The easiest way to evaluate Gaussian integral Is there any way to evaluate the integral: $$ \int_0^{1}e^{-x^2}\,\mathrm{d}x $$
using only basic integration techniques (Basic formulas, integration by parts, Substitution and change of variables)
 A: Algorithms exist (notably the Risch algorithm)
to decide whether an indefinite integral (antiderivative) can be
written as a "closed form" involving only elementary functions.
The implementation of such an algorithm is extremely complex, however, and
you cannot always guarantee that it will give a yes/no answer
to the question of whether there is a closed-form solution of
whatever integral you ask it to solve.
Methods to evaluate arbitrary definite integrals in closed form 
are even more difficult to exactly decide, 
since in addition to the obvious technique of evaluating the
antiderivative at each end of the interval of integration
(if the antiderivative can be written in closed form),
there are additional methods that can be employed on certain intervals of integration
(such as the evaluation of $\int_0^\infty e^{-x^2}\;dx$).
A formal proof that this particular integral cannot be solved is therefore
not practical (at least not for a format such as this site).
An informal argument, however, is that the integral
$F(x) = \int_0^x e^{-t^2}\;dt$ is of immense interest to mathematicians,
and it is very unlikely that interesting results such as a closed-form
expression for $F(1)$ in terms of elementary functions would have gone unnoticed.
