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 and, on the Lists of integrals wikipage, there is a sub-section on 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 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 (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.

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 and, on the Lists of integrals wikipage, there is a sub-section on 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 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 (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.

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 and, on the Lists of integrals wikipage, there is a sub-section on 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 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 (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.

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 and, on the Lists of integrals wikipage, there is a sub-section on 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 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 (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.