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?
$\int 5\sqrt{x^3+2x}\, \textrm{d}x$
7
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calculus
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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
3
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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0@Mao, the canonical forms are listed [here](http://dlmf.nist.gov/19.2.ii). – 2013-04-03