How to compute $(\int f(x) \, dx)^p$ with fractional number $p$?

Question

It is well-known that one can say $(\int f(x) \, dx)^p = \int \prod_{i=1}^p f(x_i) \, dx_i$ if $p$ isa natural number. But what is if $p$ is a fractional ore even a real number? Is it possible to set $\int f(x) \, dx = h \sum_k f(hk)$ for infinitesimal number $h$ and then use the multinomial Theorem when computing $h^p (\sum_k f(hk))^p$? How I compute $(\int f(x) \, dx)^p$ for General number $p$?

Answer 1

It sounds like you are interested in fractional calculus. The wikipedia article linked will give you some information but I also have a lengthy answer here on math stackexchange that might give you some leads: Fractional Calculus.