Question Number 62806 by mathmax by abdo last updated on 25/Jun/19 | ||
$${find}\:{the}\:{value}\:{of}\:\:\:\int_{−\infty} ^{+\infty} \:\:\:\frac{{x}+\mathrm{1}}{\left({x}^{\mathrm{4}} \:+{x}^{\mathrm{2}} \:+\mathrm{1}\right)^{\mathrm{3}} }{dx} \\ $$ | ||
Commented by mathmax by abdo last updated on 27/Jun/19 | ||
$${let}\:{A}\:=\int_{−\infty} ^{+\infty} \:\:\frac{{x}+\mathrm{1}}{\left({x}^{\mathrm{4}} \:+{x}^{\mathrm{2}} \:+\mathrm{1}\right)^{\mathrm{3}} }\:{dx}\:{let}\:{W}\left({z}\right)\:=\frac{{z}+\mathrm{1}}{\left({z}^{\mathrm{4}} \:+{z}^{\mathrm{2}} \:+\mathrm{1}\right)^{\mathrm{3}} }\:?{poles}\:{of}\:{W}? \\ $$$${z}^{\mathrm{4}} \:+{z}^{\mathrm{2}} \:+\mathrm{1}\:=\mathrm{0}\:\Rightarrow{t}^{\mathrm{2}} \:+{t}+\mathrm{1}=\mathrm{0}\:\left({with}\:{z}^{\mathrm{2}} ={t}\right) \\ $$$$\Delta\:=\mathrm{1}−\mathrm{4}\:=\left({i}\sqrt{\mathrm{3}}\right)^{\mathrm{2}} \:\Rightarrow{t}_{\mathrm{1}} =\frac{−\mathrm{1}+{i}\sqrt{\mathrm{3}}}{\mathrm{2}}\:={e}^{\frac{{i}\mathrm{2}\pi}{\mathrm{3}}} \left(={j}\right)\:{and}\:{t}_{\mathrm{2}} =\frac{−\mathrm{1}−{i}\sqrt{\mathrm{3}}}{\mathrm{2}}\:={e}^{−\frac{{i}\mathrm{2}\pi}{\mathrm{3}}} \left(\overset{−} {{j}}\right)\:\Rightarrow \\ $$$${t}^{\mathrm{2}} \:+{t}\:+\mathrm{1}\:=\left({t}−{e}^{\frac{{i}\mathrm{2}\pi}{\mathrm{3}}} \right)\left({t}−{e}^{−{i}\frac{\mathrm{2}\pi}{\mathrm{3}}} \right)\:=\left({z}^{\mathrm{2}} −{e}^{\frac{{i}\mathrm{2}\pi}{\mathrm{3}}} \right)\left({z}^{\mathrm{2}} −{e}^{−\frac{{i}\mathrm{2}\pi}{\mathrm{3}}} \right) \\ $$$$\Rightarrow{W}\left({z}\right)\:=\frac{{z}+\mathrm{1}}{\left\{\:\left({z}−{e}^{\frac{{i}\pi}{\mathrm{3}}} \right)\left({z}+{e}^{\frac{{i}\pi}{\mathrm{3}}} \right)\left({z}−{e}^{−\frac{{i}\pi}{\mathrm{3}}} \right)\left({z}+{e}^{−\frac{{i}\pi}{\mathrm{3}}} \right)\right\}^{\mathrm{3}} } \\ $$$$=\frac{{z}+\mathrm{1}}{\left({z}−{e}^{\frac{{i}\pi}{\mathrm{3}}} \right)^{\mathrm{3}} \left({z}+{e}^{\frac{{i}\pi}{\mathrm{3}}} \right)^{\mathrm{3}} \left({z}−{e}^{−\frac{{i}\pi}{\mathrm{3}}} \right)^{\mathrm{3}} \left({z}+{e}^{−\frac{{i}\pi}{\mathrm{3}}} \right)^{\mathrm{3}} }\:{so}\:{the}\:{poles}\:{of}\:\varphi\:{are}\:\overset{−} {+}{e}^{\frac{{i}\pi}{\mathrm{3}}} \:{and}\:\overset{−} {+}{e}^{−\frac{{i}\pi}{\mathrm{3}}} \:\left({triples}\right) \\ $$$${residus}\:{theorem}\:{give}\: \\ $$$$\int_{−\infty} ^{+\infty} \:{W}\left({z}\right){dz}\:=\mathrm{2}{i}\pi\:\left\{\:{Res}\left({W},{e}^{\frac{{i}\pi}{\mathrm{3}}} \right)\:+{Res}\left({W},−{e}^{−\frac{{i}\pi}{\mathrm{3}}} \right)\right\} \\ $$$${Res}\left({W},{e}^{\frac{{i}\pi}{\mathrm{3}}} \right)\:={lim}_{{z}\rightarrow{e}^{\frac{{i}\pi}{\mathrm{3}}} } \:\:\:\:\frac{\mathrm{1}}{\left(\mathrm{3}−\mathrm{1}\right)!}\left\{\:\left({z}−{e}^{\frac{{i}\pi}{\mathrm{3}}} \right)^{\mathrm{3}} {W}\left({z}\right)\right\}^{\left(\mathrm{2}\right)} \\ $$$$={lim}_{{z}\rightarrow{e}^{\frac{{i}\pi}{\mathrm{3}}} } \:\:\:\:\frac{\mathrm{1}}{\mathrm{2}}\left\{\:\:\frac{{z}+\mathrm{1}}{\left({z}+{e}^{\frac{{i}\pi}{\mathrm{3}}} \right)^{\mathrm{3}} \left({z}^{\mathrm{2}} −{e}^{−{i}\frac{\mathrm{2}\pi}{\mathrm{3}}} \right)}\right\}^{\left(\mathrm{2}\right)} \:\:\:\:...{be}\:{continued}.... \\ $$ | ||