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Question Number 166437 by mnjuly1970 last updated on 20/Feb/22

        calculate   Ω=∫_0 ^( 1) ((  ln(1−x ).ln(1+ x ))/x^( 2) ) dx=?        −−−−−−−

$$ \\ $$$$\:\:\:\:\:\:{calculate}\: \\ $$$$\Omega=\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{\:\:{ln}\left(\mathrm{1}−{x}\:\right).{ln}\left(\mathrm{1}+\:{x}\:\right)}{{x}^{\:\mathrm{2}} }\:{dx}=? \\ $$$$\:\:\:\:\:\:−−−−−−− \\ $$

Answered by qaz last updated on 21/Feb/22

∫_0 ^1 ((ln(1−x)ln(1+x))/x^2 )dx  =∫_0 ^1 ((1/x)−1)(((ln(1+x))/(x−1))+((ln(1−x))/(1+x)))dx  =−∫_0 ^1 ((ln(1+x))/x)dx+∫_0 ^1 ((ln(1−x))/x)dx−2∫_0 ^1 ((ln(1−x))/(1+x))dx  =−(1/(12))π^2 +∫_0 ^1 ((ln(1−x))/x)dx−2∫_1 ^2 ((ln(2−x))/x)dx  =−(1/(12))π^2 +∫_0 ^1 ((ln(1−x))/x)dx−2∫_(1/2) ^1 ((ln2+ln(1−x))/x)dx  =−(1/(12))π^2 −2ln^2 2+Li_2 (1)−2Li_2 ((1/2))  =−(1/(12))π^2 −3ln^2 2

$$\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\mathrm{ln}\left(\mathrm{1}−\mathrm{x}\right)\mathrm{ln}\left(\mathrm{1}+\mathrm{x}\right)}{\mathrm{x}^{\mathrm{2}} }\mathrm{dx} \\ $$$$=\int_{\mathrm{0}} ^{\mathrm{1}} \left(\frac{\mathrm{1}}{\mathrm{x}}−\mathrm{1}\right)\left(\frac{\mathrm{ln}\left(\mathrm{1}+\mathrm{x}\right)}{\mathrm{x}−\mathrm{1}}+\frac{\mathrm{ln}\left(\mathrm{1}−\mathrm{x}\right)}{\mathrm{1}+\mathrm{x}}\right)\mathrm{dx} \\ $$$$=−\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\mathrm{ln}\left(\mathrm{1}+\mathrm{x}\right)}{\mathrm{x}}\mathrm{dx}+\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\mathrm{ln}\left(\mathrm{1}−\mathrm{x}\right)}{\mathrm{x}}\mathrm{dx}−\mathrm{2}\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\mathrm{ln}\left(\mathrm{1}−\mathrm{x}\right)}{\mathrm{1}+\mathrm{x}}\mathrm{dx} \\ $$$$=−\frac{\mathrm{1}}{\mathrm{12}}\pi^{\mathrm{2}} +\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\mathrm{ln}\left(\mathrm{1}−\mathrm{x}\right)}{\mathrm{x}}\mathrm{dx}−\mathrm{2}\int_{\mathrm{1}} ^{\mathrm{2}} \frac{\mathrm{ln}\left(\mathrm{2}−\mathrm{x}\right)}{\mathrm{x}}\mathrm{dx} \\ $$$$=−\frac{\mathrm{1}}{\mathrm{12}}\pi^{\mathrm{2}} +\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\mathrm{ln}\left(\mathrm{1}−\mathrm{x}\right)}{\mathrm{x}}\mathrm{dx}−\mathrm{2}\int_{\mathrm{1}/\mathrm{2}} ^{\mathrm{1}} \frac{\mathrm{ln2}+\mathrm{ln}\left(\mathrm{1}−\mathrm{x}\right)}{\mathrm{x}}\mathrm{dx} \\ $$$$=−\frac{\mathrm{1}}{\mathrm{12}}\pi^{\mathrm{2}} −\mathrm{2ln}^{\mathrm{2}} \mathrm{2}+\mathrm{Li}_{\mathrm{2}} \left(\mathrm{1}\right)−\mathrm{2Li}_{\mathrm{2}} \left(\frac{\mathrm{1}}{\mathrm{2}}\right) \\ $$$$=−\frac{\mathrm{1}}{\mathrm{12}}\pi^{\mathrm{2}} −\mathrm{3ln}^{\mathrm{2}} \mathrm{2} \\ $$

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