It seems that the key term here may be the somewhat non-specific-sounding *special functions*.

By googling for a few examples (Erf, Si, Li) I came across a [**Table of Special Functions**](http://math2.org/math/integrals/specialfuns.htm) and, on the *Lists of integrals* wikipage, there is a sub-section on [**Special Functions**](https://en.wikipedia.org/wiki/Lists_of_integrals#Special_functions).

As a related remark, one reason that functions may be presented and/or defined in terms of integrals is that it may not be possible to express them using elementary functions. Actually *proving* that an antiderivative cannot be written using elementary functions is generally done using Liouville's criterion; in the spirit of the OP's mention of $\text{Si}(x)$, I include here a pointer to [**MSE 694915**](http://math.stackexchange.com/a/832601) in which I provided a reference with a proof that the antiderivative of $\sin(x)/x$ cannot be expressed using elementary functions alone.

For one other example, see Peter Mueller's proof that the antiderivative of $x \tan x$ cannot be written using elementary functions in [**MO 108598**](http://mathoverflow.net/a/108616) (for an additional reference proving this fact, see the answer that I provided at the same question). Although I am not aware of an important function defined directly in terms of an integral of $x \tan x$, one can use the relation between its antiderivative and the dilogarithm to show that the latter, too, cannot be expressed in terms of elementary functions.