I can not find any method to solve this, why do the typical methods fail? I have tried subtituion, parts, strange subitution, etc. How does one solve this integral?
$\int 5\sqrt{x^3+2x}\, \textrm{d}x$
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calculus
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0I think it may not have a closed form, which means it cannot be presented by elementary functions. I think it has something to do with elliptic integrals. – 2012-11-20
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3This is a very [ugly integral](http://www.wolframalpha.com/input/?i=integral%20of%205(x%5E3%20%2B%202x)%5E(1%2F2)%20dx&t=crmtb01) involving the inverse hyperbolic sine, and elliptic integrals of the first and second kind. – 2012-11-20
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0@Java, some of us consider elliptic integrals to be quite beautiful. – 2012-11-20
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0@Gerry: I do too, actually! Ugly was a bad choice of words here. – 2012-11-20
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0@JavaMan So how does one solve ellipitic integrals? – 2012-11-21
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0@GerryMyerson Know of a reference text on Elliptic integrals? – 2012-11-22
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0@GerryMyerson which one do you suggest? – 2012-12-10
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0I'm sorry, I'm not familiar with the available texts on elliptic integrals. It might not be a bad idea to post a new question to this site, asking for recommendations for learning about elliptic integrals. Be sure to include some information about how far you have already gone in mathematics, so that people who are familiar with the sources can suggest something at the right level for you. – 2012-12-10
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1@Java, the ugliness is only because Wolfram is rather crappy with the business of evaluating elliptic integrals. – 2013-04-03
1 Answers
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This is an elliptic integral. The elliptic integrals represent solutions for the arc length of an elliptic arc and they cannot be expressed in terms of elementary functions.
I do not know if the above statement is proved, or closed forms in terms of elementary functions exist but not yet discovered. Anyways, among the people who tried to find a solution was Euler and Legendre.
The elliptic integrals can be expressed (reduced) to Legendre canonical forms (there are three of them).
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0What are those 3 Legendre canonical forms? – 2012-12-12
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0@Mao, the canonical forms are listed [here](http://dlmf.nist.gov/19.2.ii). – 2013-04-03