I find myself in need of some advice on an integration problem.
Let $F(x,\lambda)=\Gamma(x,\lambda)/\Gamma(x)$, $x,\lambda>0$ be the Regularized Upper Incomplete Gamma Function, where $\Gamma(x,\lambda) = \int_\lambda^\infty e^{-s}s^{x-1}ds,$ and $\Gamma(x) = \int_0^\infty e^{-t}t^{x-1}dt$.
Let
(1) $m_k(\lambda)=\int_0^\infty x^k d F(x,\lambda)$
In some notes I'm reading I found the following identity
(2) $m_k(\lambda)=k \int_0^\infty x^{k-1} [1-F(x,\lambda)]dx$.
Any idea on how to go from (1) to (2)? Maybe I'm missing something or there's a typo somewhere I can't spot. Note: based on this result, another one is derived, that actually seems correct (although based on empirical evidence only).