# Thread: Integral in terms of elementary functions

1. ## Integral in terms of elementary functions

A recent thread in the calculus forum reminded me of the following statement, which I've seen in a book:

If $m, n, p \in\mathbb{Q}$, then $\int x^m(a+bx^n)^p\;{dx}$ can be expressed in terms of elementary functions only when:
(a) $p$ is an integer, (b) $\frac{m+1}{n}$ is an integer, or (c) $\frac{m+1}{n}+p$ is an integer. My question: how is this proven?

2. This is hardly a proof, but I understand that if m is less than n by one it can be thought of as the du of a u substitution.
I'm interested in the other cases though...

3. My understanding is that people use Differential Galois theory to prove statements like this.

Differential Galois theory - Wikipedia, the free encyclopedia

http://nd.edu/~mkamensk/lectures/diffgalois.pdf

4. I just learnt today that this actually a theorem. It's called 'Chebyshev theorem' (so the three conditions are spoken of as 'Chebyshev conditions'). There's also something called 'Chebyshev Integral', which is a special case.

5. Originally Posted by TheCoffeeMachine
A recent thread in the calculus forum reminded me of the following statement, which I've seen in a book:

If $m, n, p \in\mathbb{Q}$, then $\int x^m(a+bx^n)^p\;{dx}$ can be expressed in terms of elementary functions only when:
(a) $p$ is an integer, (b) $\frac{m+1}{n}$ is an integer, or (c) $\frac{m+1}{n}+p$ is an integer. My question: how is this proven?
This is Chebyshev's Theorem. The red only should be replaced with "if and only if". The proof of the easy direction is straightforward, the other direction less so.