Question and Answers Forum

All Questions      Topic List

Differentiation Questions

Previous in All Question      Next in All Question      

Previous in Differentiation      Next in Differentiation      

Question Number 164103 by mnjuly1970 last updated on 14/Jan/22

     prove that        Ω=∫_0 ^( 1) ln(((1+x)/(1−x)) ).(dx/(x (√( 1−x^( 2) )))) = (π^( 2) /2)       −− m.n−−

$$ \\ $$$$\:\:\:{prove}\:{that} \\ $$$$\: \\ $$$$\:\:\:\Omega=\int_{\mathrm{0}} ^{\:\mathrm{1}} \mathrm{ln}\left(\frac{\mathrm{1}+{x}}{\mathrm{1}−{x}}\:\right).\frac{{dx}}{{x}\:\sqrt{\:\mathrm{1}−{x}^{\:\mathrm{2}} }}\:=\:\frac{\pi^{\:\mathrm{2}} }{\mathrm{2}} \\ $$$$\:\:\:\:\:−−\:{m}.{n}−− \\ $$$$ \\ $$

Answered by Lordose last updated on 14/Jan/22

Ω =^(x=((1−x)/(1+x))) 2∫_0 ^( 1) ((ln((1/x)))/((1+x)^2 (((1−x)/(1+x))(√(1−(((1−x)/(1+x)))^2 ))))dx  Ω = ∫_0 ^( 1) ((ln((1/x)))/( (√x)(1−x)))dx =^(x=x^2 ) 4∫_0 ^( 1) ((ln((1/x)))/((1−x^2 )))dx  Ω = −4Σ_(k=1) ^∞ ∫_0 ^( 1) x^(2k) ln(x)dx =^(IBP) 4Σ_(k=1) ^∞ (1/((2k+1)^2 ))  Ω = Σ_(k=1) ^∞ (1/((k+(1/2))^2 ))dx = 𝛙^((1)) ((1/2))  𝛀 = (𝛑^2 /2) ▲▲▲

$$\Omega\:\overset{\mathrm{x}=\frac{\mathrm{1}−\mathrm{x}}{\mathrm{1}+\mathrm{x}}} {=}\mathrm{2}\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{\mathrm{ln}\left(\frac{\mathrm{1}}{\mathrm{x}}\right)}{\left(\mathrm{1}+\mathrm{x}\right)^{\mathrm{2}} \left(\frac{\mathrm{1}−\mathrm{x}}{\mathrm{1}+\mathrm{x}}\sqrt{\mathrm{1}−\left(\frac{\mathrm{1}−\mathrm{x}}{\mathrm{1}+\mathrm{x}}\right)^{\mathrm{2}} }\right.}\mathrm{dx} \\ $$$$\Omega\:=\:\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{\mathrm{ln}\left(\frac{\mathrm{1}}{\mathrm{x}}\right)}{\:\sqrt{\mathrm{x}}\left(\mathrm{1}−\mathrm{x}\right)}\mathrm{dx}\:\overset{\mathrm{x}=\mathrm{x}^{\mathrm{2}} } {=}\mathrm{4}\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{\mathrm{ln}\left(\frac{\mathrm{1}}{\mathrm{x}}\right)}{\left(\mathrm{1}−\mathrm{x}^{\mathrm{2}} \right)}\mathrm{dx} \\ $$$$\Omega\:=\:−\mathrm{4}\underset{\mathrm{k}=\mathrm{1}} {\overset{\infty} {\sum}}\int_{\mathrm{0}} ^{\:\mathrm{1}} \mathrm{x}^{\mathrm{2k}} \mathrm{ln}\left(\mathrm{x}\right)\mathrm{dx}\:\overset{\boldsymbol{\mathrm{IBP}}} {=}\mathrm{4}\underset{\mathrm{k}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{\left(\mathrm{2k}+\mathrm{1}\right)^{\mathrm{2}} } \\ $$$$\Omega\:=\:\underset{\mathrm{k}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{\left(\mathrm{k}+\frac{\mathrm{1}}{\mathrm{2}}\right)^{\mathrm{2}} }\mathrm{dx}\:=\:\boldsymbol{\psi}^{\left(\mathrm{1}\right)} \left(\frac{\mathrm{1}}{\mathrm{2}}\right) \\ $$$$\boldsymbol{\Omega}\:=\:\frac{\boldsymbol{\pi}^{\mathrm{2}} }{\mathrm{2}}\:\blacktriangle\blacktriangle\blacktriangle \\ $$

Commented by mnjuly1970 last updated on 14/Jan/22

mercey

$${mercey} \\ $$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com