Solution
= Solution
On bounded intervals the <exponential series> converges uniformly, allowing termwise <integration>. Since $e^{-t^2}=1-t^2+t^4/2+O(t^6)$,
$$
I(x)=\frac2{\sqrt\pi}\left(x-\frac{x^3}{3}+\frac{x^5}{10}+O(x^7)\right).
$$
The requested first two nonzero terms are therefore
$$
\boxed{I(x)=\frac{2x}{\sqrt\pi}-\frac{2x^3}{3\sqrt\pi}+O(x^5).}
$$
The function is the <error function>, with $I(0)=0$ and derivative $2e^{-x^2}/\sqrt\pi$.