Product-integration by parts

 $$\left(f(x)^{g(x)}\right)^*=f^{g'}f_*^g$$

$$f^g=\prod_{a}^{b}f^{dg}\prod_{a}^{b}f_*^{g\ dx}=f^{g(b)-g(a)}\prod_{a}^{b}f_*^{g\ dx}$$