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Question Number 42054 by SammyKT last updated on 17/Aug/18

Commented by math khazana by abdo last updated on 17/Aug/18

what s C? C is not given !

$${what}\:{s}\:{C}?\:{C}\:{is}\:{not}\:{given}\:! \\ $$

Answered by tanmay.chaudhury50@gmail.com last updated on 17/Aug/18

=(1/2)∫((2z^2 +z+2−z)/(z(2z+1)))dz  =(1/2)∫dz+(1/2)∫((2−z)/(z(2z+1)))dz  =(1/2)∫dz+(1/2)∫((z−(2z+1)+3)/(z(2z+1)))dz  =(1/2)∫dz+(1/2)∫(dz/(2z+1))−(1/2)∫(dz/z)+(3/2)∫((2z+1−2z)/(z(2z+1)))dz  =(1/2)∫dz+(1/4)∫(dz/(z+(1/2)))−(1/2)∫(dz/z)+(3/2)∫(dz/z)−3∫(dz/(2z+1))  =(1/2)∫dz+(1/4)∫(dz/(z+(1/2)))+∫(dz/z)−(3/2)∫(dz/(z+(1/2)))  =(1/2)z+(1/4)ln(z+(1/2))+lnz−(3/2)ln(z+(1/2))+c

$$=\frac{\mathrm{1}}{\mathrm{2}}\int\frac{\mathrm{2}{z}^{\mathrm{2}} +{z}+\mathrm{2}−{z}}{{z}\left(\mathrm{2}{z}+\mathrm{1}\right)}{dz} \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}\int{dz}+\frac{\mathrm{1}}{\mathrm{2}}\int\frac{\mathrm{2}−{z}}{{z}\left(\mathrm{2}{z}+\mathrm{1}\right)}{dz} \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}\int{dz}+\frac{\mathrm{1}}{\mathrm{2}}\int\frac{{z}−\left(\mathrm{2}{z}+\mathrm{1}\right)+\mathrm{3}}{{z}\left(\mathrm{2}{z}+\mathrm{1}\right)}{dz} \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}\int{dz}+\frac{\mathrm{1}}{\mathrm{2}}\int\frac{{dz}}{\mathrm{2}{z}+\mathrm{1}}−\frac{\mathrm{1}}{\mathrm{2}}\int\frac{{dz}}{{z}}+\frac{\mathrm{3}}{\mathrm{2}}\int\frac{\mathrm{2}{z}+\mathrm{1}−\mathrm{2}{z}}{{z}\left(\mathrm{2}{z}+\mathrm{1}\right)}{dz} \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}\int{dz}+\frac{\mathrm{1}}{\mathrm{4}}\int\frac{{dz}}{{z}+\frac{\mathrm{1}}{\mathrm{2}}}−\frac{\mathrm{1}}{\mathrm{2}}\int\frac{{dz}}{{z}}+\frac{\mathrm{3}}{\mathrm{2}}\int\frac{{dz}}{{z}}−\mathrm{3}\int\frac{{dz}}{\mathrm{2}{z}+\mathrm{1}} \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}\int{dz}+\frac{\mathrm{1}}{\mathrm{4}}\int\frac{{dz}}{{z}+\frac{\mathrm{1}}{\mathrm{2}}}+\int\frac{{dz}}{{z}}−\frac{\mathrm{3}}{\mathrm{2}}\int\frac{{dz}}{{z}+\frac{\mathrm{1}}{\mathrm{2}}} \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}{z}+\frac{\mathrm{1}}{\mathrm{4}}{ln}\left({z}+\frac{\mathrm{1}}{\mathrm{2}}\right)+{lnz}−\frac{\mathrm{3}}{\mathrm{2}}{ln}\left({z}+\frac{\mathrm{1}}{\mathrm{2}}\right)+{c} \\ $$$$ \\ $$$$ \\ $$

Commented by SammyKT last updated on 17/Aug/18

Thank you sir...

$${Thank}\:{you}\:{sir}... \\ $$

Commented by tanmay.chaudhury50@gmail.com last updated on 17/Aug/18

itsok...

$${itsok}... \\ $$

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