Partial integration CDF I am reading a textbook which claims that we can obtain by partial integration, for CDF $F(x)$:$$\int_{t}^{\infty} 1-F(x)  \frac{dx}{x}=\int_{t}^{\infty} (\log u -\log t) dF(u) $$
I am aware that the latter integral is a Riemann-Stieltjes integral, but I am not sure how to go from the first to the latter via the partial integration formula I am familiar with, I obtain:
$$\int_{t}^{\infty} 1-F(x)  \frac{dx}{x}=\log(x)(1-F(x)|_{t}^{\infty} -\int_{t}^{\infty}\log(x)(1-F(x) dx.$$

Hope this helps:
\begin{align}
\int_t^\infty(1-F(x))\frac{dx}{x}
& = (1-F(x))\log x|_t^\infty-\int_t^\infty (-dF(x)) \log x\\
& = -(1-F(t))\log t+\int_t^\infty\log u dF(u)\\
& = -\int_t^\infty dF(u)\log t+\int_t^\infty\log u dF(u)\\
& = \int_t^\infty(\log u-\log t)dF(u).
\end{align}
