I remember this integral being expressible in terms of pi.
The definite integral from 0 to infintity or from negative infinity to infinity is "expressible in terms of ".
Whether "approximation the only way" depends upon what you mean by "approximation".
Would you consider only done by "approximation"? If not, how would you find arctan(x) for general x?
Gaussian integral - Wikipedia, the free encyclopedia
To answer the OP, if the bounds are from negative infinity to infinity then the integral will become . If, however, the bounds are finite the integral cannot be expressed in elementary functions: http://integrals.wolfram.com/index.jsp?expr=e^(-x^2)&random=false