(Solved): 1. Gauss-Chebyshev formul of Type I and Type II are given by Type I 111x2f ...

1. Gauss-Chebyshev formulæ of Type I and Type II are given by Type I ${?}_{?1}^1\frac{f(\stackrel{?}{x})}{\sqrt{1?x^2}}dx?\frac{?}{n}{?}_{k=1}^{n}f\left(x_k\right)\text{ where }{x}_k=\cos \left(\frac{(2k?1)?}{2n}\right)$ Type II ${?}_{?1}^1\sqrt{1?x^2}f(x)dx?\frac{?}{n+1}{?}_{k=1}^n{\sin }^2\left(\frac{k?}{n+1}\right)f\left(x_k\right)\text{ where }{x}_k=\cos \left(\frac{k?}{n+1}\right)$ Write MATLAB functions to implement these integration formulæ. (a) Evaluate the integral using Type I, Type II, and Gauss-Legendre quadrature. Tabulate your results for $n=4,8,12,16$ point methods. The exact value of the integral is $?I_0(1)$ where $I_0(z)$ is the modified Bessel function of the first kind. It is available in MATLAB using besseli $(0,z)$. Comment on your findings. ${?}_{?1}^1\frac{e^x}{\sqrt{1?x^2}}dx$ (b) Evaluate the integral using Type I, Type II, and Gauss-Legendre quadrature. Tabulate your results for $n=4,8,12,16$ point methods. The exact value of the integral is $?I_1(1)$ where $I_1(z)$ is the modified Bessel function of the first kind. It is available in MATLAB using besseli $(1,z)$. Comment on your findings. ${?}_{?1}^1{e}^x\sqrt{1?x^2}dx$ 2. Consider the following integral ${?}_0^1\frac{\ln x}{1+x^2}dx=?K\text{ where }K\text{ is Catalan’s constant }$ Explain why Gauss-Legendre quadrature is appropriate for this problem whereas Simpson's- $\frac{1}{3}$ rule is not. Evaluate the integral using Gauss-Legendre quadrature for $n=4,8,16,24,32$ point methods. Tabulate the computed value of the integral and the \%-error for each case, ${?}_{??}^{?}?e^{?x^2}\cos xdx={?}_0^{?}2e^{?x^2}\cos xdx$ Tabulate the computed value of the integral and the \%-error for $n=4,8,16,24,32$ point methods. The exact value of the integral is $\sqrt{?}/e^{1/4}$. Comment on your findings. 4. There is no closed form solution for the following integral but its value is accurate to the number of figures shown ${?}_1^{?}\frac{dx}{x^2+\cos \frac{1}{x}}?0.8245401107935762$ (a) Evaluate this integral using a built-in M?TLAB function such as quad or integral. Try to find the number of function evaluations needed to achieve $0.0001\%$ accuracy. (b) Make a substitution $u=x?1$ and show that the integral is now cast into a form suitable for GaussLaguerre quadrature. Employ this method and find how many function evaluations are needed to achieve $0.0001\%$ accuracy. (c) Make a substitution $z=1/x$ and show that the integral is now cast into a form suitable for Gauss-Legendre quadrature. Employ this method and find how many function evaluations are needed to achieve $0.0001\%$ accuracy.