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

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

  • 1
    $\begingroup$ i misread and thought it said integrate tan x :S $\endgroup$ – Lost1 Apr 15 '13 at 17:13
  • $\begingroup$ Are you interested in an antiderivative or a definite integral on some particular interval? $\endgroup$ – Umberto P. Apr 15 '13 at 17:16
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    $\begingroup$ I don't know how to find an elementary function whose derivative is $(\tan x)/x$. Suspect there isn't one. $\endgroup$ – André Nicolas Apr 15 '13 at 17:16
  • $\begingroup$ mind sharing your by parts attempt? $\endgroup$ – bryanblackbee Apr 15 '13 at 17:25
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    $\begingroup$ 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. $\endgroup$ – colormegone Apr 16 '13 at 22:14

There is no solution, based on this link.


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 $\int \sin x/x$.

  • $\begingroup$ 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 $\endgroup$ – Parth Thakkar Apr 17 '13 at 16:11
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    $\begingroup$ 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! $\endgroup$ – Sharkos Apr 17 '13 at 17:36
  • $\begingroup$ 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... $\endgroup$ – colormegone Apr 18 '13 at 3:11
  • $\begingroup$ 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. $\endgroup$ – Sharkos Apr 18 '13 at 9:35

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