ex²=1+x2+x4/2!+...+∑x2n/n!+...
Therefore the integral of ex² is C+x+x3/3+x5/10+...+∑x2n+1/((2n+1)n!)+... expressed as an infinite power series, where C is a constant.
(However, this integral is officially related to the Gauss error function usually given by erf(x) and the imaginary error function erfi(x). The constant multiplier √π/2 relates the integral to the error function.)