Liouville's theorem tells us that not all indefinite integrals of elementary functions (such as $\sin(x) = \sin(x)/x$, $x^x)$ exist. However, this does not always seem to be the case when we have a definite integral.
For example, while the definite integral in terms of elementary functions for sinc(x) does not exist, the definite integral of sinc(x) from 0 through infinity is equal to Pi/2.
My question is: Suppose we have a function f(x) composed of elementary functions. When do we know the definite integral of f(x) over some interval has a closed form in terms of elementary functions?