How many points of integration are necessary to calculate the integral
$$\int_2^6\ln(x)dx$$
using Simpson's rule such that the absolute error of integration is lesser or equal to $5*10^{-4}$, knowing that the values for $\ln(x)$ have 7 significant digits?
$$E(f) = -\frac{h^4(b-a)}{180}f^{(4)}(\eta)$$ $$h = \frac{b-a}{N}$$ where N = number of intervals in between points, which should me even for Simpson's Rule.
What I tried to do:
$$h = \frac{6-2}{N} = 4/N \\ E(f) = -\frac{4^5}{N^4}*1/180*f^{(4)}(\eta) \\ f^{(4)} = -6x^{-4} \\ |E(f)| \le 5*10^{-4} \Leftrightarrow \\ |\frac{4^5}{180N^4}|\max_{6,2}|-6x^{-4}| \le 5*10^{-4} \Leftrightarrow \\ N = 0.12374 \approx 1$$
This solution doesn't make a whole lot of sense. Help?