I believe that f(x) = sqrt(1/(2pi))exp^(-x^2/2) is correct for sigma = 1 and u = 0. The expression

sqrt(1/2pi)exp[-(x-u)^2/(2*sigma)]

for the general formula has a typo. I think the correct formula is

sqrt(1/(2 pi))/sigma * exp(-(x-u)^2/(2*sigma^2)).

Magic man wanted to integrate f(x) from -x to x. As BruceZ stated, this does not have a solution among the elementary functions. If you allow more advanced functions, the integral is

I = erf(x/sqrt(2)).

For example with Excel, you can type +ERF(1/SQRT(2)) to get the 1 s.d. confidence level.