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 $p$?
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Is there a reason, why you specify $\int f dx^p$ instead of simply $a^p$ for some number $a$? Or do you mean integrating $p$ times? In that case, you might want to check out Fractional Calculus – Thomas Feb 17 '15 at 20:55
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Because I want to know whether there exist an equation $(\int f(x)dx)^p=\int \prod_{i=1}^j' f(x_i) dx_i$ for a generalized product $\prod'$ or similar. – kryomaxim Feb 17 '15 at 20:58
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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.
