Here is my problem
Let $H(x)=\Phi(\frac{Ax+b}{\sqrt{2}})$ where $A$, $b$ are some non-negative constant and $\Phi$ is the CDf of a standard normal distribution.
I want to find
$$\int x \, dH(x)$$
I thought maybe integration by parts which yields:
$$x H(x)-\int H(x) \, dx$$
And now I'm not sure how to proceed or if it's possible to take the integral of a CDf