Improper Integral Calculator

Evaluate improper integrals with infinite bounds or discontinuous integrands. Checks convergence using the p-test, comparison test, and asymptotic analysis.

Improper integral value (numerical approx.)
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Converges? —
Method used —
Extended More scenarios, charts & detailed breakdown ▾
∫₁^∞ f(x)dx (numerical approx., upper→1e6)
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Converges? —
Professional Full parameters & maximum detail ▾

Convergence Analysis

Integral value (numerical approx.) —
p-test result for ∫1/x^p from 1 to ∞ —

Advanced

Asymptotic behavior at upper bound —
Cauchy principal value note —

How to Use This Calculator

  1. Enter f(x) (e.g. 1/x^2).
  2. For infinite upper bound enter 100000000 (10⁸).
  3. The calculator evaluates the integral and checks if the result is finite.
  4. Use the p-Test tab for quick convergence analysis of ∫ 1/xᵖ.
  5. The Professional tab adds asymptotic behavior analysis and Cauchy PV note.

Formula

Type 1: ∫₁^∞ f(x)dx = lim(b→∞) ∫₁ᵇ f(x)dx  |  p-test: ∫₁^∞ 1/xᵖ dx = 1/(p−1) if p > 1

Example

∫₁^∞ 1/x² dx: p=2 > 1 → converges. Exact = 1/(2−1) = 1. Numerical ≈ 0.99999.

Frequently Asked Questions

  • An improper integral has either infinite bounds (Type 1: ∫₁^∞) or a discontinuous integrand (Type 2: ∫₀¹ 1/√x dx). Both are evaluated as limits of proper integrals.
  • For ∫₁^∞ 1/xᵖ dx: converges to 1/(p−1) when p > 1, diverges when p ≤ 1. For ∫₀¹ 1/xᵖ dx: converges to 1/(1−p) when p < 1, diverges when p ≥ 1.
  • Enter a large finite number (e.g. 100000000 for +∞). The calculator integrates from a to 10⁶ using Simpson's rule and checks whether the result is finite.
  • For integrals with symmetric singularities (like ∫₋₁¹ 1/x dx), the Cauchy principal value is the limit of ∫₋ε^ε as ε→0, which can be finite even when the ordinary integral diverges.
  • If 0 ≤ f(x) ≤ g(x) and ∫g converges, then ∫f converges. If f(x) ≥ g(x) and ∫g diverges, then ∫f diverges.

Related Calculators

∫ Definite Integral Calculator ∫ Integral Calculator → Limit Calculator
Sources & References (5) ▾
  1. Improper Integrals — Paul's Online Math Notes — Lamar University
  2. MIT OCW 18.01 Improper Integrals — MIT
  3. Improper Integrals — Khan Academy — Khan Academy
  4. Integral — Wolfram MathWorld — Wolfram MathWorld
  5. Stewart's Calculus — Improper Integrals (reference) — Cengage / Stewart