I have been working on the following problem. Given a uniform partition of the interval [-2,2] the following integral is approximated using the composite trapezoid rule $$ I = \int_{-2}^{2} ln(1 + x^2) dx.$$ I need to use the error term of the composite trapezoid rule to find the minimal number K of sub intervals that guarantees that the error $ |I - I_K|$ does not exceed $10^{-2}$.
I have been working with the following formula for the error: $$ E = - \frac{b-a}{12}H^2f’’(\epsilon) $$ for some $\epsilon \in (a,b) $. Where a and b and the lower and upper limits of the integration. I also have that $H = \frac{b-a}{K} = - \frac{4}{K}$. I have been trying to find the upper bound on the error by calculating the second derivative of $ f(x) = ln(1+x^2) $ as $f’’(x) = \frac{2(1-x^2)}{(1+x^2)^2} $ but I don’t really know where to go from here, my thinking is to try and find the maximum value of this derivative and then use that but I don’t know how to go about doing this or if this is along the right tracks. Any help would be greatly appreciated!