   Chapter 11.10, Problem 57E

Chapter
Section
Textbook Problem

Use series to approximate the definite integral to within the indicated accuracy. ∫ 0 1 / 2 x 3 arctan x   d x     ( four decimal places )

To determine

To approximate:

The definite integral to with in the indicated accuracy.

Explanation

1) Concept:

Use the Maclaurin series to evaluate the integral,

arctanx=n=0-1nx2n+12n+1 , x<1

2) Given:

012x3arctanx dx

3) Calculation:

The Maclaurin series expansion is

arctanx=n=0-1nx2n+12n+1,  x<1

Therefore,

x3arctanx=x3n=0-1nx2n+12n+1 =n=0-1nx2n+42n+1

x3arctanxdx=n=0-1nx2n+42n+1 dx

=n=0-1nx2n+52n+12n+5+C

=x51·5-x73·7+x95·9-x117·11+

Therefore,

01/2x3arctanxdx=x51·5-x73·7+x95·9-<

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