# Could a computer theoretically compute all integrals in terms of some special functions or it is not possible theoretically?

Could a computer theoretically compute all integrals in terms of some special functions or there need to be exist infinite number of such special functions to represent all integrals?

I know there theoretically could be exist a computer to prove all elementary geometry theorem.

• I think so. We don't have closed form expressions for certain integrals, like $\int\frac{\sin(x)}{x}dx$, for example, but (at least to the best of my knowledge) all well-defined integrals are computable. – Newb Jan 31 '14 at 0:26
• An uncountable set of integrals cannot be represented by a finite set of special functions. Perhaps, you meant all integrals that humans can directly consider? That would be a (theoretically) denumerable set. – Jacob Wakem Jan 31 '14 at 14:32

$$\Xi (x;t) = \sum_{i=0}^t \int_0^x \Gamma(t)dt$$