Find the Maclaurin series of $F(x)={\displaystyle \underset{0}{\overset{x}{}}}f(t)dt$ by integrating the Maclaurin series of $f$ term by term. If $f$ is not strictly defined at zero, you may substitute the value of the Maclaurin series at zero.

$F(x)={\displaystyle \underset{0}{\overset{x}{}}}{e}^{-t^2}dt$;$f(t)=e^{-t^2}={\displaystyle \underset{n=0}{\overset{}{}}}(-1{)}^n\frac{t^{2n}}{n!}$

$F(x)={\displaystyle \underset{n=0}{\overset{}{}}}(x)$