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Tolaso J Kos · #1

Δειχθήτω
$\displaystyle{\bigintsss_{0}^{\pi/2} \left ( \frac{\log \left ( \tan x +1 \right )}{\log \tan x} - \frac{{\rm Li}_2 \left ( -\cot x \right )}{\log^2 \cot x} - \frac{\zeta(2)}{2 \log^2 \tan x} \right ) \, {\rm d}x = \frac{3 \pi}{8}}$ (Cornel Ioan Valean)

pprime · #2

Tolaso J Kos έγραψε:Δειχθήτω
$\displaystyle{\bigintsss_{0}^{\pi/2} \left ( \frac{\log \left ( \tan x +1 \right )}{\log \tan x} - \frac{{\rm Li}_2 \left ( -\cot x \right )}{\log^2 \cot x} - \frac{\zeta(2)}{2 \log^2 \tan x} \right ) \, {\rm d}x = \frac{3 \pi}{8}}$ (Cornel Ioan Valean)

$\displaystyle{I=\int\limits_{0}^{\frac{\pi }{2}}{\left( \frac{\log \left( 1+\tan x \right)}{\log \tan x}-\frac{Li_{2}\left( -\cot x \right)}{\log ^{2}\cot x}-\frac{\zeta \left( 2 \right)}{2\log ^{2}\tan x} \right)dx}}$

$\displaystyle{=\int\limits_{0}^{\frac{\pi }{4}}{\left( \frac{\log \left( 1+\tan x \right)}{\log \tan x}-\frac{Li_{2}\left( -\cot x \right)}{\log ^{2}\cot x}-\frac{\zeta \left( 2 \right)}{2\log ^{2}\tan x} \right)dx}+\underbrace{\int\limits_{\frac{\pi }{4}}^{\frac{\pi }{2}}{\left( \frac{\log \left( 1+\tan x \right)}{\log \tan x}-\frac{Li_{2}\left( -\cot x \right)}{\log ^{2}\cot x}-\frac{\zeta \left( 2 \right)}{2\log ^{2}\tan x} \right)dx}}_{x=\frac{\pi }{2}-x}}$

$\displaystyle{=\int\limits_{0}^{\frac{\pi }{4}}{\left( \frac{\log \left( 1+\tan x \right)-\log \left( 1+\cot x \right)}{\log \tan x}-\frac{Li_{2}\left( -\tan x \right)+Li_{2}\left( -\cot x \right)+\zeta \left( 2 \right)}{\log ^{2}\tan x} \right)dx}}$

$\displaystyle{=\int\limits_{0}^{\frac{\pi }{4}}{\left( \frac{\log \left( 1+\tan x \right)-\log \left( 1+\cot x \right)}{\log \tan x}-\frac{-\frac{1}{2}\log ^{2}\tan x}{\log ^{2}\tan x} \right)dx}}$

$\displaystyle{=\int\limits_{0}^{\frac{\pi }{4}}{\left( \frac{\log \left( 1+\tan x \right)-\log \left( 1+\cot x \right)}{\log \tan x}+\frac{1}{2} \right)dx}=\int\limits_{0}^{\frac{\pi }{4}}{\left( \frac{\log \left( \frac{1+\tan x}{1+\cot x} \right)}{\log \tan x}+\frac{1}{2} \right)dx}}$

$\displaystyle{=\int\limits_{0}^{\frac{\pi }{4}}{\left( \frac{\log \left( \tan x\frac{\cos x+\sin x}{\sin x+\cos x} \right)}{\log \tan x}+\frac{1}{2} \right)dx}=\int\limits_{0}^{\frac{\pi }{4}}{\left( 1+\frac{1}{2} \right)dx}=\frac{3}{2}\cdot \frac{\pi }{4}=\frac{3\pi }{8}}$

Tolaso J Kos · #3

Excellent pprime !!!

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