Trick for solving an integral

Question

I am trying so solve a practical problem (i.e. not sure if my questions has any nice solution), and I reached a point where I need to solve an integral of the form $\int_0^Tf(t)e^{at}dt$, where $f(t)$ is a function for which I don't know the value at each time, $t$, but I know its time average i.e. $\frac{\int_0^Tf(t)}{T}=f_0$ (in principle $f(t)$ has a given distribution, and I know that distribution, I just don't know each individual value). Formally, in both integrals I would need $T\to \infty$, but in practice I just have a T big enough so you can assume that $T\to \infty$. I know the value of $a$. I was wondering if there is any way to simplify this integral any further (ideally obtaining something in terms of $a$, $f_0$ and $T$). Thank you!