(a) Evaluate the integral32dx.x2 + 4Your answer should be in the form kT, where k is an integer. What is the value of k?d arctan(x)(Hint:dxx²+1k -(b) Now, lets evaluate the same integral using power series. First, find the power series for the function f(x)32Then, integrate it from 0 to 2, and call itx2+4S. S should be an infinite series >-0 an .What are the first few terms of S?Aj =a2a4(c) The answer in part (a) equals the sum of the infinite series in part (b) (why?). Hence, if you divide your infinite series from (b) by k (the answer to (a)), youhave found an estimate for the value of T in terms of an infinite series. Approximate the value of T by the first 5 terms.(d) What is an upper bound for your error of your estimate if you use the first 9 terms? (Use the alternating series estimation.)

The definite integral of a function f(x) from a point x=a to x=b gives the area under the curve of…

(a) Evaluate the integral 32 -dx. x² + 4 Your answer should be in the form kr, where k is an integer. What is the value of k? d arctan(x) (Hint: Jo dz x2+1 k = (b) Now, lets evaluate the same integral using power series. First, find the power series for the function f(x) = 2.. Then, integrate it from 0 to 2, and cal x²+4 S. S should be an infinite series n-0 an · What are the first few terms of S? an = a1 = a2 = az = a4 = (c) The answer in part (a) equals the sum of the infinite series in part (b) (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 T in terms of an infinite series. Approximate the value of T by the first 5 terms. (d) What is an upper bound for your error of your estimate if you use the first 9 terms? (Use the alternating series estimation.)

(a) Evaluate the integral 32 -dx. x² + 4 Your answer should be in the form kr, where k is an integer. What is the value of k? d arctan(x) (Hint: Jo dz x2+1 k = (b) Now, lets evaluate the same integral using power series. First, find the power series for the function f(x) = 2.. Then, integrate it from 0 to 2, and cal x²+4 S. S should be an infinite series n-0 an · What are the first few terms of S? an = a1 = a2 = az = a4 = (c) The answer in part (a) equals the sum of the infinite series in part (b) (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 T in terms of an infinite series. Approximate the value of T by the first 5 terms. (d) What is an upper bound for your error of your estimate if you use the first 9 terms? (Use the alternating series estimation.)

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

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Question

(a) Evaluate the integral
32
dx.
x2 + 4
Your answer should be in the form kT, where k is an integer. What is the value of k?
d arctan(x)
(Hint:
dx
x²+1
k -
(b) Now, lets evaluate the same integral using power series. First, find the power series for the function f(x)
32
Then, integrate it from 0 to 2, and call it
x2+4
S. S should be an infinite series >-0 an .
What are the first few terms of S?
Aj =
a2
a4
(c) The answer in part (a) equals the sum of the infinite series in part (b) (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 T in terms of an infinite series. Approximate the value of T by the first 5 terms.
(d) What is an upper bound for your error of your estimate if you use the first 9 terms? (Use the alternating series estimation.)

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ISBN:

9780470458365

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Erwin Kreyszig

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