Integrate a function with a power when knowing the integral without a power

Question

So I have this function $f(i,t)$ which when integrated over $i$ gives:

$$\int_0^Nf(i,t)di=F(t)$$

How then can I then integrate this:

$$\int_0^Nf(i,t)^\frac{\epsilon-1}{\epsilon}di$$

To come out as some expression with $F(t)$ in it.

Im thinking this might be possible using the chain rule or something, but I cant quite get there. BTW this is for an economics problem, the answer is probably something simple, my maths is just quite bad

Answer 1

Fix some $t$, then you have $a = \int^N_0 g(i)\mathrm di$ and you wonder what is $\int^N_0 g^\alpha(i)\mathrm di$ for some $\alpha \neq 1$, here $g(i) = f(i, t)$. Bad news: the latter integral can be anything. There are a lot of functions for which integral is $a$, and yet their power can be all over the place.