Integrate this integral from 0 to x of e^(-t^2)

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    Integral Integrate
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Discussion Overview

The discussion revolves around the integration of the function e^(-t^2) from 0 to x. Participants explore the nature of this integral, its properties, and related concepts, including special functions and techniques for evaluation.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested, Mathematical reasoning

Main Points Raised

  • One participant requests assistance in integrating the function e^(-t^2) from 0 to x.
  • Another participant notes that the integral does not have a simple closed form and introduces the error function, erf(x), as a related concept.
  • A different participant suggests a method involving squaring the integral, renaming variables, and transforming to polar coordinates, indicating that this leads to a more manageable form.
  • It is mentioned that a closed form expression is only obtainable when the limits of integration extend to infinity, emphasizing the limitations of expressing the antiderivative of e^(-x^2) with elementary functions.

Areas of Agreement / Disagreement

Participants express differing views on the methods for evaluating the integral and the nature of its closed form. There is no consensus on a single approach or resolution to the problem.

Contextual Notes

The discussion highlights the dependence on definitions of integrals and the conditions under which certain techniques apply, such as the use of polar coordinates and the limits of integration.

hytuoc
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someone please show me how to integrate this
integral from 0 to x of e^(-t^2)
Thanks
 
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Your integral has no simple closed form. However, that particular integral appears often enough to warrant its own special designation - it's call the "error function:"

erf(x) = \frac {2}{\sqrt \pi} \int_0^{x} e^{-t^2} dt
 
Don't you square it. Rename a variable. Then transform to polar co-ords. Then you get left with something along the lines of...

I^2 = 2pi.int^x_0 r.e^(-r^2)dr

which is easy.

Think it's also called the guassian integral or probability integral and must be one of the most common integrals, comes up all the time in stats etc...
 
Only when the limits of integration extend to infinity can we get a closed form expression by using that polar-coordinate trick.

What Tide means is that the antiderivative of e^{-x^2} can't be expressed with elementary functions alone.
 

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