Compare tricky integrands against known benchmark functions. Review direct guidance, ratio clues, and worked outputs. Download clean reports and example tables for revision anytime.
| Target integrand | Comparison function | Interval | Method | Expected result |
|---|---|---|---|---|
| ln(x) / x2 | 1 / x3/2 | [2, ∞) | Direct comparison | Convergent |
| 1 / (x + 1) | 1 / x | [1, ∞) | Limit comparison | Divergent |
| 1 / √x | 1 / x1/2 | near x = 0 | Power benchmark | Divergent |
| 1 / x3 | 1 / x2 | [1, ∞) | Direct comparison | Convergent |
| e-2x | e-x | [0, ∞) | Direct comparison | Convergent |
Direct comparison for convergence: If 0 ≤ f(x) ≤ g(x) and ∫g(x)dx converges, then ∫f(x)dx converges.
Direct comparison for divergence: If 0 ≤ g(x) ≤ f(x) and ∫g(x)dx diverges, then ∫f(x)dx diverges.
Limit comparison: If lim f(x)/g(x) = L with 0 < L < ∞, then ∫f(x)dx and ∫g(x)dx have the same behavior.
Power benchmark: On [a, ∞), ∫1/xp dx converges when p > 1. Near a finite singularity, ∫1/|x-c|p dx converges when p < 1.
Log power benchmark: On [a, ∞), ∫1/(xp(ln x)q) dx converges if p > 1, or if p = 1 and q > 1.
Exponential benchmark: On infinite intervals, ∫e-kx dx converges when k > 0.
An improper integral comparison test calculator helps you judge convergence faster. It is useful in calculus, analysis, and exam practice. Many integrals are hard to integrate directly. Comparison methods give a cleaner path. You compare a difficult integrand with a simpler benchmark function. Then you transfer convergence or divergence from the known model.
This page organizes the comparison process into clear steps. You enter the interval type, the comparison family, and the needed parameters. You can test direct comparison or limit comparison. The tool then explains the decision in plain language. It also reminds you that positivity matters. That detail is essential for valid comparison arguments.
The direct comparison test uses inequalities. If 0 ≤ f(x) ≤ g(x) and the integral of g converges, then the integral of f converges. If 0 ≤ g(x) ≤ f(x) and the integral of g diverges, then the integral of f diverges. The limit comparison test uses the ratio f(x)/g(x). When the limit is finite and positive, both integrals share the same behavior.
This calculator supports common benchmark families. These include p-test forms, shifted power forms, logarithmic corrections, and exponential decay models. Such examples appear often in homework and competitive exams. They also appear in proofs from advanced calculus courses. Choosing the right benchmark is usually the main skill.
The result box gives a compact explanation. The example table shows realistic cases. CSV and PDF exports help with revision notes. The formula section keeps the theory close to the calculator. This makes the page practical for students, tutors, and self-learners. It is designed for quick checking without visual clutter.
Another benefit is structure. Students often know the theorem but miss the setup. They may ignore the singular point, choose the wrong side of an inequality, or forget that the comparison function must already have known behavior. This calculator reduces those mistakes. It turns theory into a checklist. That supports accuracy, confidence, and faster problem solving during timed practice.
Use it to test ideas, verify benchmarks, and build stronger intuition about asymptotic growth near infinity or singular endpoints daily.
Use direct comparison when you can prove one function stays below or above a known benchmark. Use limit comparison when the ratio approaches a finite positive constant near the problematic point.
The calculator gives theorem-based guidance, not a symbolic proof. You must still justify the inequality, positivity, and asymptotic behavior in homework, quizzes, or formal solutions.
For many common positive functions, yes. You can compare near infinity, near a vertical singularity, or on an endpoint where the integral becomes improper.
A positive benchmark is standard because comparison theorems rely on nonnegative integrands. If signs change, split the problem or analyze absolute convergence first.
The power model 1/x^p converges on infinity intervals when p is greater than 1. Near a finite singularity, it converges when p is less than 1.
When the limit of f(x)/g(x) is finite and positive, both integrals have the same convergence behavior. That is the core limit comparison rule.
CSV export is useful for saving structured result fields. PDF export is useful for sharing a neat summary, formulas, and calculator notes.
Yes. The example table gives realistic comparison setups. Change the inputs to match your own integrand, chosen benchmark, and interval type.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.