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?

Have you ever heard of Galois theory? It is a theory that studies the solutions of equations over fields.
As it turns out, there is a special type of Galois theory called Differential Galois theory, which studies fields with a differential operator on them:
http://en.wikipedia.org/wiki/Differential_galois_theory
Using this theory, one can prove that functions like $\frac{\sin(x)}{x}$ and $x^x$ don't have an indefinite integral.
