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# Composite Trapezoidal Rule, Simpson's Rule and Gaussian Quadratures

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1. Use the composite Trapezoidal Rule with indicated values of n=4 to approximate the following integrals

See Attached file for integrals.

2. Use the Excel programs for Simpson's composite rule to evaluate integrals in Problem 1.

3. Use Gaussian Quadratures with n = 2, n = 4, n = 5 to evaluate integrals in Problem 1.

4. Find exact values of the integrals given in Problem 1. Compare errors in approximate values of the integrals.

5. Give a comparative analysis of errors made by the Trapezoidal Rule, and Simpson rule. Explain for what kind of functions The Simson's rule gives an exact result for the integral. Explain why.

6. For the improper integral of the following type , (See attached File) please explain in what case it is convergent, and in what case its divergent. Bring an example. Also explain for what value of p we can apply the Simpson's rule?