   Chapter 11.10, Problem 59E

Chapter
Section
Textbook Problem

Use series to approximate the definite integral to within the indicated accuracy. ∫ 0 0.4 1 + x 4   d x     ( | error | < 5 × 10 − 6 )

To determine

To approximate:

The definite integral to with in the indicated accuracy.

Explanation

1) Concept:

Use the Maclaurin series to evaluate the integral.

1+xk=n=0knxn, for x<1

2) Given:

00.41+x4dx , (error<5×10-6)

3) Calculation:

The Maclaurin series expansion of 1+xk=n=0knxn, for x<1

Therefore,

1+x4=1+x412=n=01/2 nx4n

(Replacing, x by x4)

1+x4dx=n=01/2 nx4ndx

1+x4dx=1/2 nx4ndx

By the power rule of integration,

=n=01/2nx4n+14n+1+C

=1/20x+1/21x55+1/22x99+1/23x1313+)

Therefore,

00.41+x4dx

=1/20x+1/21x55+1/22x99+1/23x1313+00

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