The integral is the following and I want to produce it in terms of $r$. Is there a theory that the partial producer enters the integral?
$$\frac{\partial }{\partial r} \int_{F^{-1}(p)}^{\infty} \left(\frac{\overline{F}(x)}{1-p} \right)^r dx,\quad\overline{F}(x)=e^{-(\lambda x)^\alpha}$$