Exercise 17.3

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 composite Simpsons rule.

Calculate the values for \(n = 1, 2, \dots, 10\) sub-intervals and compare the approximations to the exact value by plotting the value of the integral versus \(n\). Be sure to represent the exact solution in the plot as well.