In general, what is the integral $\displaystyle \int_{0}^{\infty} f^n(z) dn$ where n is the nth derivative of f(z)?
What, exactly, does $\displaystyle f^n(z)$ mean? Is it the nth power of f or the nth derivative (more commonly written $\displaystyle f^{(n)}(z)$)? If it means the nth derivative, and you are integrating with respect to its nth derivative that is exactly the same as $\displaystyle \int x dx= (1/2)x^2$ evaluated at $\displaystyle f^{(n)}(z)$, $\displaystyle \frac{1}{2}\left[\lim_{x\to\infty} (f^{(n)}(x))^2- f^{(n)}(0)\right]$.
If you mean f to the power n, then it will depend more strongly on exactly what f is.
I am really having trouble with the concept of this problem. If you integrated with respect to the variable n by taking the antiderivative of n, and the limits of the integral are set to 1 and 0, you come up with a 1/2 derivative which makes no sense at all. I cannot take the antiderivative of the function as a whole because I am not integrating with respect to z, since z is a constant. If anyone could point me in the right direction I would appreciate it very much.