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Integral of e^(-x^2)

Why e^(-x^2) has no closed-form antiderivative, and its numeric value over any range.

Kind of integral
e−x2e^{-x^{2}}e−x2
A number, pi/2, inf or -inf.

Numeric estimate

∫01e−x2 dx\int_{0}^{1} e^{-x^{2}}\,dx∫01​e−x2dx
≈0.7468241328\approx 0.7468241328≈0.7468241328

This is a numeric estimate, computed by sampling the function, not an exact value.

  • f(x)
Table of values
Graph of e^(-x^2) with the area shaded: sampled values across the visible range
xf(x)
-0.250.93941
-0.10.99005
0.050.9975
0.20.96079
0.350.88471
0.50.7788
0.650.65541
0.80.52729
0.950.40555
1.10.2982
1.250.20961

Step-by-step working

  1. 1Numeric integration
    ∫01e−x2 dx\int_{0}^{1} e^{-x^{2}}\,dx∫01​e−x2dx
    ↓
    ≈0.7468241328\approx 0.7468241328≈0.7468241328

    No elementary antiderivative was found, so the value is estimated with tanh-sinh (double-exponential) quadrature, which copes with singular ends. Infinite bounds use exp-sinh.

Worked solution: integral of e^(-x^2)

Worked steps

  1. 1Numeric integration
    ∫01e−x2 dx\int_{0}^{1} e^{-x^{2}}\,dx∫01​e−x2dx
    ↓
    ≈0.7468241328\approx 0.7468241328≈0.7468241328

    No elementary antiderivative was found, so the value is estimated with tanh-sinh (double-exponential) quadrature, which copes with singular ends. Infinite bounds use exp-sinh.

Answer

∫01e−x2 dx≈0.7468241328\int_{0}^{1} e^{-x^{2}}\,dx \approx 0.7468241328∫01​e−x2dx≈0.7468241328
  1. Tools
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  4. Integral of e^(-x^2)

About Integral of e^(-x^2)

The integral of e^(-x^2) cannot be written with elementary functions. There is no antiderivative of e^(-x^2) made from elementary functions. The definite integral can still be computed numerically, and the whole real line gives exactly the square root of pi. It is free, needs no sign-up and runs in your browser, so nothing you enter is uploaded.

The calculator below opens with the definite integral from 0 to 1 and labels the result as a numeric estimate, not an exact value. Change the bounds to inf and -inf to get the whole-line value, which the calculator estimates as 1.7725, the square root of pi.

When the antiderivative does not exist in closed form, a definite integral is the useful question. The estimate is found by sampling the function over the interval, so quote it to a handful of digits.

How to use Integral of e^(-x^2)

  1. 1Type the function in the Function box, for example x*e^x, or use the key pad. The preview shows how it was read.
  2. 2Choose Indefinite for an antiderivative, or Definite to enter a lower and an upper bound.
  3. 3Write bounds as numbers or constant expressions such as pi/2. Type inf or -inf for an improper integral.
  4. 4Read the result card. A definite integral shows the exact value when one exists and a labelled numeric estimate otherwise.
  5. 5Check the graph: the function is drawn with its area shaded between the bounds, or with its antiderivative for an indefinite integral.
  6. 6Open the steps to see which method was used, and copy the answer with the copy button.

Features

  • Indefinite integrals with the constant of integration.
  • Definite integrals with exact values, including bounds written with pi and with inf or -inf.
  • Step-by-step working that names the rule: power rule, substitution, integration by parts, partial fractions and trigonometric identities.
  • A shaded area under the curve for definite integrals, on a graph you can zoom and drag.
  • A clear label when the result is a numeric estimate instead of an exact value.
  • An honest "no elementary rule found" message instead of a guess.
  • A live preview and a math key pad, so input is easy on a phone.
  • Checks improper integrals and says when one does not settle on a finite value.

What is known exactly

There is no antiderivative of e^(-x^2) made from elementary functions. The definite integral can still be computed numerically, and the whole real line gives exactly the square root of pi.

integral of e^(-x^2) from 0 to 1 = 0.74682413 (numeric)
integral of e^(-x^2) from -inf to inf = sqrt(pi)

Worked examples

  • Integral of e^(-x^2) from 0 to 1 → gives about 0.74682413

    A numeric estimate, not an exact value.

  • Integral of e^(-x^2) from -inf to inf → gives about 1.7725

    The exact value is the square root of pi.

Privacy: how it works

  • The integral is worked out by code running in your browser tab. Your functions are not sent to Ulkaa or to any other server, and there is nothing to sign in to.

Tips

  • Differentiate your answer with the derivative calculator to check it. You should get the original function back.
  • For an area between a curve and the x axis, remember that parts below the axis count as negative in a definite integral.
  • An improper integral with a bound at infinity only has a value if the function dies away fast enough. 1/x^2 does, 1/x does not.
  • Use * between factors when you can, for example x*e^x, and write powers of e as e^x.

Frequently asked questions

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Does it run in my browser?

Yes. Your functions are processed by code running in your browser tab, on your own device.

Is my data uploaded?

No. Your functions are never sent to Ulkaa or to anyone else. You can check this yourself by watching the Network tab in your browser developer tools while you use the tool.

What is the integral of e^(-x^2)?

There is no elementary antiderivative. The definite integral from 0 to 1 is about 0.74682413, shown as a numeric estimate.

Why can this integral only be estimated?

It has been proved that e^(-x^2) has no antiderivative expressible with elementary functions. The error function erf(x) is defined as this integral, so the numeric value is shown as an estimate. From minus infinity to infinity the exact value is the square root of pi, about 1.7725.

What does the calculator say for an integral it cannot do?

For an indefinite integral it says that no elementary rule was found instead of guessing. For a definite integral it gives a numeric estimate and labels it as one.

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  • Integral of sec(x)Step-by-step integral of sec(x), with the method named and a graph of the function and its antiderivative.
  • Integral of x*sin(x)Step-by-step integral of x*sin(x), with the method named and a graph of the function and its antiderivative.
  • Integral of 1/(1+x^2)Step-by-step integral of 1/(1+x^2), with the method named and a graph of the function and its antiderivative.
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  • Integral of 1/(x^2-1)Step-by-step integral of 1/(x^2-1), with the method named and a graph of the function and its antiderivative.
  • Integral of x/(x^2+1)Step-by-step integral of x/(x^2+1), with the method named and a graph of the function and its antiderivative.

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