2 16 (a) Evaluate the integral: da x² + 4 Your answer should be in the form kT, where k is an integer. What is the value of k? d 1 -arctan(x) dx Hint: x² + 1 k = (b) Now, let's evaluate the same integral using a power series. First, find the power series for the 16 function f(x) = Then, integrate it from 0 to 2, and call the result S. S should be an infinite x2 + 4 series. What are the first few terms of S? aj = a2 = az = a4 = (C) 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 T in terms of an infinite series. Approximate the value of r by the first 5 terms.
2 16 (a) Evaluate the integral: da x² + 4 Your answer should be in the form kT, where k is an integer. What is the value of k? d 1 -arctan(x) dx Hint: x² + 1 k = (b) Now, let's evaluate the same integral using a power series. First, find the power series for the 16 function f(x) = Then, integrate it from 0 to 2, and call the result S. S should be an infinite x2 + 4 series. What are the first few terms of S? aj = a2 = az = a4 = (C) 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 T in terms of an infinite series. Approximate the value of r by the first 5 terms.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
Part C
of the attach question.
Transcribed Image Text:2
16
(a) Evaluate the integral:
da
x² + 4
Your answer should be in the form ka, where k is an integer. What is the value of k?
d
1
-arctan(x)
dx
Hint:
x² + 1
k =
(b) Now, let's evaluate the same integral using a power series. First, find the power series for the
16
function f(x) =
. Then, integrate it from 0 to 2, and call the result S. S should be an infinite
2² + 4'
series.
What are the first few terms of S?
= Op
aj =
a2 =
az =
a4 =
(c) 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 T in terms of an infinite series.
Approximate the value of r by the first 5 terms.
(d) What is the upper bound for your error of your estimate if you use the first 7 terms? (Use the
alternating series estimation.)
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