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".
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Would you consider only done by "approximation"? If not, how would you find arctan(x) for general x?
You're thinking of the Gaussian integral: 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