# $\int \frac{\tan x}{x} dx$

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

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

• 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
• 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? – bryanblackbee Apr 15 '13 at 17:25
• 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

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