Q$8.$ a$)$ Evaluate the following integral

${\displaystyle \underset{0}{\overset{}{}}}xco{s}^2(x)dx$

using Gauss quadrature integration with $4$ points with the following coordinates and weights $($the ones given in our textbook

are not accurate enough$).$

Provide the result and the true error. For the true value, either perform analytical integration, or use MATLAB's integral

function.

b$)$ Write a MATLAB code that calculates the integral using trapezoidal rule and determine the number of segments necessary

to have roughly the same accuracy.

c$)$ Write a MATLAB code that calculates the integral using Simpson's $1/3$ rule and determine the number of segments necessary

to have roughly the same accuracy.

d$)$ Write a MATLAB code that calculates the integral using Simpson's $3/8$ rule and determine the number of segments necessary

to have roughly the same accuracy.