I am writing some Calculus content, and I would like a "big list" of useful functions which are defined by definite integrals, but are not elementary functions.

Two examples of such functions are

$$ \mathrm{Erf}(x) = \frac{2}{\sqrt{\pi}}\int_0^x e^{-t^2}\mathrm{d} t $$

which is fundamentally important to statistics, and

$$ \mathrm{Si}(x) = \int_0^x \frac{\sin(t)}{t} \mathrm{d} t $$

which comes up all the time in signal processing.

I would like to be able to sketch such functions, express some definite integrals (like $\int_0^1 e^{-4t^2} dt$) in terms of such functions, etc.

So what other functions are important enough to have their own name, and are given as integrals of elementary functions?