac{1}{R}\right) = 1.$

Example∫₁^∞ 1/x dx diverges

$\int_1^\infty \frac{1}{x} \, dx = \lim_{R \to \infty} \ln(R) = \infty.$

The integral diverges.

Example∫₀¹ 1/√x dx converges

$\int_0^1 \frac{1}{\sqrt{x}} \, dx = \lim_{\epsilon \to 0^+} \int_\epsilon^1 x^{-1/2} \, dx = \lim_{\epsilon \to 0^+} [2\sqrt{x}]_\epsilon^1 = 2.$

Convergence tests

Theorem6.1Comparison test

If $0 \leq f(x) \leq g(x)$ for $x \geq a$ , then:

If $\int_a^\infty g$ converges, then $\int_a^\infty f$ converges.
If $\int_a^\infty f$ diverges, then $\int_a^\infty g$ diverges.

ExampleUsing comparison test

Does $\int_1^\infty \frac{\sin^2(x)}{x^2} \, dx$ converge? Since $0 \leq \sin^2(x) \leq 1$ , we have

$\frac{\sin^2(x)}{x^2} \leq \frac{1}{x^2}.$

Since $\int_1^\infty \frac{1}{x^2} \, dx = 1$ converges, by comparison, $\int_1^\infty \frac{\sin^2(x)}{x^2} \, dx$ converges.

Summary

Improper integrals extend integration to unbounded domains or functions:

Type I: Infinite intervals $\int_a^\infty f$ .
Type II: Unbounded functions near a point.
Convergence tests: comparison, limit comparison, absolute convergence.

See Riemann Integral for the foundation.