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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

2 Answers 2