Some elementary functions, such as f(x) = sin(x2 ), do not have antiderivatives that are elementary functions. Joseph Liouville proved that                                      ∫ ex/ x dx                                                                                                          does not have an elementary antiderivative. Use this fact to prove that           ∫ 1 / lnx  dx                                                                                                   does not have an elementary antiderivative

Calculus: Early Transcendentals
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Chapter1: Functions And Models
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Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Some elementary functions, such as f(x) = sin(x2 ), do not have antiderivatives that are elementary functions. Joseph Liouville proved that                                      ∫ ex/ x dx                                                                                                          does not have an elementary antiderivative. Use this fact to prove that           ∫ 1 / lnx  dx                                                                                                   does not have an elementary antiderivative

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let x-necxdx for n a positive integer and c a non zero constant is non elementary since x-n=R'(x)+cR(x) has no solution R(x) in the field of rational functions over C.

As exxdx has no elementary antiderivative thus using the cases of integral of the form xneaxmdx where a is a non zero constant and m and n are integers.

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