Absolute-value bars, "ln()" v "log()", etc.

[lookagain]

\(\displaystyle \begin{cases}u=x & dv=(\cos^2(x))^{-1}dx=\sec^2(x)dx \\ du=dx & v=\tan(x)\end{cases}\)
Thus
\(\displaystyle uv - \int {\tan (x)dx = x\tan (x) + \log (\cos (x)) + c} \)

Some people use "log" for "natural log" as above. I'm using "ln."

The argument needs absolute value bars:


\(\displaystyle uv \ - \ \int {\tan (x)dx \ = \ x\tan (x) \ + \ \ln|\cos (x)| \ + \ C} \)




Source:
http://math2.org/math/integrals/more/tan.htm

 

[pka]

EDIT: It says in my book that the answer is ln(cosx) + xtanx + C so perhaps looking at that someone could solve it and explain how to do it?

See no absolute value

Some people use "log" for "natural log" as above. I'm using "ln."
The argument needs absolute value bars:
\(\displaystyle uv \ - \ \int {\tan (x)dx \ = \ x\tan (x) \ + \ \ln|\cos (x)| \ + \ C}\)

\(\displaystyle
You really should read the thread before interfering. Also it has been the consensus of even mathed group to use \(\displaystyle \log(x)\) since the mid-1980's. You are behind the times.\)

 

[ksdhart2]

Well, for what it's worth, literally every text book I've ever learned from used ln(x) for natural log, reserving log(x) for base 10. I assume they were all published post-1980 as well. Just based on this anecdotal evidence, I'd be hesitant to say that either way is either "right" or "wrong," choosing instead to say that there are multiple equally valid notations (as is often the case, I've found).

 

[pka]

Well, for what it's worth, literally every text book I've ever learned from used ln(x) for natural log, reserving log(x) for base 10. I assume they were all published post-1980 as well. Just based on this anecdotal evidence, I'd be hesitant to say that either way is either "right" or "wrong," choosing instead to say that there are multiple equally valid notations (as is often the case, I've found).

I agree with you: the is no right or wrong notation. But what is the harm in going for consistency? It seems that it is always the schools of engineering as well as departments of chemistry that raise the most objections to that new convention. I know one case where the chair of mathematics became the academic provost. When faced with their objections, she asked then to yield to a higher intelligence.

BTW: When did you first see absolute value used with integration??

 

[ksdhart2]

I agree with you: the is no right or wrong notation. But what is the harm in going for consistency? It seems that it is always the schools of engineering as well as departments of chemistry that raise the most objections to that new convention. I know one case where the chair of mathematics became the academic provost. When faced with their objections, she asked then to yield to a higher intelligence.

BTW: When did you first see absolute value used with integration??

It sure would be nice if we had a consistent, standardized notation. In high school, I saw ln(x) used in my math textbooks, but then I kind of had to relearn it as log(x) because Javascript and other programming languages use some variant of Math.log(x). Then I went back to college and my textbooks once again used ln(x). Go figure :/

As for the absolute values in integration, my Calc textbook used them, as in \(\displaystyle \displaystyle \int tan(x)dx=-log|cos(x)|+C\). I don't actually know if they're strictly necessary or not though, because WolframAlpha doesn't use them.

 

[lookagain]

You really should read the thread before interfering.

Stop making wrong presumptions about me reading the thread. I didn't "interfere," I corrected you.
The OP's book answer is wrong for not having the absolute value bars.

Also it has been the consensus of even mathed group to use \(\displaystyle \log(x)\)
since the mid-1980's. You are behind the times.

Your allegation is irrelevant here. All I see is you becoming unglued and throwing out
unsubstantiated "facts" to try to bolster your post that I amended.

But what is the harm in going for consistency?

There is no consistency, so your comment makes no sense.

BTW: When did you first see >> absolute value used with integration < < ??

You need to stop spreading ignorance about this to students.


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absolute value used with integration:
http://math2.org/math/integrals/tableof.htm
https://www.reference.com/math/integral-tan-x-53efea2c4c44dac8


"Notational conventions" and further down there is the integral of tan(x):
https://en.wikipedia.org/wiki/Natural_logarithm


* * And all of these sources readily use "ln." The last states it's not exclusive.