Absolutely convergent
Both the improper integral and the integral of |f(x)| converge. This is stronger than ordinary convergence.
Improper Integral Convergence Calculator
Test whether an integral with infinite bounds or isolated singularities converges. The calculator splits every required one-sided limit, looks for a verified antiderivative, checks absolute convergence, and shows finite-cutoff evidence when a symbolic conclusion is unavailable. Calculations stay in your browser.
Enter an improper integral
Advertisement
Convergence conclusion
Enter an improper integral to begin.
The conclusion will appear here.
Every improper part must have a finite limit.
Absolute convergence
The separate |f(x)| check will appear here.| Required part | Conclusion | Contribution | Method |
|---|---|---|---|
| One-sided parts will appear here. | |||
Truncated-integral convergence trace
Analyze an integral to compare successively more extreme cutoffs. A flattening trace is evidence, not a general proof.
Calculator methodology and limits
- Developed and reviewed by
- Starlight Robotics calculator team
- Last reviewed
- Primary method
- Split the interval at each improper boundary → evaluate antiderivative limits when supported → otherwise compare a sequence of truncated adaptive-Simpson estimates.
- Classification policy
- Exact limit conclusions are labeled exact. Finite cutoff behavior is labeled numerical evidence; slow or irregular cases remain inconclusive.
- Input limits
- One real variable, a 300-character expression, ordered bounds, and up to 8 isolated singular points. Trigonometric inputs use radians.
- Important limit
- Sampling cannot prove that no unlisted singularity exists. Enter every known interior singularity; do not treat a Cauchy principal value as ordinary convergence.
- Reference
- Definitions and comparison principles follow OpenStax Calculus Volume 2, §3.7.
- Privacy
- Your function, bounds, results and exported data remain on your device.
How to test improper integral convergence
- Enter the integrand and select its variable.
- Use
infor-inffor an infinite bound. For a finite discontinuity, list the point even if it is also an endpoint. - Select Analyze convergence. The calculator splits interior singularities into separate left- and right-hand limits.
- Check the exact/numerical label, every required part, and the absolute-convergence result. Download the cutoff sequence if you want to inspect it elsewhere.
How to interpret the result
Conditionally convergent
The signed integral converges, but the absolute-value integral diverges. Cancellation is essential.
Divergent
At least one required one-sided antiderivative limit is non-finite or fails to exist. One bad part makes the whole improper integral diverge.
Inconclusive
The available exact rules and finite cutoff sequence do not justify a confident classification. A comparison, limit comparison, or another analytic test may be needed.
Core definitions and p-tests
| Form | Definition or convergence rule |
|---|---|
| Infinite upper bound | ∫ₐ∞ f(x)dx = lim R→∞ ∫ₐᴿ f(x)dx |
| Singular lower endpoint | ∫ₐᵇ f(x)dx = lim ε→0⁺ ∫ₐ₊εᵇ f(x)dx |
| Tail p-integral | ∫₁∞ 1/xᵖ dx converges exactly when p > 1. |
| Endpoint p-integral | ∫₀¹ 1/xᵖ dx converges exactly when p < 1. |
Improper integral convergence FAQs
When does an improper integral converge?
It converges only when every limit introduced by an infinite bound or singular point exists as a finite real number. If the interval is split, every piece must converge.
What is absolute convergence?
An improper integral converges absolutely when the corresponding integral of |f(x)| converges. Absolute convergence guarantees ordinary convergence; the converse need not hold.
Does a Cauchy principal value prove convergence?
No. Symmetric cancellation can produce a finite principal value even when the left- and right-hand improper integrals diverge. This calculator tests the sides independently.
Why can the result be inconclusive?
Finite numerical samples cannot settle every limit, especially slowly converging tails and highly oscillatory functions. The calculator reports uncertainty instead of turning a cutoff estimate into a false proof.
Do I need to list interior singularities?
Yes. The calculator performs a limited scan, but it cannot prove that it found every discontinuity. Enter all known isolated singularities so each one-sided integral is tested.
Why is a regular finite interval rejected?
This tool is specifically for improper integrals. Use the Integral Calculator when both bounds are finite and the integrand is continuous throughout.
Does my expression leave the browser?
No. Parsing, integration, limit checks, graphing, copying, permalink construction and CSV generation happen locally.