# $\int x\tan x$ and the Clausen Function

I have been attempting to evaluate $\int x \tan x \;\mathrm{d} x$.

My first instinct was integration by parts, which produces $-x \ln|\cos x|+\int \ln|\cos x| \;\mathrm{d} x$.

I have read online that the order 2 Clausen function can be used to complete the integration, but the Clausen function applies to the standard logarithm function, and I am unclear on how it extends to the natural logarithm. An explanation of this and a guide as to how I can finish the integral would be greatly appreciated.

• I think the log in the Clausen function is not the $10$-fingered people logarithm, but the natural logarithm. Anyway, the two differ by a constant factor. – André Nicolas Jun 26 '14 at 2:36

The $\log$ function in the order 2 Clausen function is the natural logarithm.

$$\mathrm{Cl}_2(\phi) = -\int\limits_{0}^{\phi} \log_e |2 \sin \frac{x}{2}| \mathrm{d}x$$

Logarithms in calculus expressions are always base $e$, rather than base $10$, unless explicitly stated otherwise.  The subscript is often omitted by lazy authors because this convention is so generally understood.

(Though the whole point of having the $\ln$ function was to avoid such confusion.)

• Is there any reason to use base 10 logarithms at all anymore? :P – user41281 Jun 26 '14 at 3:04
• Thank you. I had no clue. – Gotthold Jun 26 '14 at 3:04
• @user41281 Well, it has applications in chemistry. – Graham Kemp Jun 26 '14 at 3:06
• @GrahamKemp Aren't those instances entirely contrived? If I recall correctly, they go out of their way to change a naturally occuring natural logarithm into $2.3\log_{10}(x)$. (Actually, I just thought of the decibel scale, which is a use I suppose). – user41281 Jun 26 '14 at 3:08
• Technically all logarithms may be useful in manmade formulae... However, e^x is the perfect curve such that it is it's own derivative and antiderivative. So... Id imagine it comes up in nature often. – The Great Duck Jul 1 '16 at 4:37