In some probability book I've come across this notation: $E(X)=\int_{-\infty}^{\infty}x d F(x)$, and it's very confusing, when I see other books defining the same concept as: $E(X)=\int_{-\infty}^{\infty}xf(x) dx$
I wonder... what does that kind of notation mean?
In this case, $f(x)$ would be the probability density function and $F(x)$ would be the cumulative distribution function. They are related by $$f(x)=\frac{dF(x)}{dx}\ ,$$ so $dF(x)=f(x)\,dx$ and the two integrals are equivalent.
In practice, to evaluate the first integral you would probably begin by transforming it to the second anyway.