Calculator form
Example data table
| Example integral | Family | Key parameter rule | Conclusion |
|---|---|---|---|
| ∫1∞ 1/x² dx | p-Integral tail | p = 2 > 1 | Convergent |
| ∫1∞ 1/x dx | p-Integral tail | p = 1 | Divergent |
| ∫01 1/√x dx | Endpoint near 0 | p = 0.5 < 1 | Convergent |
| ∫01 1/x² dx | Endpoint near 0 | p = 2 ≥ 1 | Divergent |
| ∫e∞ 1/(x(ln x)²) dx | Log tail | p = 2 > 1 | Convergent |
| ∫0∞ e-2x dx | Exponential tail | k = 2 > 0 | Convergent |
| ∫1∞ x²/x⁵ dx | Rational tail | n − m = 3 > 1 | Convergent |
| ∫02 1/(x-0)1.2 dx | Shifted endpoint | p = 1.2 ≥ 1 | Divergent |
Formula used
1) p-Integral at infinity
∫a∞ 1/xp dx converges exactly when p > 1. The case p = 1 is the harmonic boundary and diverges.
2) p-Integral near zero
∫0b 1/xp dx converges exactly when p < 1. The threshold p = 1 creates a logarithmic singularity.
3) Log-modified tail
∫a∞ 1 / (x(ln x)p) dx converges exactly when p > 1, provided a > 1.
4) Exponential tail
∫a∞ e-kx dx converges when k > 0. If k ≤ 0, the tail does not decay enough.
5) Rational tail rule
For xm/xn = xm-n, the tail converges when n − m > 1. This is the p-test after simplification.
6) Shifted endpoint rule
If the integrand looks like 1/(x-a)p or 1/(b-x)p, shift the endpoint and apply the near-zero rule: convergence occurs when p < 1.
How to use this calculator
- Select the integral family that matches the structure of your improper integral.
- Enter the needed parameters, such as bounds, exponent values, and decay rate.
- Set epsilon for endpoint cutoffs and horizon for finite tail previews.
- Press Evaluate convergence to display the result above the form.
- Read the status, exact value, classification, finite-window estimate, and reason.
- Inspect the Plotly graph to understand tail decay or endpoint growth behavior.
- Use the CSV and PDF buttons to export the current result summary.
- Compare your setup with the example table when checking classroom or homework problems.
FAQs
1) What makes an integral improper?
An integral is improper when it has an infinite interval, an unbounded integrand, or both. You evaluate it through limits rather than ordinary finite-area formulas.
2) Why is p = 1 a special boundary?
The value p = 1 produces logarithmic growth. That boundary separates finite area from infinite area in the classic power-test models near zero and at infinity.
3) Does a finite truncated estimate prove convergence?
No. A finite-window estimate only shows area over a chosen cutoff. True convergence depends on the limiting behavior as the interval extends or approaches the singular point.
4) When should I use the rational tail option?
Use it when the dominant behavior of your integrand at large x is a ratio of powers. The calculator simplifies the degrees and applies the power-test condition.
5) Why does the graph cap very large values?
Near singularities, function values can explode visually and flatten the rest of the plot. Capping keeps the chart readable while still showing where the blow-up happens.
6) Can this tool solve every symbolic improper integral?
No. It is designed for major comparison-test families and teaching examples. It evaluates standard convergence patterns rather than performing general symbolic integration.
7) What does “classification” mean in the result table?
Classification explains the convergence style, such as harmonic boundary, exponential decay, or strong endpoint singularity. It helps you interpret why the final verdict occurs.
8) How do I choose epsilon?
Pick a small positive cutoff that avoids division by zero, such as 0.001 or smaller. It affects only the preview and truncated estimate, not the formal convergence rule.