Question Number 208245 by Shrodinger last updated on 08/Jun/24
$${K}=\int_{\mathrm{0}} ^{\frac{\mathrm{4}}{\pi}} {ln}\left({cosx}\right){dx} \\ $$
Answered by mathzup last updated on 09/Jun/24
$${K}=\int_{\mathrm{0}} ^{\frac{\mathrm{4}}{\pi}} {ln}\left(\frac{{e}^{{ix}} +{e}^{−{ix}} }{\mathrm{2}}\right){dx}\:\:\:{let}\:\lambda=\frac{\mathrm{4}}{\pi} \\ $$$$=\int_{\mathrm{0}} ^{\lambda} \left(\right. \\ $$$$\left.{ln}\left({e}^{{ix}} \right)+{ln}\left(\mathrm{1}+{e}^{−\mathrm{2}{ix}} \right)−{ln}\mathrm{2}\right){dx} \\ $$$$=\int_{\mathrm{0}} ^{\lambda} {ix}\:{dx}+\int_{\mathrm{0}} ^{\lambda} {ln}\left(\mathrm{1}+{e}^{−\mathrm{2}{ix}} \right){dx}−\lambda{ln}\mathrm{2} \\ $$$$={i}\frac{\lambda^{\mathrm{2}} }{\mathrm{2}}−\lambda{ln}\mathrm{2}\:+\int_{\mathrm{0}} ^{\lambda} {ln}\left(\mathrm{1}+{e}^{−\mathrm{2}{ix}} \right){dx} \\ $$$$\left({ln}\left(\mathrm{1}+{z}\right)^{'} =\frac{\mathrm{1}}{\mathrm{1}+{z}}=\sum_{{n}=\mathrm{0}} ^{\infty} \left(−\mathrm{1}\right)^{{n}} {z}^{{n}} \:\Rightarrow\right. \\ $$$${ln}\left(\mathrm{1}+{z}\right)=\sum_{{n}=\mathrm{0}} ^{\infty} \frac{\left(−\mathrm{1}\right)^{{n}} }{{n}+\mathrm{1}}{z}^{{n}+\mathrm{1}} \:+{c}\:\:\left({c}=\mathrm{0}\right) \\ $$$$=\sum_{{n}=\mathrm{1}} ^{\infty} \frac{\left(−\mathrm{1}\right)^{{n}−\mathrm{1}} }{{n}}{z}^{{n}} \:\Rightarrow \\ $$$$\int_{\mathrm{0}} ^{\lambda} {ln}\left(\mathrm{1}+{e}^{−\mathrm{2}{ix}} \right){dx}=\int_{\mathrm{0}} ^{\lambda} \sum_{{n}=\mathrm{1}} ^{\infty} \frac{\left(−\mathrm{1}\right)^{{n}−\mathrm{1}} }{{n}}\:{e}^{−\mathrm{2}{inx}} {dx} \\ $$$$=\sum_{{n}=\mathrm{1}} ^{\infty} \frac{\left(−\mathrm{1}\right)^{{n}−\mathrm{1}} }{{n}}\:\int_{\mathrm{0}} ^{\lambda} \:{e}^{−\mathrm{2}{inx}} {dx} \\ $$$$=\sum_{{n}=\mathrm{1}} ^{\infty} \frac{\left(−\mathrm{1}\right)^{{n}−\mathrm{1}} }{{n}}\left[−\frac{\mathrm{1}}{\mathrm{2}{in}}\:{e}^{−\mathrm{2}{ix}} \right]_{\mathrm{0}} ^{\lambda} \\ $$$$=\frac{{i}}{\mathrm{2}}\sum_{{n}=\mathrm{1}} ^{\infty} \frac{\left(−\mathrm{1}\right)^{{n}−\mathrm{1}} }{{n}^{\mathrm{2}} }\left(\:{e}^{−\mathrm{2}{i}\lambda} −\mathrm{1}\right) \\ $$$$=\frac{{i}}{\mathrm{2}}\sum_{{n}=\mathrm{1}} ^{\infty} \frac{\left(−\mathrm{1}\right)^{{n}−\mathrm{1}} }{{n}^{\mathrm{2}} }\left(\:{e}^{−\mathrm{2}{i}\lambda} −\mathrm{1}\right) \\ $$$$=−\frac{{i}}{\mathrm{2}}\sum_{{n}=\mathrm{1}} ^{\infty} \frac{\left(−\mathrm{1}\right)^{{n}−\mathrm{1}} }{{n}^{\mathrm{2}} } \\ $$$$+\frac{{i}}{\mathrm{2}}\sum_{{n}=\mathrm{1}} ^{\infty} \frac{\left(−\mathrm{1}\right)^{{n}−\mathrm{1}} }{{n}^{\mathrm{2}} }\left({cos}\left(\mathrm{2}\lambda\right)−{isin}\left(\mathrm{2}\lambda\right)\right) \\ $$$${or}\:{I}\:{is}\:{real}\:{so} \\ $$$${I}=−\lambda{ln}\mathrm{2}+\frac{\mathrm{1}}{\mathrm{2}}\sum_{{n}=\mathrm{1}} ^{\infty} \frac{\left(−\mathrm{1}\right)^{{n}−\mathrm{1}} }{{n}^{\mathrm{2}} }{sin}\left(\mathrm{2}\lambda\right) \\ $$$$=−\frac{\mathrm{4}}{\pi}{ln}\mathrm{2}\:+\frac{\mathrm{1}}{\mathrm{2}}\sum_{{n}=\mathrm{1}} ^{\infty} \frac{\left(−\mathrm{1}\right)^{{n}−\mathrm{1}} }{{n}^{\mathrm{2}} }{sin}\left(\frac{\mathrm{8}}{\pi}\right) \\ $$$${rest}\:{to}\:{find}\:{the}\:{value}\:{of}\:{this}\:{serie} \\ $$$$…{be}\:{continued}… \\ $$
Commented by Shrodinger last updated on 10/Jun/24
$${thanks}\:{sir} \\ $$