The integrals of some functions can simply not be written in terms of elementary functions, and we have to introduce new notation to denote these "special" functions. This means that no matter how much you try, you'll never be write the integral of some functions using algebraic, logarithmic, exponential or trigonometric expressions with the ordinary +,- and x operations. Take for example, the function ##f(x)=e^{-x^2}##. You cannot write its integral in a conventional manner. You can always use numerical approximations for definite integrals or use values to which improper functions tend to, but there is no way around the indefinite integral of these functions other than giving them special names, in this case, the error function ##erf(x)##(actually the integral for this function will be ##\frac{\sqrt{π}}{2} erf(x) +c## , where ##c## is an arbitrary constant). Another class of these special integrals are called elliptical integrals, and you can read about them on the net.
