Question Number 141623 by qaz last updated on 21/May/21
$$\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\left(\int_{\mathrm{0}} ^{\mathrm{1}} \frac{{x}^{{n}} }{\mathrm{1}+{x}}{dx}\right)^{\mathrm{2}} ={ln}\:\mathrm{2} \\ $$
Answered by mindispower last updated on 21/May/21
$$\left(\int_{\mathrm{0}} ^{\mathrm{1}} \frac{{x}^{{n}} }{\mathrm{1}+{x}}{dx}\right)^{\mathrm{2}} =\int_{\mathrm{0}} ^{\mathrm{1}} \int_{\mathrm{0}} ^{\mathrm{1}} \frac{\left({xy}\right)^{{n}} }{\left(\mathrm{1}+{x}\right)\left(\mathrm{1}+{y}\right)}{dxdy} \\ $$$$\Leftrightarrow\underset{{n}\geqslant\mathrm{0}} {\sum}\int\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\left({xy}\right)^{{n}} }{\left(\mathrm{1}+{x}\right)\left(\mathrm{1}+{y}\right)}{dxdy} \\ $$$$=\int\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\mathrm{1}}{\left(\mathrm{1}+{x}\right)\left(\mathrm{1}+{y}\right)\left(\mathrm{1}−{xy}\right)}{dxdy} \\ $$$${S}=\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\mathrm{1}}{\left(\mathrm{1}+{y}\right)}\int_{\mathrm{0}} ^{\mathrm{1}} \frac{{dx}}{\left(\mathrm{1}+{x}\right)\left(\mathrm{1}−{xy}\right)}{dy} \\ $$$$\int_{\mathrm{0}} ^{\mathrm{1}} \frac{{dx}}{\left(\mathrm{1}+{x}\right)\left(\mathrm{1}−{xy}\right)}=\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\mathrm{1}}{\left(\mathrm{1}+{y}\right)}.\frac{\mathrm{1}}{\mathrm{1}+{x}}+\frac{{y}}{\mathrm{1}+{y}}.\frac{\mathrm{1}}{\mathrm{1}−{xy}}{dx} \\ $$$$=\frac{{ln}\left(\mathrm{2}\right)}{\mathrm{1}+{y}}−\frac{\mathrm{1}}{\mathrm{1}+{y}}{ln}\left(\mathrm{1}−{y}\right) \\ $$$${S}=\int_{\mathrm{0}} ^{\mathrm{1}} \frac{{ln}\left(\mathrm{2}\right)}{\left(\mathrm{1}+{y}\right)^{\mathrm{2}} }−\frac{\mathrm{1}}{\left(\mathrm{1}+{y}\right)^{\mathrm{2}} }{ln}\left(\mathrm{1}−{y}\right){dy} \\ $$$$\left.=\left[−\frac{{ln}\left(\mathrm{2}\right)}{\mathrm{1}+{y}}\right]_{\mathrm{0}} ^{\mathrm{1}} +\underset{{x}\rightarrow\mathrm{1}} {\mathrm{lim}}\frac{{ln}\left(\mathrm{1}−{y}\right)}{\mathrm{1}+{y}}\right]_{\mathrm{0}} ^{{x}} +\int_{\mathrm{0}} ^{{x}} \frac{\mathrm{1}}{\left(\mathrm{1}+{y}\right)\left(\mathrm{1}−{y}\right)}{dy} \\ $$$$=\frac{{ln}\left(\mathrm{2}\right)}{\mathrm{2}}+\underset{{x}\rightarrow\mathrm{1}} {\mathrm{lim}}\frac{{ln}\left(\mathrm{1}−{x}\right)}{\mathrm{1}+{x}}+\frac{\mathrm{1}}{\mathrm{2}}{ln}\left(\frac{\mathrm{1}+{x}}{\mathrm{1}−{x}}\right) \\ $$$$=\frac{{ln}\left(\mathrm{2}\right)}{\mathrm{2}}+\underset{{x}\rightarrow\mathrm{1}} {\mathrm{lim}}\frac{{ln}\left(\mathrm{1}+{x}\right)}{\mathrm{2}}+{ln}\left(\mathrm{1}−{x}\right)\left\{\frac{\mathrm{1}}{\mathrm{1}+{x}}−\frac{\mathrm{1}}{\mathrm{2}}\right\} \\ $$$$={ln}\left(\mathrm{2}\right)+\underset{{x}\rightarrow\mathrm{1}} {\mathrm{lim}}\frac{\left(\mathrm{1}−{x}\right){ln}\left(\mathrm{1}−{x}\right)}{\mathrm{2}\left(\mathrm{1}+{x}\right)}={ln}\left(\mathrm{2}\right) \\ $$$$\Leftrightarrow\underset{{n}\geqslant\mathrm{0}} {\sum}\left(\int_{\mathrm{0}} ^{\mathrm{1}} \frac{{x}^{{n}} }{\mathrm{1}+{x}}{dx}\right)^{\mathrm{2}} ={ln}\left(\mathrm{2}\right) \\ $$
Commented by qaz last updated on 22/May/21
$${thanks}\:{sir}\:{power} \\ $$