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Question Number 124360 by Lordose last updated on 02/Dec/20

∫_( 0) ^( ∞) ((ln(x))/(x^4 +x^2 +1))dx

$$\int_{\:\mathrm{0}} ^{\:\infty} \frac{\mathrm{ln}\left(\mathrm{x}\right)}{\mathrm{x}^{\mathrm{4}} +\mathrm{x}^{\mathrm{2}} +\mathrm{1}}\mathrm{dx} \\ $$

Answered by mathmax by abdo last updated on 02/Dec/20

we use ∫_0 ^∞ q(x)ln(x)dx=−(1/2)Re(Σ Res(q(z)ln^2 z ,a_i ) q(x)=(1/(x^4 +x^2 +1)) ⇒q(z)ln^2 (z)=((ln^2 z)/(z^4 +x^2 +1))=w(z) z^4 +z^2 +1=0 ⇒u^2 +u+1 (u=z^2 ) Δ=1−4=−3 ⇒u_1 =((−1+i(√3))/2)=e^((i2π)/3) and u_2 =((−1−i(√3))/2)=e^(−((i2π)/3)) ⇒w(z) =((ln^2 z)/((z^2 −e^((i2π)/3) )(z^2 −e^(−((i2π)/3)) ))) =((ln^2 z)/((z−e^((iπ)/3) )(z+e^((iπ)/3) )(z−e^(−((iπ)/3)) )(z+e^(−((iπ)/3)) ))) Σ Res(w.)=Res(w,e^((iπ)/3) ) +Res(w,−e^((iπ)/3) )+Res(w,e^(−((iπ)/3)) )+Res(w,−e^(−((iπ)/3)) ) we have Res(w,e^((iπ)/3) )=(((((iπ)/3))^2 )/(2e^((iπ)/3) (2isin(((2π)/3))))) =−(π^2 /(36)) (e^(−((iπ)/3)) /((√3)/2)) =−(π^2 /(18i(√3)))e^(−((iπ)/3)) Res(w,−e^((iπ)/3) ) =(((iπ+((iπ)/3))^2 )/(−2e^((iπ)/3) (2isin(((2π)/3)))))=−(((((4π)/3))^2 )/(−4i×((√3)/2)))e^(−((iπ)/3)) =((16π^2 )/(18i(√3)))e^(−((iπ)/3)) =((8π^2 )/(9i(√3)))e^(−((iπ)/3)) Res(w,e^(−((iπ)/3)) ) =(((−((iπ)/3))^2 )/(−2isin(((2π)/3))2e^(−((iπ)/3)) )) =(π^2 /(36i×((√3)/2)))e^((iπ)/3) =(π^2 /(18i(√3)))e^((iπ)/3) Res(w,−e^(−((iπ)/3)) ) =(((iπ−((iπ)/3))^2 )/(−2e^(−((iπ)/3)) (−2isin(((2π)/3)))))=−(((((2π)/3))^2 )/(4i .((√3)/2))) e^((iπ)/3) =−((4π^2 )/(18i(√3)))e^((iπ)/3) =−((2π^2 )/(9i(√3)))e^((iπ)/3) ⇒ Σ Res(w,z_i ) =((iπ^2 )/(18(√3)))e^(−((iπ)/3)) −i((8π^2 )/(9(√3)))e^(−((iπ)/3)) −i(π^2 /(18(√3))) e^((iπ)/3) +i((2π^2 )/(9(√3))) e^((iπ)/3) =−((iπ^2 )/(18(√3)))(2isin((π/3)))+e^((iπ)/3) (−((iπ^2 )/(18(√3))) +((4iπ^2 )/(18(√3)))) =(π^2 /(9(√3)))×((√3)/2) +((1/2)+((i(√3))/2))(((iπ^2 )/( 6(√3)))) =(π^2 /(18)) +((iπ^2 )/(12(√3))) −(π^2 /(12)) ⇒ I =−(1/2)((π^2 /(18))−(π^2 /(12)))=(π^2 /(24))−(π^2 /(36))

$$\mathrm{we}\:\mathrm{use}\:\int_{\mathrm{0}} ^{\infty} \mathrm{q}\left(\mathrm{x}\right)\mathrm{ln}\left(\mathrm{x}\right)\mathrm{dx}=−\frac{\mathrm{1}}{\mathrm{2}}\mathrm{Re}\left(\Sigma\:\mathrm{Res}\left(\mathrm{q}\left(\mathrm{z}\right)\mathrm{ln}^{\mathrm{2}} \mathrm{z}\:,\mathrm{a}_{\mathrm{i}} \right)\right. \\ $$$$\mathrm{q}\left(\mathrm{x}\right)=\frac{\mathrm{1}}{\mathrm{x}^{\mathrm{4}} +\mathrm{x}^{\mathrm{2}} \:+\mathrm{1}}\:\Rightarrow\mathrm{q}\left(\mathrm{z}\right)\mathrm{ln}^{\mathrm{2}} \left(\mathrm{z}\right)=\frac{\mathrm{ln}^{\mathrm{2}} \mathrm{z}}{\mathrm{z}^{\mathrm{4}} \:+\mathrm{x}^{\mathrm{2}} \:+\mathrm{1}}=\mathrm{w}\left(\mathrm{z}\right) \\ $$$$\mathrm{z}^{\mathrm{4}} \:+\mathrm{z}^{\mathrm{2}} \:+\mathrm{1}=\mathrm{0}\:\Rightarrow\mathrm{u}^{\mathrm{2}} \:+\mathrm{u}+\mathrm{1}\:\:\left(\mathrm{u}=\mathrm{z}^{\mathrm{2}} \right) \\ $$$$\Delta=\mathrm{1}−\mathrm{4}=−\mathrm{3}\:\Rightarrow\mathrm{u}_{\mathrm{1}} =\frac{−\mathrm{1}+\mathrm{i}\sqrt{\mathrm{3}}}{\mathrm{2}}=\mathrm{e}^{\frac{\mathrm{i2}\pi}{\mathrm{3}}} \:\mathrm{and}\:\mathrm{u}_{\mathrm{2}} =\frac{−\mathrm{1}−\mathrm{i}\sqrt{\mathrm{3}}}{\mathrm{2}}=\mathrm{e}^{−\frac{\mathrm{i2}\pi}{\mathrm{3}}} \\ $$$$\Rightarrow\mathrm{w}\left(\mathrm{z}\right)\:=\frac{\mathrm{ln}^{\mathrm{2}} \mathrm{z}}{\left(\mathrm{z}^{\mathrm{2}} −\mathrm{e}^{\frac{\mathrm{i2}\pi}{\mathrm{3}}} \right)\left(\mathrm{z}^{\mathrm{2}} \:−\mathrm{e}^{−\frac{\mathrm{i2}\pi}{\mathrm{3}}} \right)}\:=\frac{\mathrm{ln}^{\mathrm{2}} \mathrm{z}}{\left(\mathrm{z}−\mathrm{e}^{\frac{\mathrm{i}\pi}{\mathrm{3}}} \right)\left(\mathrm{z}+\mathrm{e}^{\frac{\mathrm{i}\pi}{\mathrm{3}}} \right)\left(\mathrm{z}−\mathrm{e}^{−\frac{\mathrm{i}\pi}{\mathrm{3}}} \right)\left(\mathrm{z}+\mathrm{e}^{−\frac{\mathrm{i}\pi}{\mathrm{3}}} \right)} \\ $$$$\Sigma\:\mathrm{Res}\left(\mathrm{w}.\right)=\mathrm{Res}\left(\mathrm{w},\mathrm{e}^{\frac{\mathrm{i}\pi}{\mathrm{3}}} \right)\:+\mathrm{Res}\left(\mathrm{w},−\mathrm{e}^{\frac{\mathrm{i}\pi}{\mathrm{3}}} \right)+\mathrm{Res}\left(\mathrm{w},\mathrm{e}^{−\frac{\mathrm{i}\pi}{\mathrm{3}}} \right)+\mathrm{Res}\left(\mathrm{w},−\mathrm{e}^{−\frac{\mathrm{i}\pi}{\mathrm{3}}} \right) \\ $$$$\mathrm{we}\:\mathrm{have}\:\mathrm{Res}\left(\mathrm{w},\mathrm{e}^{\frac{\mathrm{i}\pi}{\mathrm{3}}} \right)=\frac{\left(\frac{\mathrm{i}\pi}{\mathrm{3}}\right)^{\mathrm{2}} }{\mathrm{2e}^{\frac{\mathrm{i}\pi}{\mathrm{3}}} \left(\mathrm{2isin}\left(\frac{\mathrm{2}\pi}{\mathrm{3}}\right)\right)}\:=−\frac{\pi^{\mathrm{2}} }{\mathrm{36}}\:\frac{\mathrm{e}^{−\frac{\mathrm{i}\pi}{\mathrm{3}}} }{\frac{\sqrt{\mathrm{3}}}{\mathrm{2}}}\:=−\frac{\pi^{\mathrm{2}} }{\mathrm{18i}\sqrt{\mathrm{3}}}\mathrm{e}^{−\frac{\mathrm{i}\pi}{\mathrm{3}}} \\ $$$$\mathrm{Res}\left(\mathrm{w},−\mathrm{e}^{\frac{\mathrm{i}\pi}{\mathrm{3}}} \right)\:=\frac{\left(\mathrm{i}\pi+\frac{\mathrm{i}\pi}{\mathrm{3}}\right)^{\mathrm{2}} }{−\mathrm{2e}^{\frac{\mathrm{i}\pi}{\mathrm{3}}} \left(\mathrm{2isin}\left(\frac{\mathrm{2}\pi}{\mathrm{3}}\right)\right)}=−\frac{\left(\frac{\mathrm{4}\pi}{\mathrm{3}}\right)^{\mathrm{2}} }{−\mathrm{4i}×\frac{\sqrt{\mathrm{3}}}{\mathrm{2}}}\mathrm{e}^{−\frac{\mathrm{i}\pi}{\mathrm{3}}} =\frac{\mathrm{16}\pi^{\mathrm{2}} }{\mathrm{18i}\sqrt{\mathrm{3}}}\mathrm{e}^{−\frac{\mathrm{i}\pi}{\mathrm{3}}} =\frac{\mathrm{8}\pi^{\mathrm{2}} }{\mathrm{9i}\sqrt{\mathrm{3}}}\mathrm{e}^{−\frac{\mathrm{i}\pi}{\mathrm{3}}} \\ $$$$\mathrm{Res}\left(\mathrm{w},\mathrm{e}^{−\frac{\mathrm{i}\pi}{\mathrm{3}}} \right)\:=\frac{\left(−\frac{\mathrm{i}\pi}{\mathrm{3}}\right)^{\mathrm{2}} }{−\mathrm{2isin}\left(\frac{\mathrm{2}\pi}{\mathrm{3}}\right)\mathrm{2e}^{−\frac{\mathrm{i}\pi}{\mathrm{3}}} }\:=\frac{\pi^{\mathrm{2}} }{\mathrm{36i}×\frac{\sqrt{\mathrm{3}}}{\mathrm{2}}}\mathrm{e}^{\frac{\mathrm{i}\pi}{\mathrm{3}}} \\ $$$$=\frac{\pi^{\mathrm{2}} }{\mathrm{18i}\sqrt{\mathrm{3}}}\mathrm{e}^{\frac{\mathrm{i}\pi}{\mathrm{3}}} \\ $$$$\mathrm{Res}\left(\mathrm{w},−\mathrm{e}^{−\frac{\mathrm{i}\pi}{\mathrm{3}}} \right)\:=\frac{\left(\mathrm{i}\pi−\frac{\mathrm{i}\pi}{\mathrm{3}}\right)^{\mathrm{2}} }{−\mathrm{2e}^{−\frac{\mathrm{i}\pi}{\mathrm{3}}} \left(−\mathrm{2isin}\left(\frac{\mathrm{2}\pi}{\mathrm{3}}\right)\right)}=−\frac{\left(\frac{\mathrm{2}\pi}{\mathrm{3}}\right)^{\mathrm{2}} }{\mathrm{4i}\:.\frac{\sqrt{\mathrm{3}}}{\mathrm{2}}}\:\mathrm{e}^{\frac{\mathrm{i}\pi}{\mathrm{3}}} \\ $$$$=−\frac{\mathrm{4}\pi^{\mathrm{2}} }{\mathrm{18i}\sqrt{\mathrm{3}}}\mathrm{e}^{\frac{\mathrm{i}\pi}{\mathrm{3}}} \:=−\frac{\mathrm{2}\pi^{\mathrm{2}} }{\mathrm{9i}\sqrt{\mathrm{3}}}\mathrm{e}^{\frac{\mathrm{i}\pi}{\mathrm{3}}} \:\Rightarrow \\ $$$$\Sigma\:\mathrm{Res}\left(\mathrm{w},\mathrm{z}_{\mathrm{i}} \right)\:=\frac{\mathrm{i}\pi^{\mathrm{2}} }{\mathrm{18}\sqrt{\mathrm{3}}}\mathrm{e}^{−\frac{\mathrm{i}\pi}{\mathrm{3}}} −\mathrm{i}\frac{\mathrm{8}\pi^{\mathrm{2}} }{\mathrm{9}\sqrt{\mathrm{3}}}\mathrm{e}^{−\frac{\mathrm{i}\pi}{\mathrm{3}}} −\mathrm{i}\frac{\pi^{\mathrm{2}} }{\mathrm{18}\sqrt{\mathrm{3}}}\:\mathrm{e}^{\frac{\mathrm{i}\pi}{\mathrm{3}}} \:+\mathrm{i}\frac{\mathrm{2}\pi^{\mathrm{2}} }{\mathrm{9}\sqrt{\mathrm{3}}}\:\mathrm{e}^{\frac{\mathrm{i}\pi}{\mathrm{3}}} \\ $$$$=−\frac{\mathrm{i}\pi^{\mathrm{2}} }{\mathrm{18}\sqrt{\mathrm{3}}}\left(\mathrm{2isin}\left(\frac{\pi}{\mathrm{3}}\right)\right)+\mathrm{e}^{\frac{\mathrm{i}\pi}{\mathrm{3}}} \left(−\frac{\mathrm{i}\pi^{\mathrm{2}} }{\mathrm{18}\sqrt{\mathrm{3}}}\:+\frac{\mathrm{4i}\pi^{\mathrm{2}} }{\mathrm{18}\sqrt{\mathrm{3}}}\right) \\ $$$$=\frac{\pi^{\mathrm{2}} }{\mathrm{9}\sqrt{\mathrm{3}}}×\frac{\sqrt{\mathrm{3}}}{\mathrm{2}}\:+\left(\frac{\mathrm{1}}{\mathrm{2}}+\frac{\mathrm{i}\sqrt{\mathrm{3}}}{\mathrm{2}}\right)\left(\frac{\mathrm{i}\pi^{\mathrm{2}} }{\:\mathrm{6}\sqrt{\mathrm{3}}}\right) \\ $$$$=\frac{\pi^{\mathrm{2}} }{\mathrm{18}}\:+\frac{\mathrm{i}\pi^{\mathrm{2}} }{\mathrm{12}\sqrt{\mathrm{3}}}\:−\frac{\pi^{\mathrm{2}} }{\mathrm{12}}\:\Rightarrow\:\mathrm{I}\:=−\frac{\mathrm{1}}{\mathrm{2}}\left(\frac{\pi^{\mathrm{2}} }{\mathrm{18}}−\frac{\pi^{\mathrm{2}} }{\mathrm{12}}\right)=\frac{\pi^{\mathrm{2}} }{\mathrm{24}}−\frac{\pi^{\mathrm{2}} }{\mathrm{36}} \\ $$

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