I am working on $E[\Phi(\frac{z_{\alpha/2}}{\sqrt{n}}+\bar{x}-k)- \Phi(-\frac{z_{\alpha/2}}{\sqrt{n}}+\bar{x}-k)]$, where $x_1, ..., x_n \sim N(\mu, 1)$, $\mu$ is unknown, k is known.
$\bar{x} \sim N (\mu, \frac{1}{n})$. Thus
$E[\Phi(\frac{z_{\alpha/2}}{\sqrt{n}}+\bar{x}-k)- \Phi(-\frac{z_{\alpha/2}}{\sqrt{n}}+\bar{x}-k)] \\= \int_{-\infty}^{\infty}[\Phi(\frac{z_{\alpha/2}}{\sqrt{n}}+\bar{x}-k)- \Phi(-\frac{z_{\alpha/2}}{\sqrt{n}}+\bar{x}-k)]\frac{\sqrt{n}}{\sqrt{2\pi}}exp(-\frac{n(x-\mu)^2}{2})dx \\=\int_{-\infty}^{\infty}\int_{-\frac{z_{\alpha/2}}{\sqrt{n}}+\bar{x}-k}^{\frac{z_{\alpha/2}}{\sqrt{n}}+\bar{x}-k}\frac{1}{\sqrt{2\pi}}exp(-\frac{y^2}{2})dy\frac{\sqrt{n}}{\sqrt{2\pi}}exp(-\frac{n(x-\mu)^2}{2})dx$
However, I get stuck here since the integration is very complex. Can anyone give me a hint on how to do the integration? And is there any easy ways to do this problem? Thank you very much.