Improper Integral
An improper integral is a definite integral that has either an infinite limit of integration or an integrand that becomes unbounded (has a vertical asymptote) within the interval of integration. These integrals are evaluated using limits, replacing the problematic bound with a parameter and taking the limit as the parameter approaches the infinity or the singularity. Improper integrals arise frequently in probability theory, Fourier analysis, and physics when computing total accumulated quantities over unbounded domains.
Key Formula
integral from a to infinity of f(x) dx = lim(b→∞) integral from a to b of f(x) dx
LaTeX: \int_a^{\infty} f(x)\,dx = \lim_{b \to \infty} \int_a^b f(x)\,dx
| Symbol | Meaning | Unit |
|---|---|---|
| a | Lower limit of integration | dimensionless |
| b | Upper limit parameter approaching infinity | dimensionless |
| f(x) | Integrand function | depends on context |
Worked Example
Problem
Evaluate the improper integral: ∫ from 1 to ∞ of (1/x²) dx.
Solution
Step 1: Replace ∞ with parameter b: lim(b→∞) ∫₁ᵇ x⁻² dx. Step 2: Integrate: lim(b→∞) [−x⁻¹]₁ᵇ = lim(b→∞) (−1/b + 1/1). Step 3: Take the limit as b → ∞: lim(b→∞) (1 − 1/b) = 1 − 0 = 1.
Answer
The improper integral converges to 1.
Types of Improper Integrals and Their Convergence
| Integral | Type | Convergence | Value / Condition |
|---|---|---|---|
| ∫₁^∞ 1/xᵖ dx | Infinite upper limit | Converges if p > 1 | 1/(p−1) |
| ∫₀^∞ e^(−x) dx | Infinite upper limit | Always converges | 1 |
| ∫₀¹ 1/√x dx | Integrand singularity at 0 | Converges | 2 |
| ∫₀¹ 1/x dx | Integrand singularity at 0 | Diverges | ∞ (ln x → −∞) |
| ∫₋∞^∞ e^(−x²) dx | Both limits infinite | Converges | √π |
Interactive Tools
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Related Terms
From Latin "improprius" (not proper or suitable). The term distinguishes these integrals from "proper" Riemann integrals defined on closed, bounded intervals. The concept was formalized in the 18th–19th centuries as mathematicians (including Cauchy) extended integration theory to unbounded domains.