Question Number 49144 by AdqhK ÐQeQqQ last updated on 03/Dec/18
$$\int\frac{\mathrm{1}}{{x}^{{n}} +\mathrm{1}}{dx}=?? \\ $$
Commented by MJS last updated on 04/Dec/18
$${x}^{{n}} +\mathrm{1}=\underset{{k}=\mathrm{0}} {\overset{{n}−\mathrm{1}} {\prod}}\left({x}−\mathrm{e}^{\mathrm{i}\frac{\pi}{{n}}\left(\mathrm{2}{k}+\mathrm{1}\right)} \right) \\ $$$$\int\frac{{dx}}{{x}^{{n}} +\mathrm{1}}=\underset{{k}=\mathrm{0}} {\overset{{n}−\mathrm{1}} {\sum}}\left(\int\frac{{A}_{{k}} }{{x}−\mathrm{e}^{\mathrm{i}\frac{\pi}{{n}}\left(\mathrm{2}{k}+\mathrm{1}\right)} }{dx}\right)=\underset{{k}=\mathrm{0}} {\overset{{n}−\mathrm{1}} {\sum}}\left({A}_{{k}} \mathrm{ln}\:\mid{x}−\mathrm{e}^{\mathrm{i}\frac{\pi}{{n}}\left(\mathrm{2}{k}+\mathrm{1}\right)} \mid\right)+{C} \\ $$$${A}_{{k}} =\frac{\mathrm{1}}{\underset{{l}=\mathrm{0};\:{l}\neq{k}} {\overset{{n}−\mathrm{1}} {\prod}}\left(\mathrm{e}^{\mathrm{i}\frac{\pi}{{n}}\left(\mathrm{2}{k}+\mathrm{1}\right)} −\mathrm{e}^{\mathrm{i}\frac{\pi}{{n}}\left(\mathrm{2}{l}+\mathrm{1}\right)} \right)} \\ $$
Commented by Abdo msup. last updated on 04/Dec/18
$${let}\:{decompose}\:{inside}\:{R}\left({x}\right)\:{the}\:{fraction}\:{F}\left({x}\right)=\frac{\mathrm{1}}{{x}^{{n}} \:+\mathrm{1}} \\ $$$${roots}\:{of}\:{z}^{{n}} +\mathrm{1}\:=\mathrm{0}\:\Rightarrow{z}^{{n}} ={e}^{{i}\pi} \:\Rightarrow\:{if}\:{z}={r}\:{e}^{{i}\theta} \:{we}\:{get} \\ $$$${r}\:=\mathrm{1}\:{and}\:{n}\theta=\left(\mathrm{2}{k}+\mathrm{1}\right)\pi\:\Rightarrow\theta\:=\frac{\left(\mathrm{2}{k}+\mathrm{1}\right)\pi}{{n}}?\Rightarrow\:{the}\:{roots}\:{are} \\ $$$${z}_{{k}} =\:{e}^{\frac{{i}\left(\mathrm{2}{k}+\mathrm{1}\right)\pi}{{n}}} \:\:{k}\in\left[\left[\mathrm{0},{n}−\mathrm{1}\right]\right]\:\Rightarrow \\ $$$${F}\left({z}\right)\:=\frac{\mathrm{1}}{\prod_{{k}=\mathrm{0}} ^{{n}−\mathrm{1}} \left({z}−{z}_{{k}} \right)}\:=\sum_{{k}=\mathrm{0}} ^{{n}−\mathrm{1}} \:\:\frac{\lambda_{{k}} }{{z}−{z}_{{k}} } \\ $$$$\lambda_{{k}} =\:\frac{\mathrm{1}}{{n}\:{z}_{{k}} ^{{n}−\mathrm{1}} }\:=\frac{{z}_{{k}} }{−{n}}\:=−\frac{{z}_{{k}} }{{n}}\:\Rightarrow{F}\left({z}\right)\:=−\frac{\mathrm{1}}{{n}}\sum_{{k}=\mathrm{0}} ^{{n}−\mathrm{1}} \:\:\frac{{z}_{{k}} }{{z}−{z}_{{k}} }\:\Rightarrow \\ $$$$\int\:{F}\left({x}\right){dx}\:=−\frac{\mathrm{1}}{{n}}\:\sum_{{k}=\mathrm{0}} ^{{n}−\mathrm{1}} \:\:{z}_{{k}} \:\int\:\:\frac{{dx}}{{x}−{z}_{{k}} }\:{let}\:{determine} \\ $$$$\int\:\:\:\frac{{dx}}{{x}−{z}_{{k}} }\:\:={A}_{{k}} \\ $$$${A}_{{k}} =\:\int\:\:\frac{{dx}}{{x}−{e}^{{i}\theta_{{k}} } }\:\:=\int\:\:\:\frac{{dx}}{{x}−{cos}\left(\theta_{{k}} \right)−{isin}\left(\theta_{{k}} \right)} \\ $$$$=\:\int\:\:\:\:\frac{{x}−{cos}\left(\theta_{{k}} \right)+{i}\:{sin}\left(\theta_{{k}} \right)}{\left({x}−{cos}\theta_{{k}} \right)^{\mathrm{2}} +{sin}^{\mathrm{2}} \theta_{{k}} }{dx} \\ $$$$=\int\:\:\:\frac{{x}−{cos}\theta_{{k}} }{{x}^{\mathrm{2}} \:−\mathrm{2}{xcos}\theta_{{k}} +\mathrm{1}}{dx}\:+{isin}\theta_{{k}} \:\int\:\:\:\frac{{dx}}{\left({x}−{cos}\theta_{{k}} \right)^{\mathrm{2}} \:+{sin}^{\mathrm{2}} \theta_{{k}} } \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}{ln}\left({x}^{\mathrm{2}} −\mathrm{2}{xcos}\theta_{{k}} +\mathrm{1}\right)\:+{isin}\theta_{{k}} \:\:\int\:\:\:\frac{{dx}}{\left({x}−{cos}\theta_{{k}} \right)^{\mathrm{2}} \:+{sin}^{\mathrm{2}} \theta_{{k}} } \\ $$$${but}\:\int\:\:\:\frac{{dx}}{\left({x}−{cos}\theta\right)^{\mathrm{2}} \:+{sin}^{\mathrm{2}} \theta_{{k}} }\:=_{{x}−{cos}\theta_{{k}} ={u}\:{sin}\theta_{{k}} } \\ $$$$=\int\:\:\:\frac{{sin}\theta_{{k}} }{{sin}^{\mathrm{2}} \theta_{{k}} \left(\mathrm{1}+{u}^{\mathrm{2}} \right)}{du}\:=\frac{\mathrm{1}}{{sin}\theta_{{k}} }\:{arctan}\left(\frac{{x}−{cos}\theta_{{k}} }{{sin}\theta_{{k}} }\right)\:\Rightarrow \\ $$$${A}_{{k}} =\frac{\mathrm{1}}{\mathrm{2}}{ln}\left({x}^{\mathrm{2}} \:−\mathrm{2}{xcos}\theta_{{k}} \:+\mathrm{1}\right)+{i}\:{arctan}\left(\frac{{x}−{cos}\theta_{{k}} }{{sin}\theta_{{k}} }\right)\:+{C}…. \\ $$$$ \\ $$