Chapter 7.8, Problem 69ES

A convergent improper integral over an infinite interval can be approximated by first replacing the infinite limit(s) of integration by finite limit(s), then using a numerical integration technique, such as Simpson’s rule, to approximate the integral with finite limit(s). This technique is illustrated in these exercises.

(a) It can be shown that

${\displaystyle {\int }_0^{+\infty }\frac{1}{x^6+1}dx=\frac{\pi }{3}}$

Approximate this integral by applying Simpson’s rule with $n=20$ subdivisions to the integral

${\displaystyle {\int }_0^4\frac{1}{x^6+1}dx}$

Round your answer to three decimal places and compare it to $\frac{\pi }{3}$ rounded to three decimal places.

(b) Use the result that you obtained in Exercise 52 and the fact that $1/(x^6+1)<1/x^6$ for $x\ge 4$ to show that the truncation error for the approximation in part (a) satisfies $0<E<2\times {10}^{-4}.$