# 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:

• 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?

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.