   Chapter 11.10, Problem 58E

Chapter
Section
Textbook Problem

# Use series to approximate the definite integral to within the indicated accuracy.58. ∫ 0 1 sin ( x 4 )   d x (four decimal places)

To determine

To approximate: The definite integral 01sin(x4)dx correct to four decimal places.

Explanation

Result used:

For all x the expansion of sinx is n=0(1)nx2n+1(2n+1)! (1)

Calculation:

Obtain 01sin(x4)dx .

Substitute x4 for x in equation (1),

sin(x)4 =n=0(1)n(x4)2n+1(2n+1)!

sin(x4)=n=0(1)nx8n+4(2n+1)! (2)

Integrate on both sides,

sin(x4)dx=n=0(1)nx8n+4(2n+1)!dx=n=0(1)n(2n+1)![x8n+58n+5]+C

Compute the boundaries,

01sin(x4)dx=[n=0(1)n(2n+1)!x8n+5(8n+5)]01  =[n=0(1)n(2n+1)!(1)8n+58n+5][n=0(

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