Exercise 18.4#
Consider the integral
\[
\int_0^{0.9} \frac{dx}{1 - x^2}
\]
which has the exact solution:
\[
\frac{1}{2}\ln\left(\frac{x + 1}{1 - x}\right)~~\Bigg|_{x = 0.9}
\]
Approximate the value of the integral using the trapezoid and improved trapezoidal rules for \(n = 1, 2, \dots, 10\) sub-intervals.
On the same set of axes, plot the following vs \(n\):
The true error of the trapezoidal rule
The true error of the improved trapezoidal rule
The upper-bound of the theoretical error from (20)
The error estimation from (24)
How do each of these compare?