$\int_a^b u(x)dF(x)$ (1)$ = u(t)F(t)|_a^b - \int_a^b F(t)u^\prime(t)dt$ (2)$ = u(b)-\int_a^b F(t)u^\prime(t)dt$
$= u(b)-(\Phi(t)u^\prime(t)|_a^b-\int_a^b \Phi(t)u^{\prime\prime}(t)dt=u(b)-\Phi(b)u^\prime(b)+\int_a^b \Phi(t)u^{\prime\prime}(t)dt$.
From (1) to (2), I know that it is a partial derivative, but shoudn't it be $u(t)F(t)|_a^b - \int_a^b F(t)u^\prime(t)dF(t)$?? How can we transform $dF(t)$ to $dt$?
I also want to know why $\Phi(b)$ is not equal to 1.
I attach the original paper in case the information is not enough.