Add new solution

The current solution is definitely right. But I am not sure whether $f^{\prime }=-2xe^{-x^{2}}$ is something easy to think for students in 100 level courses.

I guess most student use substitution first $u=x^{2}$, then use integration by parts.

If anyone agree, can you please move the current solution to the second one and write the new first hint and first solution?