Improper integral of exponential calculator
The area under 1·e^(−2x) from 0 to infinity is exactly 0.5, because the integral of a·e^(−kx) over that range equals a/k whenever k is positive. The antiderivative and limits are shown too. Free in your browser.
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Worked example
- A
- 1
- K
- 2
- Value
- 0.5
- Integrand
- 1*e^(-2x)
- Limits
- [0, +∞)
- Antiderivative
- -(a/k)*e^(-kx)
Improper integrals of decaying exponentials from zero to infinity are a standard first-year calculus result and a workhorse model in probability, physics, and engineering. The integral of a times e to the minus k x over the half-line converges exactly when the decay rate k is positive, and the value is the simple ratio a over k obtained from the fundamental theorem and a one-sided limit. Students checking homework, tutors building answer keys, and notebooks that need a deterministic Laplace-style mass without a computer-algebra system all benefit from the same closed form.
What the improper integral ∫₀^∞ a e^{-kx} dx means
For the family of exponential integrands a times e to the power of negative k x, the antiderivative is elementary. Differentiating or integrating the exponential shows that F of x equals negative a over k times e to the power of negative k x, provided k is not zero. Evaluating F at b and at zero gives F of b minus F of zero equals negative a over k times e to the minus k b, plus a over k.
When k is strictly positive, the exponential e to the minus k b vanishes as b goes to infinity, so the improper integral equals a over k. When k is zero the integrand is the constant a and the integral of a constant over an infinite ray diverges unless a is also zero in a trivial sense that is not useful here. When k is negative the exponential grows without bound and the integral diverges even more clearly.
This capability therefore accepts only k greater than zero, returns converges true with value a over k, and treats non-positive decay rates as invalid input rather than returning infinity or a pseudo-numeric sentinel. Amplitude a may be any finite real number, including zero and negatives, because the same closed form holds by linearity of the integral.
How the exact value is worked out
On success the body echoes a and k, provides an integrand display string of the form a times e to the minus k x, states the limits as the half-open ray from zero to positive infinity, returns value equal to a divided by k rounded to twelve decimal places for stable goldens across runtimes, includes the antiderivative string negative a over k times e to the minus k x, restates the formula integral from zero to infinity of a e to the minus k x equals a over k when k is positive, and sets converges to true.
Example: a equals one and k equals two yields integrand one times e to the minus two x and value one half. Example: a equals three and k equals one half yields value six. No numerical quadrature is performed; the answer is purely algebraic from the fundamental theorem and the vanishing limit of the exponential tail.
Where this integral shows up in practice
Classrooms and tutoring platforms use an improper integral of exponential calculator so learners can verify the classic textbook result against hand work with limits and antiderivatives without flipping a sign on the factor of one over k. Exam-prep generators need a deterministic oracle that marks divergent cases as invalid rather than silently returning zero or a fabricated finite number. Failed validation does not consume balance.
Keep requests small, pin the rounded value in tests, and you get reproducible improper-integral answers for homework, LMS plugins, and edge workers without standing up symbolic math infrastructure of your own.
What you can do with it
Calculus homework and limit checks
Students enter amplitude a and positive decay rate k and verify that ∫₀^∞ a e^{-kx} dx equals a/k against hand evaluation of lim_{b→∞}[F(b)−F(0)].
Exam banks and LMS auto-grading
Mark improper-integral items with a deterministic oracle that rejects non-positive k as divergent and never invents a finite value for growing exponentials.
Probability and decay notebooks
Confirm exponential density normalization and simple Laplace-style masses in scripts without importing a computer-algebra library.
FAQ
How do I evaluate ∫₀^∞ a e^{-kx} dx?
Send finite numbers a and k with k>0.
What happens if k is zero or negative?
For k≤0 the improper integral of a e^{-kx} from 0 to ∞ diverges (or is not a convergent decaying case), so it is not billed as a successful evaluation.
Can amplitude a be negative or zero?
Yes. Any finite a is accepted; the integral is linear in a, so the value is still a/k when k>0. Zero amplitude yields value zero.
Does the calculator use numerical quadrature?
No. It uses only the closed-form result a/k for convergent decaying exponentials. There is no sampling, random seed, or network call.
What is the integral of e^(−2x) from 0 to infinity?
It is 0.5. With amplitude 1 and decay rate 2, the closed form a/k gives 1 divided by 2. The antiderivative −(a/k)·e^(−kx) tends to zero at infinity, so only its value at 0 remains.
Related tools
For developers — API access
Everything on this page is available programmatically. This section is for teams who want to wire it into their own systems; everyone else can just use the tool above.
API endpoint
Prefer to automate it? One authenticated POST creates the task; the result comes back by webhook or a signed link. The same capability also runs here on the web, by email and from Telegram — and soon from our app too.
Call it from your stack
curl -X POST https://api.kit.forhosting.com/calculus/improper-integral-exponential \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"a":1,"k":2}'const res = await fetch("https://api.kit.forhosting.com/calculus/improper-integral-exponential", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"a": 1,
"k": 2
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/calculus/improper-integral-exponential",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"a": 1,
"k": 2
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/calculus/improper-integral-exponential", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"a":1,"k":2}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"a":1,"k":2}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/calculus/improper-integral-exponential", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"a": 1,
"k": 2
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "calculus.improper_integral_exponential",
"status": "queued",
"_links": {
"result": "/tasks/tsk_…/result"
}
}The API is asynchronous: the call returns a task_id immediately and the result arrives by webhook. Polling is capped at 1 req/s per task.
Pricing
Published price — no tokens, no invented credits. A failed task is never charged.
Errors
| HTTP | Code | Meaning |
|---|---|---|
401 | unauthorized | Missing or invalid API key. |
402 | insufficient_balance | Your balance doesn't cover the task price. |
404 | unknown_type | That task type doesn't exist. |
429 | rate_limited | Too many requests. Use the webhook instead of polling. |
Technical notes
This improper integral of exponential calculator accepts finite amplitude a and positive decay rate k, returns the numeric value together with display strings for the integrand, limits, antiderivative, and formula, and fails with invalid_input when k is not strictly positive so divergent cases are never billed as success.
What the improper integral ∫₀^∞ a e^{-kx} dx means
An ordinary definite integral over a finite interval is the signed area under a continuous curve between two finite endpoints. Extending the upper endpoint to positive infinity produces an improper integral of the first kind: by definition it is the limit as b tends to infinity of the ordinary integral from zero to b.
Closed-form evaluation, inputs, and response shape
POST a JSON body with two required finite numbers: a, the amplitude of the integrand, and k, the strictly positive decay rate in the exponent. Each field may arrive as a JSON number or as a numeric string; missing fields, empty strings, and non-finite values fail with invalid_input naming the field. Pair the result with your own plotter if you need a sketch of the decaying curve and the accumulated area; this endpoint evaluates the closed form only.
Where an improper exponential integral API earns its keep
Probability courses treat the exponential density as k e to the minus k x on the positive reals, whose total mass is exactly the special case a equals k of this integral; calling a tiny parse API confirms normalization without importing a full CAS.
Technical questions
What does the improper integral of exponential calculator cost?
The browser tool is free. API requests use prepaid KIT balance at $0.002 per successful call; invalid input is not billed.