<Dd> y = x s d y = x d s . (\ displaystyle (\ begin (aligned) y& = xs \ \ dy& = x \, ds. \ end (aligned))) </Dd> <P> Since the limits on s as y → ± ∞ depend on the sign of x, it simplifies the calculation to use the fact that e is an even function, and, therefore, the integral over all real numbers is just twice the integral from zero to infinity . That is, </P> <Dl> <Dd> ∫ − ∞ ∞ e − x 2 d x = 2 ∫ 0 ∞ e − x 2 d x . (\ displaystyle \ int _ (- \ infty) ^ (\ infty) e ^ (- x ^ (2)) \, dx = 2 \ int _ (0) ^ (\ infty) e ^ (- x ^ (2)) \, dx .) </Dd> </Dl> <Dd> ∫ − ∞ ∞ e − x 2 d x = 2 ∫ 0 ∞ e − x 2 d x . (\ displaystyle \ int _ (- \ infty) ^ (\ infty) e ^ (- x ^ (2)) \, dx = 2 \ int _ (0) ^ (\ infty) e ^ (- x ^ (2)) \, dx .) </Dd>

Integral of e^(-x^2) from negative infinity to infinity