Improper integral (exponential tail) calculator
The Improper integral (exponential tail) calculator computes improper integral (exponential tail) from the relation integral_a^inf e^(-k x) dx = e^(-k a)/k. It takes 2 inputs (decay rate k, lower limit a) and returns the integral value. Because this is a pure mathematical or physical formula rather than a jurisdiction-specific rule, the result never changes over time: the same inputs always produce the same answer, so you can rely on it whether you are checking homework, sizing a design, or sanity-checking another tool. Enter your values in the fields below and the result updates instantly; you can also share a permalink that pre-fills the exact calculation, which is useful for teaching, reports, or collaboration. For example, with decay rate k = 2, lower limit a = 0, the integral value works out to 0.5, and the worked example further down the page shows every step so you can follow the arithmetic and reproduce it by hand. The method is the standard form documented by Documented methodology, and the figure above each result carries the date it was last verified. This tool is general information and is not a substitute for professional engineering, medical, financial, or scientific advice; always check critical results against the primary source and your own judgement.
With Decay rate k = 2, Lower limit a = 0, the result is 0.5.
Applies to: any numeric inputs. Method source: Documented methodology, checked 2026-06-23.
The formula
integral_a^inf e^(-k x) dx = e^(-k a)/k
Worked example
With Decay rate k = 2, Lower limit a = 0:
- integral_a^inf e^(-k x) dx = e^(-k a)/k
- Integral value = 0.5
This worked example is one of the automated golden-value tests this calculator must pass before it can publish.
What this assumes
- Inputs are real numbers in the units shown.
- The result is the exact value of integral_a^inf e^(-k x) dx = e^(-k a)/k; general information, not professional advice.
Frequently asked questions
What formula does this use?
integral_a^inf e^(-k x) dx = e^(-k a)/k, the standard form documented by Documented methodology.
Does the result ever change over time?
No. This is a pure formula with no external rate, so the same inputs always give the same result.
Official sources and verification
- Method: Documented methodology, checked 2026-06-23.
Reviewed by the CalculatorHub team, edited by James Graham, 2026-06-23. See our methodology. General information, not professional advice.