Let
$$ f(x) = 1 - (1-c\log (F(x)))^s\\ g(x) = \log(\frac{a}{x}) $$
for some positive integers $s$, $c$, and some real-valued function $F(x)$. $\log$ denotes the natural logarithm. One can think about $f(x)$ as the minimum outcome of $s$ independent draws of a random variable with distribution $c\log (F(x))$.
I'm trying to get the expected value of $g(x)$, given the minimum of $s$ draws. That is, I need to integrate
$$ \int f'(x)g(x) dx$$
for some positive real number $a$. I know that $g'(x) = \frac{1}{x}$, and I am aware of the integral trick $\int u'v = uv - \int v'u$. However, I fail to see how I could use that. There's no substitution trick that I could think of that makes sense here.
How could I proceed? Note that this is not a homework assignment, so a closed form solution to the expression is not guaranteed.
I don't believe it is useful (because it doesn't have a nice structure), but here's F(x) for two real numbers $d,e$:
$$ F(x) = \frac{dx}{x-e}$$
FooBar.
We have: $ f(x) = 1 - \left( 1 - c \cdot \log(F(x)) \right)^s \Leftrightarrow f'(x) =-s \cdot (1 - c \cdot \log(F(x)))^{s-1} \cdot \left( - \frac{c}{F(x) \cdot \ln (10) } \right) \cdot F'(x) $, assuming $ \log(F(x)) = \log_{10}( F(x)) $.
Then:
$ \int f'(x) \cdot g(x) dx = \frac{s}{\ln(10)}\int (1 - c \cdot \log(F(x)))^{s-1} \cdot \frac{c \cdot F'(x)}{F(x)} \cdot \log \left( \frac{a}{x} \right) dx $
Now: $ F(x) = \frac{dx}{x-e} \Leftrightarrow F'(x) = \frac{d \cdot (x-e) - dx \cdot 1}{(x-e)^2} = \frac{dx - de - dx}{(x-e)^2} = -\frac{de}{(x-e)^2} $
And thus we have:
$ \large{\frac{s}{\ln(10)} \cdot \int (1 - c \cdot \log \left( \frac{dx}{x-e} \right))^{s-1} \cdot -\frac{c \cdot \left( \frac{de}{(x-e)^2}\right)}{\frac{dx}{x-e}} \cdot \log \left( \frac{a}{x} \right) dx \\\\ = -\frac{s \cdot c}{\ln(10)} \cdot \int \left(1 - c \cdot \log \left( \frac{dx}{x-e} \right)\right)^{s-1} \cdot \frac{e}{(x-e)^2 \cdot x} \cdot \log \left( \frac{a}{x} \right) dx}$
I don't see how one can proceed from here on out without knowing the values of the variables.
Kind regards, Pedro