Let $F$ be a cumulative density function of a random variable, define on $[0,t]$:
$$G(u) = \frac{\int_{0}^{u} xe^{-x} dF(\frac{x}{t})}{\int_{0}^{t} xe^{-x} dF(\frac{x}{t})}$$
How can I show that
$$\int_{0}^{t} u^k dG(u) = \frac{\int_{0}^{1} (tx)^{k+1}e^{-tx} dF(x)}{\int_{0}^{1} (tx)e^{-tx} dF(x)}?$$
I am very new to this kind of integral, I am not even sure how to interpret $dF(\frac{x}{t})$.