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How can I integrate $$\int_{0}^{\large\pi}\ln (\tan x)\,\text{dx}\;\;\;?$$

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Is your function defined on the interval $[0,\pi]$ ? –  Lubin Jan 24 '13 at 15:29
    
$\ln(\tan(x))$ isn't defined for $x \in [\pi/2,\pi]$. –  Raskolnikov Jan 24 '13 at 15:29
4  
What do you guys mean, $\ln(\tan(x))$ is defined for $ \frac{\pi}{2} < x< \pi$, but it is not real-valued there. –  Sasha Jan 24 '13 at 15:31
    
the value is roughly 4.93i and the indefinite integral is possible but very difficult. It involves the polylogarithmic function. I am looking for an easier way. –  kaine Jan 24 '13 at 15:37
    
I wonder if the integration limits are correct. –  Chris's sis Jan 24 '13 at 15:49

2 Answers 2

up vote 6 down vote accepted

Since $\tan\left(\pi -x\right) = - \tan(x)$, and $\tan(x)$ is positive for $0<x<\frac{\pi}{2}$ we have: $$ \int_0^\pi \ln\left(\tan x\right) \mathrm{d}x = i \pi \int_0^{\pi/2} \mathrm{d} x + 2 \int_0^{\pi/2} \ln\left(\tan x\right) \mathrm{d}x $$ Furthermore, using $\tan\left(\frac{\pi}{2}-x\right) = \frac{1}{\tan(x)}$ we see that $$\begin{eqnarray} 2 \int_0^{\pi/2} \ln\left(\tan x\right) \mathrm{d}x &=& \int_0^{\pi/2} \log\left(\tan x \right) \mathrm{d} x + \int_0^{\pi/2} \log\left(\tan y \right) \mathrm{d} y \\ &\stackrel{y=\pi/2 - x}{=}& \int_0^{\pi/2} \log\left(\tan x \right) \mathrm{d} x + \int_0^{\pi/2} \log\left(\frac{1}{\tan x} \right) \mathrm{d} x = 0 \end{eqnarray} $$ Thus $$ \int_0^\pi \ln\left(\tan x\right) \mathrm{d}x = i \frac{\pi^2}{2} $$

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just curious, is it correct to solve it by noticing that $\log(\tan(x)) =\log ( \sin(x)) - \log( \cos(x)) = \Im(\log e^{ix})- \Re(\log e^{ix})= \Im(ix)- \Re(ix)$ and then integrating over $x \in [0, \pi]$? –  Alex Jan 24 '13 at 15:48
    
@Alex No, because $\log(\sin(x)) = \log(\Im \mathrm{e}^{ix}) \not= \Im\left(\log \mathrm{e}^{i x}\right)$. –  Sasha Jan 24 '13 at 15:58
    
Thanks, now I see. –  Alex Jan 24 '13 at 16:03
    
Thanks Sasha but i did not understand the line $\displaystyle\int_0^\pi \ln\left(\tan x\right) \mathrm{d}x = i \pi \int_0^{\pi/2} \mathrm{d} x + 2 \int_0^{\pi/2} \ln\left(\tan x\right) \mathrm{d}x$ –  juantheron Jan 25 '13 at 18:44
    
@juantheron $ \int_0^\pi \ln(\tan(x)) \mathrm{d}x = \int_0^{\pi/2} \ln(\tan(x)) \mathrm{d}x + \int_{\pi/2}^\pi \ln(\tan(y)) \mathrm{d}y$. Then change variables $y=\pi - x$, use $\tan\left(\pi - x\right) = -\tan(x)$. Now, since $\tan(x)>0$ for $0<x<\pi/2$, use $\ln(-\tan(x)) = i \pi + \ln(\tan(x))$. –  Sasha Jan 25 '13 at 19:14

If looking for the real value only, it's enough to consider

$$I=\int_{0}^{\pi/2}\ln (\tan x)dx$$ and let $x=\pi/2-y$ $$I=\int_{0}^{\pi/2}\ln (\tan x)dx=\int_{0}^{\pi/2}\ln (\cot x)dx$$ $$2I=\int_{0}^{\pi/2}\ln (1)dx$$ $$I=0$$

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