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Evaluate the integral: $$\int \frac{\tan x}{x} dx$$

I tried integration by parts, got stuck. Ideas/ suggestions please.

Martin Sleziak
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dajoker
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    i misread and thought it said integrate tan x :S – Lost1 Apr 15 '13 at 17:13
  • Are you interested in an antiderivative or a definite integral on some particular interval? – Umberto P. Apr 15 '13 at 17:16
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    I don't know how to find an elementary function whose derivative is $(\tan x)/x$. Suspect there isn't one. – André Nicolas Apr 15 '13 at 17:16
  • mind sharing your by parts attempt? – bryan.blackbee Apr 15 '13 at 17:25
  • Seconding that there is probably no such antiderivative, based on Mathematica. @Lost1 - that answer is wrong as well as badly written! – not all wrong Apr 15 '13 at 18:06
  • @Sharkos I didnt read it, a closed form doesn't exist right? a min effort question deserves a min effort answer :P – Lost1 Apr 15 '13 at 18:08
  • @Lost1 - "there is probably no such antiderivative" – not all wrong Apr 15 '13 at 18:09
  • have you tried expressing sin and cos as a rational function? Usually this is done by the strange substitution $t=\tan(x/2)$. There is an article on that on the german wikipedia: http://de.wikipedia.org/wiki/Generalsubstitution Basically it says that from the prior choice of $t$ the results for sin and cos follow. – Quickbeam2k1 Apr 15 '13 at 18:43
  • @AndréNicolas,@Sharkos, I agree, may be there's no such anti-derivative. I thought there might be something like a tan integral just like the sine integral. Even wolfram alpha doesn't answer this one! – dajoker Apr 16 '13 at 08:33
  • @bryansis2010, we (Vijay, me and other people) had already tried by parts, it didn't work out. The problem in that was no matter what you choose as 'u' or 'v', you get another complicated function to be integrated. – Parth Thakkar Apr 16 '13 at 14:36
  • @Sharkos, is there any way to differentiate integrable and non-integrable functions? (pun intended! :D) – Parth Thakkar Apr 16 '13 at 15:11
  • @Parth I'm not sure what you mean at all. Integrability doesn't tell you much about differentiability, though non-integrability implies discontinuity which implies non differentiability at least somewhere. What does this have to do with this problem? $tan x/x$ is definitely integrable away from the tan singularities. – not all wrong Apr 16 '13 at 16:31
  • @Vijay Well you can certainly define the integral from 0 to $x \in (-\pi/2,\pi/2)$ to be a special function P: Whether it has a relationship to any other special functions I don't know, though Mathematica is pretty good at finding these. – not all wrong Apr 16 '13 at 16:33
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    Gee, according to the "best answer" on Yahoo!Answers, it's $\ln x \cdot \tan x$ (there is a comment that this gives the wrong derivative...) Since $\int \frac{\sin x}{x} dx$ and $\int \frac{\cos x}{x} dx$ are the Fresnel integrals, which don't have anti-derivative functions, it probably isn't surprising that this one doesn't either. (Free WolframAlpha just "times out" on it and offers to give you more computation time with a Pro membership...) If you'll be satisfied with a power series, you could divide the Maclaurin series for $\tan x$ by $x$ and integrate the general term. – colormegone Apr 16 '13 at 22:14
  • @Sharkos, I think you got me wrong. What I meant was is there any way in which we can differentiate between (i ate up this word, sorry!!) integrable and non-integrable functions - that is, can we tell whether a function will have an anti-derivative or not? – Parth Thakkar Apr 17 '13 at 15:09
  • @Parth - ah, right! I've posted an answer with a little context in case you're interested. – not all wrong Apr 17 '13 at 15:51
  • Related: Check if antiderivative is elementary. – Martin Sleziak Jan 01 '20 at 11:14

2 Answers2

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There is no solution, based on this link.

Glorfindel
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bryan.blackbee
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2

If you want to prove things like the inexpressibility of various integrands in terms of some set of functions, you want to look at Liouville's theorem and possibly some differential Galois theory. Check out https://mathoverflow.net/questions/58966/solvability-in-differential-galois-theory for some reading in this area.

As others have said, it is likely no elementary integral exists, since this holds for similar integrals like $\int \sin x/x$.

Community
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not all wrong
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  • guess what? I can't understand a thing that's written on that page!! Probably something that I'll learn in college...! Seriously, too much to digest. Moral of the story: take it that tanx/x is not integrable. :D – Parth Thakkar Apr 17 '13 at 16:11
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    Fair enough! It's not in my area of expertise, and not something that's taught very often at all, since it's not of great interest to most people! I think of it as a curiosity. It's remarkable to me that there is any theory in this area at all. It is of most interest to people writing software like Mathematica I think! – not all wrong Apr 17 '13 at 17:36
  • Since almost anyone likely to be working with applications that require integration has ready access to significant amounts of computation time nowadays, they tend to simply use a numerical integrator and not even worry about whether the integrand has an anti-derivative function in closed form. Even in the early days of electronic computation, when "time" was expensive, it was preferable to be able to use the Fundamental Theorem of Calculus, rather than even compute terms in a power series. Now, you'll see some people use computers even when an anti-derivative is known... – colormegone Apr 18 '13 at 03:11
  • That isn't really completely true. When I've done some many-parameter computations and been interested in special cases, it would have been vastly preferable to have a closed form in terms of functions I understand. – not all wrong Apr 18 '13 at 09:35