subharpe said:
Can someone tell me which function is integrable (analytically, not numerically) and which function is not and how to trace from the expression of the function?
The answers above seem to answer the question: when is a function integrable?
One answer is e.g. that all continuos functions are Riemann integrable on a closed interval. If we use another definition of integral e.g. Lebesgue integral then we will get a another set of integrable functions.
I think your actually asking the question: how can I tell if a function has an
integral function consisting of elementary functions (which is a different question)? An elementary functions is a functions that is an alebraic expression of polynomials, trig functions, log and exponential function. E.g. 1/tan(x) or log(x³-1)/cos(x).
One thing to note about the definition of elementary function is that its a bit arbitrary. There is really no reason why we should call this set 'the elementary functions'.
An example of an integral that does not have an anti-derivative made up of elementary functions is e^(-t²) (this can proved but you have to read a bit of math to understand it). So we can define a new function
f(x) = integral from 0 to x of e^(-t²)
which is not an elementary function. But there is really no reason why this function should be regarded as non elementary, other than that it might be a bit more unfamiliar than sin(x), cos(x) ,tan(x) and so on.
Anyway it's not an easy matter telling if a function has an analytical expression made up of elementary functions.