Expected value of composition of two normal distributions

Question

Let $X$ be a normal distribution with mean $\mu$ and variance $\sigma$. If $N(u)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^u e^{-\frac{s^2}{2}}ds$, compute $E[N(X)]$. I know how to do it for $X$ being the standard random variable, but I do not if $X$ is general. For the standard random variable is just computing $\int_{-\infty}^{\infty}N(u)N'(u)du$ which is easy by the FTC.

Answer 1

Hint:

Let $U$ have standard normal distribution and let $X$ and $U$ be independent.

Then: $$\mathbb{E}\left[N\left(X\right)\right]=\int N\left(x\right)f_{X}\left(x\right)dx=\int P\left(U\leq x\right)f_{X}\left(x\right)dx=$$$$\int P\left(U\leq X\mid X=x\right)f_{X}\left(x\right)dx=P\left(U\leq X\right)=P\left(U-X\leq0\right)$$

where $U-X$ has normal distribution.