Answered: Some elementary functions, such as f(x)…

Calculus For The Life Sciences

2nd Edition

ISBN:9780321964038

Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Chapter6: Applications Of The Derivative

Section6.3: Implicit Differentiation

Problem 44E

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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 ∫ e^x/ x dx does not have an elementary antiderivative. Use this fact to prove that ∫ 1 / lnx dx does not have an elementary antiderivative

Expert Solution

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let $\int x^{- n} e^{c x} d x$ for n a positive integer and c a non zero constant is non elementary since $x^{- n} = R' (x) + c R (x)$ has no solution $R (x)$ in the field of rational functions over C.

As $\int \frac{e^{x}}{x} d x$ has no elementary antiderivative thus using the cases of integral of the form $\int x^{n e^{a x^{m}}} d x$ where a is a non zero constant and m and n are integers.

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