Calculator
Plotly Graph
Example Data Table
| Example | Integral Family | Parameters | Expected Result | Why |
|---|---|---|---|---|
| 1 | Power tail | c = 1, s = 0, p = 2, L = 1 | Convergent | p is greater than 1 |
| 2 | Power tail | c = 1, s = 0, p = 1, L = 1 | Divergent | Logarithmic growth appears |
| 3 | Power singularity | c = 1, a = 0, p = 0.5, U = 1 | Convergent | p is less than 1 near the singularity |
| 4 | Exponential tail | c = 3, k = 0.8, L = 0 | Convergent | Positive exponential decay dominates |
| 5 | Rational balance | c = 1, m = 0, n = 3 | Convergent | m > -1 and n - m > 1 |
Formula Used
1) Power tail test: ∫[L,∞] c/(x+s)^p dx converges only when p > 1.
2) Power singularity test: ∫(a,U] c/(x-a)^p dx converges only when p < 1.
3) Exponential tail test: ∫[L,∞] ce-kx dx converges only when k > 0.
4) Logarithmic tail test: ∫[L,∞] c/(x(ln x)^p) dx converges only when p > 1 and L > 1.
5) Rational balance test: ∫[0,∞] cxm/(1+x)n dx converges when m > -1 and n - m > 1.
The calculator checks the correct rule for the chosen family, states whether the improper integral converges or diverges, and returns a finite value when a closed form exists.
How to Use This Calculator
- Select the integral family that matches your improper integral.
- Enter the relevant constants, exponents, bounds, and graph limit.
- Press Analyze Integral to get the verdict above the form.
- Read the test used, convergence reason, exact value, and sample truncated area.
- Review the graph to see how the integrand behaves over the displayed interval.
- Use the CSV and PDF buttons to save the current result summary.
FAQs
1) What does converge mean for an improper integral?
It means the limiting area approaches a finite number. Even if the interval is infinite or the function has a singularity, the total accumulated area stays bounded.
2) What does diverge mean?
It means the limit does not settle to a finite value. The accumulated area may grow forever, fail near a singularity, or oscillate without producing a stable improper integral.
3) Why does the power tail depend on p > 1?
For tails like 1/xp, the denominator must grow fast enough. When p is greater than 1, the function shrinks rapidly and the area becomes finite.
4) Why does a singular power need p < 1?
Near a singularity, the function can blow up. If p is less than 1, the blow-up is mild enough for the local area to remain finite.
5) Does multiplying by a constant change convergence?
No. A nonzero constant changes the final value or sign, but it does not change whether the improper integral converges or diverges.
6) Why is there a sample area in the result?
It gives a finite truncated comparison over a chosen window. This helps you see the growth or stabilization pattern before taking the full improper limit.
7) Can this tool test every possible integral?
No. This version covers important families used in calculus courses and comparison tests. It is designed for fast decisions on common improper-integral models.
8) Why does the logarithmic family require L > 1?
Because ln(x) must stay defined and nonzero in the tested interval. Starting above 1 avoids the singular behavior created by the logarithm near x = 1.