Evaluate the integral 16/x2+4 dx Your answer should be in the form kpi, where k

Question

Evaluate the integral 16/x2+4 dx Your answer should be in the form kpi, where k is an integer. What is the value of k? Hint:darctan(x) dx=1/x2+1 k = Now, lets evaluate the same integral using power series. First, find the power series for the functionf(x) =16/x2+4 .Then,integrate it from 0 to 2, and call it S. S should be an infinite series. What are the first few terms of S ? ao = a1= a2= a3= a4= The answers to part (a) and (b) are equal (why?). Hence, if you divide your infinite series from (b) by k (the answer to (a)), you have found an estimate for the value of n in terms of an infinite series. Approximate the value of pi by the first 5 terms. What is the upper bound for your error of your estimate if you use the first 10 terms? (Use the alternating series estimation.)

Explanation / Answer

a> 2

b> a0 = 8, a1 = -8/3, a2 = 8/5, a3 = -8/7, a4 = 8/9

c> 3.33968254

d> 0.1904761905

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