Question Number 141346 by mnjuly1970 last updated on 17/May/21
Answered by qaz last updated on 17/May/21
$$\int_{\mathrm{0}} ^{\mathrm{1}} \frac{{ln}^{\mathrm{2}} {x}}{\mathrm{1}−{x}^{\mathrm{2}} }{dx} \\ $$$$=\int_{\mathrm{0}} ^{\infty} \frac{{y}^{\mathrm{2}} {e}^{−{y}} }{\mathrm{1}−{e}^{−\mathrm{2}{y}} }{dy} \\ $$$$=\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\int_{\mathrm{0}} ^{\infty} {y}^{\mathrm{2}} {e}^{−\left(\mathrm{2}{n}+\mathrm{1}\right){y}} {dy} \\ $$$$=\mathrm{2}\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{\left(\mathrm{2}{n}+\mathrm{1}\right)^{\mathrm{3}} } \\ $$$$=\mathrm{2}\left(\mathrm{1}−\mathrm{2}^{−\mathrm{3}} \right)\zeta\left(\mathrm{3}\right) \\ $$$$=\frac{\mathrm{7}}{\mathrm{4}}\zeta\left(\mathrm{3}\right) \\ $$
Commented by mnjuly1970 last updated on 18/May/21
$$\:{thanks}\:{alot}\:... \\ $$