?
Solved

derivates and anti-derivatives of ln x and 1/x (respectively)

Posted on 2011-05-10
5
Medium Priority
?
432 Views
Last Modified: 2012-05-11
So I know derivative of ln(x) = 1/x, but the anti-derivative of 1/x = ln(|x|), why the absolute value? It is possible to have negative values for x in ln, so why isn't it just x?
0
Comment
Question by:Zenoture
5 Comments
 
LVL 85

Expert Comment

by:ozo
ID: 35726736
the values of ln for negative x are not the values of the anti-derivative of 1/x for negative x
0
 
LVL 18

Accepted Solution

by:
deighton earned 2000 total points
ID: 35727284
F(x) = ln(|x|)

means that for x<0

F(x) = ln(-x)

F'(x) = (-1) / (-x) = 1/x

so the anti-derivative is shown to be valid

but also of course

d/dx(ln(x)) = 1/x

_________________________

in the real numbers ln(x) is not defined for x<0, but in complex numbers

-x = x e^(pi i)

since e^(pi i) = -1

so for example

ln(-5) = ln(5 e^(pi i)) = ln(5) + pi i

and in general

ln(-x) = ln(x) + pi i

therefore the difference between ln(-x) and ln(x) is a constant which can be absorbed into a constant of integration, and ultimately would not affect a definite integral


______________________________

integral ln(x) is found by considering ln(x) = 1 ln(x) , then using integration by parts to give

x ln(x) - x
0
 
LVL 5

Expert Comment

by:jarrod_williams
ID: 35727354
Hi Zenoture.

Im sure you will find the following link interesting.

http://en.wikipedia.org/wiki/Integration_by_parts
0
 
LVL 27

Expert Comment

by:aburr
ID: 35730713
deighton gave the key to the answer to your question when he said
 "in the real numbers ln(x) is not defined for x<0"
He then went on to give a nice illustration of what the situation is if one considers complex numbers.
0
 

Author Closing Comment

by:Zenoture
ID: 35828891
Sorry, forgot to award points.
0

Featured Post

Industry Leaders: We Want Your Opinion!

We value your feedback.

Take our survey and automatically be enter to win anyone of the following:
Yeti Cooler, Amazon eGift Card, and Movie eGift Card!

Question has a verified solution.

If you are experiencing a similar issue, please ask a related question

This article seeks to propel the full implementation of geothermal power plants in Mexico as a renewable energy source.
This article provides a brief introduction to tissue engineering, the process by which organs can be grown artificially. It covers the problems with organ transplants, the tissue engineering process, and the current successes and problems of the tec…
This is a video describing the growing solar energy use in Utah. This is a topic that greatly interests me and so I decided to produce a video about it.
Although Jacob Bernoulli (1654-1705) has been credited as the creator of "Binomial Distribution Table", Gottfried Leibniz (1646-1716) did his dissertation on the subject in 1666; Leibniz you may recall is the co-inventor of "Calculus" and beat Isaac…
Suggested Courses
Course of the Month17 days, 9 hours left to enroll

830 members asked questions and received personalized solutions in the past 7 days.

Join the community of 500,000 technology professionals and ask your questions.

Join & Ask a Question