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mikfig

Indefinite integral I can't figure out

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mikfig    114
Does anyone know how to find the antiderivative of this function: e^(2x-x^2) I can see how you can split up the e like this: e^(2x)*e^(-x^2) But since the derivative of e^x is just itself, I don't think it can be split up w/ integration by parts. I also don't see how it can be done with u-substitution as it really doesn't simplify it much. Unless there is one value for u that I just didn't try yet. When I tried it in mathematica it gave me this: 1/2 * e * Pi^(1/2) * Erf[-1 + x] It says "Erf[z] is the integral of the Gaussian distribution" So can anyone show me how this function could be integrated, or if it is even possible to integrate it using stuff from Calculus I/II. Thanks, mikfig

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SPuntte    158
There's no way to integrate that analytically. You have to either do it numerically or approximate the gaussian density function with some simpler function or series like the Taylor expansion. You can read more about the "error function" here.

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mikfig    114
Ok thanks, because I got this as part of a problem on a past AP Calculus exam that my calc teacher wanted us to try out. I was just sitting there trying everything to integrate this function forever :P

Thanks a lot :D,
mikfig

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Jesper T    322
You'll get the erf function as mathematica says.

What they (probably) want you to show is that you can factorize the expression so that you get from

exp(2x - x^2)

to

exp(1)*exp(-(1 - x)^2)

And do a substitution, for example

1 - x = t

And integrate

exp(-t^2)

Where you'll get the erf(t) function (times the constant and 0.5*sqrt(pi))

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mikfig    114
Ok thanks for all the replies.

Quote:
Original post by Jesper T
You'll get the erf function as mathematica says.

What they (probably) want you to show is that you can factorize the expression so that you get from

exp(2x - x^2)

to

exp(1)*exp(-(1 - x)^2)

And do a substitution, for example

1 - x = t

And integrate

exp(-t^2)

Where you'll get the erf(t) function (times the constant and 0.5*sqrt(pi))


Do you think that the AP Calculus BC exam would expect you to know the erf function? What is it's purpose?

Thanks again :D,
mikfig

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Jesper T    322
I'm not familiar with the US education system, but I guess the first time a student will need the erf function is while working with probability and normal distributions.

In Norway thats high school level. However they don't call it the erf function at that stage. Instead students are taught to use a table or numeric integration on the calculator to evaluate integrals.

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