Hard Light Productions Forums
Off-Topic Discussion => General Discussion => Topic started by: Kosh on December 16, 2008, 12:58:36 am
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ln(tan(x)) dx
I'm not even sure where to start with this. Anyone got some pointers so I can get going on it? Thanks.
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Well, you'll have to use the Chain Rule, I'm pretty sure.
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the chain rule is for derivatives, this is an integration problem.
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Where did you get this problem - it looks ugly.
Try mathematica:
]&random=false]http://integrals.wolfram.com/index.jsp?expr=Log[+Tan- ]&random=false[/url]
It is ugly. :eek2:
(http://integrals.wolfram.com/index.jsp?expr=Log[+Tan[x)
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We just did integrals in our final Maths B test...
...but that was only with Logs of base 10. What is this for, exactly?
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You can't express this in terms of elementary functions. There are various additional, standard functions that allow you to write things like this in closed form. My guess is it would involve a polylog or Lerch function somewhere, just from looking at it.
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We just did integrals in our final Maths B test...
...but that was only with Logs of base 10. What is this for, exactly?
You sure it wasn't base e - these are much easier.
It's been too long since my last maths units....
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No, we did logs of base 10. Although we just did simple stuff. Like log xn = n log x and so forth. And most of the time on the calculator. We haven't done ln (base e) yet. Knowing some of the people in my class, I'm sure they had trouble with these logs, which, on our scientific calculators, can ONLY do base 10 logs. ;)
This is Maths B in Queensland, and we haven't got a good reputation when it comes to smartness IIRC. :hopping:
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Just wait 'til you see the new WA curriculum :)
I thought you meant integrals of log10
Don't forget that ln x = log(x)/log(e)... Which scientific calculator do you have?
Keep up the good work - I'm originally from Queensland!
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A Casio fx-82MS.
And thank you, the same to you!
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We just did integrals in our final Maths B test...
...but that was only with Logs of base 10. What is this for, exactly?
Calculus homework, and it isn't log(tanx), it's ln(tanx).
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That's still just as ugly. Mathematica uses Log to mean the natural logarithm (as opposed to using ln(x)). Still Ugly.
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yeah I know.... :doubt:
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I think you'd have to use integration by parts, not sure though.
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Undoubtedly, but even then that thing is a *****. Hopefully solving that problem is not a requirement for passing the course.
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Integration by parts isn't suitable for this question since you've got f(g(x)), not f(x)g'(x).
I think CP5670 summed it up best:
You can't express this in terms of elementary functions. There are various additional, standard functions that allow you to write things like this in closed form. My guess is it would involve a polylog or Lerch function somewhere, just from looking at it.
I've got no idea what he's on about (I really haven't done enough maths... and I can't remember all of it), but mathematica seems to spit out an answer along those lines (you gotta trust somebody :p)
Agreed with hoping that that's not a requirement for your course.
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Integration by parts isn't suitable for this question since you've got f(g(x)), not f(x)g'(x).
Whoops. Correct you are. Pardon our confusion.
Edit: Hmm. I consulted The Holy BookTM and started to work on the following (if I'm spewing out lies, hit me with a stick or something). If we have g(f(x)), deriving it would be Dg(f(x)) = g'(f(x))f'(x). From there, we could extrapolate that
g(f(x)) = ∫ g ' (f(x)) f ' (x) dx
, where f(x) = tan(x) and g(x) = lnx. That way we'd get
ln(tan(x)) = ∫(1 / tanx) * (1 + tan2(x))dx
or something. Brainfreeze.
Edit: Damn, ignore that. Dunno what I was thinking.
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Integration by parts IS suitable here, I think.
What I meant is that (ln(tanx)) is the f(x), and you have to introduce a g'(x) equal to 1. Then start integrating by parts.
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Skip it.
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Are you supposed to find the value of this integral in some interval or actually get an antiderivative? I'm guessing it's the former, since as I said earlier the latter cannot be done using only elementary functions.
That's still just as ugly. Mathematica uses Log to mean the natural logarithm (as opposed to using ln(x)). Still Ugly.
It's conventionally written as log in pretty much all math beyond the elementary level. They only differ by a constant factor anyway.
I've got no idea what he's on about (I really haven't done enough maths... and I can't remember all of it), but mathematica seems to spit out an answer along those lines (you gotta trust somebody )
Yeah, I just put it in there and it does involve a polylog as I expected. This (http://en.wikipedia.org/wiki/Polylog) is the thing I am referring to.
This is one of numerous "special functions" that can't be expressed using the standard, elementary functions. They come up a lot in various places though and are useful enough to be considered as basic objects in themselves.
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Arrived in slightly different special function, maybe someone could explain why?
And how the hell you can put up those neato integrate signs here?
Integrate:
ln(tan x) dx || Substituted tan(x)=u and solved dx = 1/(1+u^2) du
Arrived at:
INT { ln(u)*[1/(1+u^2)] }du, partial integration resulted in
ln(u)*arctan(u) - INT { arctan(u)/u }du, which results in a special function also.
The handbook of mathematical functions, (Abramowitz-Stegun) states that:
INT { arctan(u) / u } du = INT from 0 to INF { exp(-u*t)*si(t) }dt,
where si is a special function also, defined on the same book.
It was probably something like INT from Z to INF { (sin t) / t }dt.
Mika
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Arrived in slightly different special function, maybe someone could explain why?
Many of them are related to each other. There are three or four fundamental classes of functions (hypergeometric, zeta/Lerch and elliptic come to mind) and the others are all basically combinations or special cases of those. The si (sine integral) and polylog are both of the hypergeometric variety.
There are a lot of essentially redundant functions in use since they often came from different contexts. I know si for example is used in speech recognition, even though it's just a slight variant of the arguably more basic ei.
And how the hell you can put up those neato integrate signs here?
You can find it in charmap. Getting the superscripts and subscripts to line up can be annoying though. I think it's clearer to just write something like Integral[ log(tan(x)) , {x,0,pi}].
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This is exactly why I despised integrals so much. Derivatives are such beautiful, elegant things...learn a few simple rules, and you can derive out even the most complex function. But as soon as you get into integration, you realize that the rules are largely useless, as there are gobs of types out there that are flat-out impossible to do. This is about the point in my schooling when I decided to yell a very hearty "**** you!" to math.
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This is exactly why I despised integrals so much. Derivatives are such beautiful, elegant things...learn a few simple rules, and you can derive out even the most complex function. But as soon as you get into integration, you realize that the rules are largely useless, as there are gobs of types out there that are flat-out impossible to do. This is about the point in my schooling when I decided to yell a very hearty "**** you!" to math.
The funny thing is that this is not true in general. It just happens that all the elementary functions (rational functions, exp and log, and combinations of those) have derivatives that can be easily written out.
The most basic function you encounter beyond that setting is the gamma function (factorial), and there you already run into a situation where you have to introduce a new function for every derivative.
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Oh, kind of like: Heaviside(x)'= delta(x) ?
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Yes, that's another example, although the derivative there is only meaningful in a weak/distributional sense.
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ln(tan x) dx || Substituted tan(x)=u and solved dx = 1/(1+u^2) du
I should have figured it was something like that. Thanks.
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No problem. Physicists know these type of integrals by heart.
Mika
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No problem. Physicists know these type of integrals by heart.
Good physicists do. I, unfortunately, do not.
And I suppose it's a good thing I got out of math when I did, CP. I don't need another excuse to dislike its higher workings. :p
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****. I'll have to do this in AP Calculus BC for my Senior year, won't I?
Though, I have done harder things... Such as creating a single formula that calculates the exact horizontal speed that a projectile must have in order to hit a certain point, taking into effect the change in elevation, gravity, actual speed of the projectile, etc (not including air resistance and such).
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Though, I have done harder things... Such as creating a single formula that calculates the exact horizontal speed that a projectile must have in order to hit a certain point, taking into effect the change in elevation, gravity, actual speed of the projectile, etc (not including air resistance and such).
I don't think that is yet too bad. Try solving the equation of the projectile in rotating or accelerating coordinate system. Especially rotating coordinate system (like Earth) introduces several rather interesting forces into the play. Did it once, spherical coordinate system was a must on Earth's case. Also do try showing that the effect of the Coriolis force is neglible when the projetile travels 300 m for example...
Lagrange's formulation of mechanics helps a lot of these analysis. The only inherent problem in it is that it cannot handle friction.
Mika