Why most of our math is built around the multiplication, addition, logarithm, exponential, trigonometric etc. functions? What makes those so special, other than being convenient for human usage?
I find it particularly interesting that so many integrals can't be expressed in terms of those functions - which makes it feel like calculus is, somehow, broken. Is there an alternative math with a different set of elementary functions where this "integral problem" doesn't happen?
The exponential (and its scalar multiples) has the property that it is its own derivative; if that doesn't make it a natural function for calculus... The sine and cosine functions are its real and imaginary parts, and also functions that equal to minus their second derivatives and are basically derivatives of each other. The logarithm is the inverse function of the exponential.
Addition and multiplication are the two natural operations one does with numbers, and differentiation/integration behave in a natural way with respect to them.
You seem to expect it as natural that many integrals should be expressible in terms of a few functions. Why would it be so?
What Martin said is all good. The exponentials (and logs) are deeply embedded in calculus.
I suspect the expectation of elementary function solutions to integrals comes from the fact that it seems easy to do derivatives and get elementary functions. The easy part is largely because your textbook or prof pick problems which are suitable for students; they are not all so easy.
Regardless, remember that inverse functions are never as easy to calculate as the original function (the integral being the inverse of derivative). arcsin is harder calculate than sin; logx is harder to calculate than e$^x$; even division is harder than multiplication and was an intractable problem until the use of the place decimal system. Just try dividing in Roman numerals, you will see. People had tables of division, just as we now have tables of logs (or computerized versions thereof).
With all that said, the elementary functions are indeed those of convenience. There are many, many other functions in constant use out there: gamma functions, zeta functions, Bessel functions, and thousands more special functions which have had reason to be named. (And what is "log" anyway except a name?)
Some of the integrals you can't "solve" can be expressed in terms of one or more of these special functions, and some just cannot (unless we invent a new special function and give it a name -- although that is kind of running around in circles).
For practical purposes it can all be dealt with. If you have a definite integral it can be calculated to any desired accuracy with numeric methods. If in the middle of a proof you have denoted something as a indefinite integral, you can always replace the integrand with some kind of approximation that does have a nice solution, or known behavior.
Many students ask your question. I guess in general one shouldn't hope anything will be easy, mathematically or otherwise. Then if it is you get a nice surprise.
