How can you prove that a function has no closed form integral?

In the past, I’ve come across statements along the lines of “function $f(x)$ has no closed form integral”, which I assume means that there is no combination of the operations:

  • addition/subtraction
  • multiplication/division
  • raising to powers and roots
  • trigonometric functions
  • exponential functions
  • logarithmic functions

which when differentiated gives the function $f(x)$. I’ve heard this said about the function $f(x) = x^x$, for example.

What sort of techniques are used to prove statements like this? What is this branch of mathematics called?


Merged with “How to prove that some functions don’t have a primitive” by Ismael:

Sometimes we are told that some functions like $\dfrac{\sin(x)}{x}$ don’t have an indefinite integral, or that it can’t be expressed in term of other simple functions.

I wonder how we can prove that kind of assertion?

Answer

It is a theorem of Liouville, reproven later with purely algebraic methods, that for rational functions $f$ and $g$, $g$ non-constant, the antiderivative of

$$f(x)\exp(g(x)) \, \mathrm dx$$

can be expressed in terms of elementary functions if and only if there exists some rational function $h$ such that it is a solution of

$$f = h’ + hg’$$

$e^{x^2}$ is another classic example of such a function with no elementary antiderivative.

I don’t know how much math you’ve had, but some of this paper might be comprehensible in its broad strokes:
https://ksda.ccny.cuny.edu/PostedPapers/liouv06.pdf

Liouville’s original paper:

Liouville, J. Suite du Mémoire sur la classification des Transcendantes, et sur l’impossibilité d’exprimer les racines de certaines équations en fonction finie explicite des coefficients.” J. Math. Pure Appl. 3, 523-546, 1838.

Michael Spivak’s book on Calculus also has a section with a discussion of this.

Attribution
Source : Link , Question Author : Simon Nickerson , Answer Author : Charles Brillon

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