real analysis – Solving $int_0^infty N(1-F(t))^{N-1}tf(t)dt$ when the expected value is known


Suppose that $f:mathbb R_{geq 0} rightarrow mathbb R_{geq 0}$ is a probability density function, and $F$ is a cumulative distribution function (i.e. $F(t)=int_0^t kf(k)dk$). Also, assume that the expected value, $E = int_0^infty tf(t)dt$ is known.

(Question 1) Is there a way to solve $A_N:=int_0^infty N(1-F(t))^{N-1}tf(t)dt$, where $N$ is a positive integer, so that the result only depends on $N$ and $E$ (e.g. $Ee^{-N}+const$)? If so, how?

(Question 2) Or if solving exactly is hard or impossible, then would it possible to get an approximate solution?

Also, from the problem I am working on, it is safe to assume that $A_Nrightarrow C$ as $Nrightarrow infty$ where $0<C<infty$ is some constant, and $A_{N}$ is decreasing. Not sure whether these are helpful, though…

So far, I tried to approximate $(1-F(t))^{N-1}$ as $e^{-F(T)(N-1)}$, but haven’t gotten much so far…
Thank you in advance 🙂