I have an integral inte1 as below:-

f[x_] := PDF[NormalDistribution[0, 1], x];
F[x_] := CDF[NormalDistribution[0, 1], x];
inte1 = Integrate[f[x]*x^2, {x, -\[Infinity], k}] // FullSimplify

Which is:-

But I want to show it in terms of F[x]:-

How can I achieve this? Many thanks!