What is an improper integral?
A definite integral with an infinite limit or an integrand singularity.
Improper Integral Calculator
Take limits at infinity for exponential decay or p-integrals on [a, ∞).
Choose the integrand
Select exponential or power type and enter parameters with lower limit a.
About this calculator
How to use Improper Integral Calculator, plus the formulas and logic behind the results.
Tool description
Improper integrals with infinite upper limits are defined as limits of proper integrals as B → ∞.
How to use
- Select exponential or power integrand.
- Enter k or p and the lower limit a.
- Select Calculate to view the convergent value or an error if divergent.
- Check the convergence conditions stated in the formulas.
Formulas & reference
∫_a^∞ e^{−kx} dx = e^{−ka}/k (k > 0)∫_a^∞ x^{−p} dx = a^{1−p}/(p−1) (p > 1, a > 0)
Improper integral tips
- k ≤ 0 makes the exponential diverge
- p ≤ 1 makes the power integral diverge at infinity
- a must be positive for x^{−p}
- Compare with comparison/limit tests in theory class
Calculation logic
Applies closed-form limits for the two convergent families; rejects parameter values that diverge.
Only these standard convergent cases are supported.
Improper Integral Calculator FAQ
Common questions about this calculator and how to use the results.
When does ∫ e^{−kx} converge?
When k > 0.
When does ∫_a^∞ x^{−p} converge?
When p > 1 and a > 0.
What about ∫_1^∞ 1/x?
Diverges (p = 1 harmonic case).
Is the lower limit free?
For exponentials yes; for powers keep a > 0.
Is this free?
Yes.