Optimizing over an Integral

Question

I have to solve the following optimization problem:

$max_\tau \int_\underline{\epsilon}^\bar{\epsilon} \tau(1-\tau)^\epsilon d\epsilon$

How does one solve these with the integral. I think I may be able to do it point-wise but I am not sure.

Answer 1

Try integrating: $$\tau\int_{a}^{b}{(1-\tau)^xdx} =\left. \frac{\tau(1-\tau)^x}{\log(1-\tau)} \right|_a^b$$ Then take the derivative with respect to $\tau$ to find the extrema.