Convergent Divergent Integral Calculator
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Reviewed by Calculator Editorial Team
Determine whether an integral converges or diverges using our calculator. This tool helps you analyze improper integrals and understand their behavior at infinity.
What is a Convergent Divergent Integral?
An integral is said to be convergent if its value is a finite number. If the value grows without bound, the integral is divergent. This concept is crucial in calculus and mathematical analysis.
For an improper integral of the form:
∫a∞ f(x) dx
We say it converges if the limit exists and is finite:
limb→∞ ∫ab f(x) dx = L (finite)
Convergence tests help determine whether an improper integral converges or diverges. Common tests include the Comparison Test, Ratio Test, and Integral Test.
Methods to Determine Convergence
Comparison Test
The Comparison Test compares the integrand to a known integral. If the known integral converges, so does the test integral.
Example
Consider ∫1∞ (1/x²) dx. Compare to ∫1∞ (1/x) dx, which is known to diverge. Since 1/x² < 1/x for x > 1, and the integral of 1/x diverges, the integral of 1/x² must also diverge.
Ratio Test
The Ratio Test examines the limit of the absolute value of the function as x approaches infinity.
If limx→∞ |f(x)| = 0, the integral may converge.
Integral Test
The Integral Test applies to positive, decreasing functions. If the integral converges, the series does as well.
Worked Examples
Let's examine two integrals to determine their convergence.
| Integral | Test Used | Result |
|---|---|---|
| ∫1∞ (1/x²) dx | Comparison Test | Converges (value = 1) |
| ∫0∞ e-x dx | Integral Test | Converges (value = 1) |
These examples demonstrate how different tests can be applied to determine convergence.
FAQ
- What does it mean for an integral to converge?
- An integral converges when its value is a finite number. This means the area under the curve is bounded.
- How do I know which test to use?
- Consider the form of the integrand and whether it resembles known convergent or divergent integrals.
- Can all integrals be tested for convergence?
- No, only improper integrals (those with infinite limits or singularities) need convergence testing.