   Chapter 8, Problem 14PS

Chapter
Section
Textbook Problem

Elementary Functions Some elementary functions. such as f ( x ) = sin ( x 2 ) , do not have antiderivatives that are elementary functions. Joseph Liouville proved that ∫ e x x d x does not have an elementary antiderivative. Use this fact to prove that ∫ 1 ln x d x does not have an elementary antiderivative.

To determine

To prove: The integral 1lnxdx does not have an antiderivative.

Explanation

Given:

The integral exxdx does not have an elementary anti-derivative.

Formula used:

The derivatives of logarithmic function:

ddx(lnx)=1x

Proof:

An elementary anti-derivative function does not have an anti-derivative which can be determined by the elementary function.

Let u=lnx.

Therefore, du=1xdx.

Now,

u=lnxx=eu

So, the integralnow becomes,

1lnxdx=

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