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Sine integral function The function is called the sine

Calculus: Early Transcendentals | 1st Edition | ISBN: 9780321570567 | Authors: William L. Briggs, Lyle Cochran, Bernard Gillett ISBN: 9780321570567 2

Solution for problem 72E Chapter 9.4

Calculus: Early Transcendentals | 1st Edition

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Calculus: Early Transcendentals | 1st Edition | ISBN: 9780321570567 | Authors: William L. Briggs, Lyle Cochran, Bernard Gillett

Calculus: Early Transcendentals | 1st Edition

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Problem 72E

Sine integral function The function  is called the sine integral function.

a. Expand the integrand in a Taylor series about 0.

b. Integrate the series to find a Taylor series for Si.

c. Approximate Si (0.5) and Si (1). Use enough terms of the series so the error in the approximation does not exceed 10−3

Step-by-Step Solution:

Solution 72E:

Step 1:

Given , the function “Si(x) =  dt” is called the sine integral function.

Integrand is ;

             

In this problem we need to expand the  integrand in a taylor series about “0”.

We know that the taylor series about “0” is ;

                   

f(x) = f(0) +x ++................

Consider , f(t) =  , then f(0) =  = 1

 , then = 0

= , then   = -1, since = = -1

  = , then   = 0

 = , then  = 1, since = = 1

……………………………………

Therefore , the taylor series about zero is ;

                             

f(t) =  =  1 +t ++................

                                    = 1 - +-................

Step 2:

b)  Now , we need to find taylor series of Si(x) , by using integration method .

From the above step   f(t) =  =  1 - +-................

Integrating on both sides we get ,

 dt = ( 1 - +-................)dt

                               =

                              = ( x- +- ..........)-(0)

                             = x- +-..............

Therefore , Si(x) = x- +-+...................

Step 3 of 4

Chapter 9.4, Problem 72E is Solved
Step 4 of 4

Textbook: Calculus: Early Transcendentals
Edition: 1
Author: William L. Briggs, Lyle Cochran, Bernard Gillett
ISBN: 9780321570567

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Sine integral function The function is called the sine

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