TITLE: Why are some non elementary integrals defined and others are not?
QUESTION [20 upvotes]: Why are some integrals that cannot be integrated in elementary terms defined and given names, while others aren’t? Based on what criteria are they chosen? Applicability to real life? And what is the point if we cannot solve them?
For example:
$$\int\frac{\sin(x)}{x}\,dx=\text{Si}(x), \quad -\int_{-x}^\infty \frac{e^{-t}}{t}\,dt=\text{Ei}(x), \quad \int \cos\left(x^2\right)\,dx = \sqrt{\frac\pi2} \text{C}\left( \sqrt{\frac2\pi}x\right)$$
while others like 
$$\int x^x \,dx$$
are not defined.

REPLY [9 votes]: "Cannot solve them" might not be the right point of view. Example: consider 
$$\tag1
\text{erf}(x)=\frac2{\sqrt\pi}\,\int_{0}^xe^{-t^2}\,dt.
$$
This is the error function, the prototypical function that cannot be expressed in terms of elementary functions. Now, if you had a formula, it would probably involve the exponential; and any time you want to obtain numbers from the exponential, you need to use its Taylor series. So, approximations of the series 
$$\tag2
\text{erf}\,(x)=\frac2{\sqrt\pi}\sum_{n=0}^\infty\frac{(-1)^nx^{2n+1}}{n!(2n+1)}
$$
are actually the mosta direct way of calculating it (and fairly efficient for $x$ not too big). 
More importantly, just the fact that a function has a name, doesn't really make it "easier": compare $(2)$ with 
$$\tag3
\sin x=\sum_{n=0}^\infty\frac{(-1)^nx^{2n+1}}{(2n+1)!}.
$$
Do you think that having $\sin x$ in a formula is much better than having $\text{erf}\,(x)$?