(a) Approximate this integral using Simpson's Rule with n = 4. (b) Find an upper bound for the error in your estimation from part a). For reference, the fourth and fifth derivatives of f(x) = ln(1+x²) are -12(x4 – 6x² + 1) (1+ x²)4 f(4) (x) = and f) (x): 48x(x4 – 10x2 + 5) (1+x²)5

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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4. Consider the integral
In(1+ x²) dx.
(a) Approximate this integral using Simpson's Rule with n = 4.
(b) Find an upper bound for the error in your estimation from part a). For reference,
the fourth and fifth derivatives of f(x) = ln(1+x²) are
-12(x – 6x? + 1)
(1+x²)4
f(4) (x)
and
48x (x4 – 10x? + 5)
(1+x²)5
f(5) (x)
(c) Compute this integral explicitly, and compare the actual value with your estimate
in part a). How does this difference compare with the upper bound for the error
in part b)?
Transcribed Image Text:4. Consider the integral In(1+ x²) dx. (a) Approximate this integral using Simpson's Rule with n = 4. (b) Find an upper bound for the error in your estimation from part a). For reference, the fourth and fifth derivatives of f(x) = ln(1+x²) are -12(x – 6x? + 1) (1+x²)4 f(4) (x) and 48x (x4 – 10x? + 5) (1+x²)5 f(5) (x) (c) Compute this integral explicitly, and compare the actual value with your estimate in part a). How does this difference compare with the upper bound for the error in part b)?
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