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Question Number 39165 by math khazana by abdo last updated on 03/Jul/18

calculate  ∫_0 ^∞     (dt/(1+t^(4 )  +t^6 ))

$${calculate}\:\:\int_{\mathrm{0}} ^{\infty} \:\:\:\:\frac{{dt}}{\mathrm{1}+{t}^{\mathrm{4}\:} \:+{t}^{\mathrm{6}} } \\ $$

Commented by tanmay.chaudhury50@gmail.com last updated on 03/Jul/18

Commented by tanmay.chaudhury50@gmail.com last updated on 03/Jul/18

    1.5×1>∫_0 ^∞ (dt/(1+t^4 +t^6 ))>3×(0.5)^2   1.5>∫_0 ^∞ (dt/(1+t^4 +t^6 ))>0.75

$$\:\:\:\:\mathrm{1}.\mathrm{5}×\mathrm{1}>\int_{\mathrm{0}} ^{\infty} \frac{{dt}}{\mathrm{1}+{t}^{\mathrm{4}} +{t}^{\mathrm{6}} }>\mathrm{3}×\left(\mathrm{0}.\mathrm{5}\right)^{\mathrm{2}} \\ $$$$\mathrm{1}.\mathrm{5}>\int_{\mathrm{0}} ^{\infty} \frac{{dt}}{\mathrm{1}+{t}^{\mathrm{4}} +{t}^{\mathrm{6}} }>\mathrm{0}.\mathrm{75} \\ $$$$ \\ $$

Commented by math khazana by abdo last updated on 04/Jul/18

let  I =∫_0 ^∞      (dt/(1+t^4  +t^6 ))  2I  = ∫_(−∞) ^(+∞)     (dt/(1+t^4  +t^6 )) let ϕ(z)= (1/(1+z^4  +z^8 ))  poles of ϕ?  1+z^4  +z^8  =0 ⇔ 1+z^4  +(z^4 )^2 =0   ⇔ ((1−(z^4 )^3 )/(1−z^4 )) =0  and z^4 ≠1 ⇔ z^(12)  =1 and z^4 ≠1  z^4 =1 ⇒4θ=2kπ ⇒θ_k =((kπ)/2)  and k∈[[0,3]]  z_k =e^(i((kπ)/2))    let  z=re^(iα)    so z^(12) =1 ⇒r=1 and  12α =2kπ ⇒ α_k =((kπ)/6)  with k∈[[0,11]]⇒  z_k =e^(i((kπ)/6))    and k∈[[0,5]]  z_0 =1 (non pole of ϕ)  z_1 =e^(i(π/6))     z_1 ^4  =e^(i((2π)/3))   ≠1 ⇒z_1 pole of ϕ  z_2 =e^(i(π/3))     (z_2 ^4  ≠1 ⇒z_2 pole of ϕ)  z_3 =e^(i(π/2))  (z_3 ^4 =1 so z_3  non pole of ϕ)  z_4  =e^(i((2π)/3))  ( z_4 ^4  ≠1 so z_4 pole of ϕ)  z_5  =e^(i ((5π)/6))    (z_5 ^4 ≠1 so z_5 is pole of ϕ)  z_6  = e^(iπ) ( z_6 ^4  =1 non pole of ϕ)  z_7 =e^(i((7π)/6))  =−e^(i(π/6))   (pole of ϕ)  z_8 =e^(i((8π)/6))  = e^(i((4π)/3))    (pole of ϕ)  z_9 =e^(i((9π)/6))  =e^(i((3π)/2))  (non pole of ϕ)  z_(10) =e^(i((10π)/6))  = e^(i((5π)/3)) (pole ofϕ)  z_(11) = e^(i((11π)/6))  =−e^(−i(π/6))   (pole of ϕ)  be continued...

$${let}\:\:{I}\:=\int_{\mathrm{0}} ^{\infty} \:\:\:\:\:\frac{{dt}}{\mathrm{1}+{t}^{\mathrm{4}} \:+{t}^{\mathrm{6}} } \\ $$$$\mathrm{2}{I}\:\:=\:\int_{−\infty} ^{+\infty} \:\:\:\:\frac{{dt}}{\mathrm{1}+{t}^{\mathrm{4}} \:+{t}^{\mathrm{6}} }\:{let}\:\varphi\left({z}\right)=\:\frac{\mathrm{1}}{\mathrm{1}+{z}^{\mathrm{4}} \:+{z}^{\mathrm{8}} } \\ $$$${poles}\:{of}\:\varphi? \\ $$$$\mathrm{1}+{z}^{\mathrm{4}} \:+{z}^{\mathrm{8}} \:=\mathrm{0}\:\Leftrightarrow\:\mathrm{1}+{z}^{\mathrm{4}} \:+\left({z}^{\mathrm{4}} \right)^{\mathrm{2}} =\mathrm{0}\: \\ $$$$\Leftrightarrow\:\frac{\mathrm{1}−\left({z}^{\mathrm{4}} \right)^{\mathrm{3}} }{\mathrm{1}−{z}^{\mathrm{4}} }\:=\mathrm{0}\:\:{and}\:{z}^{\mathrm{4}} \neq\mathrm{1}\:\Leftrightarrow\:{z}^{\mathrm{12}} \:=\mathrm{1}\:{and}\:{z}^{\mathrm{4}} \neq\mathrm{1} \\ $$$${z}^{\mathrm{4}} =\mathrm{1}\:\Rightarrow\mathrm{4}\theta=\mathrm{2}{k}\pi\:\Rightarrow\theta_{{k}} =\frac{{k}\pi}{\mathrm{2}}\:\:{and}\:{k}\in\left[\left[\mathrm{0},\mathrm{3}\right]\right] \\ $$$${z}_{{k}} ={e}^{{i}\frac{{k}\pi}{\mathrm{2}}} \:\:\:{let}\:\:{z}={re}^{{i}\alpha} \:\:\:{so}\:{z}^{\mathrm{12}} =\mathrm{1}\:\Rightarrow{r}=\mathrm{1}\:{and} \\ $$$$\mathrm{12}\alpha\:=\mathrm{2}{k}\pi\:\Rightarrow\:\alpha_{{k}} =\frac{{k}\pi}{\mathrm{6}}\:\:{with}\:{k}\in\left[\left[\mathrm{0},\mathrm{11}\right]\right]\Rightarrow \\ $$$${z}_{{k}} ={e}^{{i}\frac{{k}\pi}{\mathrm{6}}} \:\:\:{and}\:{k}\in\left[\left[\mathrm{0},\mathrm{5}\right]\right] \\ $$$${z}_{\mathrm{0}} =\mathrm{1}\:\left({non}\:{pole}\:{of}\:\varphi\right) \\ $$$${z}_{\mathrm{1}} ={e}^{{i}\frac{\pi}{\mathrm{6}}} \:\:\:\:{z}_{\mathrm{1}} ^{\mathrm{4}} \:={e}^{{i}\frac{\mathrm{2}\pi}{\mathrm{3}}} \:\:\neq\mathrm{1}\:\Rightarrow{z}_{\mathrm{1}} {pole}\:{of}\:\varphi \\ $$$${z}_{\mathrm{2}} ={e}^{{i}\frac{\pi}{\mathrm{3}}} \:\:\:\:\left({z}_{\mathrm{2}} ^{\mathrm{4}} \:\neq\mathrm{1}\:\Rightarrow{z}_{\mathrm{2}} {pole}\:{of}\:\varphi\right) \\ $$$${z}_{\mathrm{3}} ={e}^{{i}\frac{\pi}{\mathrm{2}}} \:\left({z}_{\mathrm{3}} ^{\mathrm{4}} =\mathrm{1}\:{so}\:{z}_{\mathrm{3}} \:{non}\:{pole}\:{of}\:\varphi\right) \\ $$$${z}_{\mathrm{4}} \:={e}^{{i}\frac{\mathrm{2}\pi}{\mathrm{3}}} \:\left(\:{z}_{\mathrm{4}} ^{\mathrm{4}} \:\neq\mathrm{1}\:{so}\:{z}_{\mathrm{4}} {pole}\:{of}\:\varphi\right) \\ $$$${z}_{\mathrm{5}} \:={e}^{{i}\:\frac{\mathrm{5}\pi}{\mathrm{6}}} \:\:\:\left({z}_{\mathrm{5}} ^{\mathrm{4}} \neq\mathrm{1}\:{so}\:{z}_{\mathrm{5}} {is}\:{pole}\:{of}\:\varphi\right) \\ $$$${z}_{\mathrm{6}} \:=\:{e}^{{i}\pi} \left(\:{z}_{\mathrm{6}} ^{\mathrm{4}} \:=\mathrm{1}\:{non}\:{pole}\:{of}\:\varphi\right) \\ $$$${z}_{\mathrm{7}} ={e}^{{i}\frac{\mathrm{7}\pi}{\mathrm{6}}} \:=−{e}^{{i}\frac{\pi}{\mathrm{6}}} \:\:\left({pole}\:{of}\:\varphi\right) \\ $$$${z}_{\mathrm{8}} ={e}^{{i}\frac{\mathrm{8}\pi}{\mathrm{6}}} \:=\:{e}^{{i}\frac{\mathrm{4}\pi}{\mathrm{3}}} \:\:\:\left({pole}\:{of}\:\varphi\right) \\ $$$${z}_{\mathrm{9}} ={e}^{{i}\frac{\mathrm{9}\pi}{\mathrm{6}}} \:={e}^{{i}\frac{\mathrm{3}\pi}{\mathrm{2}}} \:\left({non}\:{pole}\:{of}\:\varphi\right) \\ $$$${z}_{\mathrm{10}} ={e}^{{i}\frac{\mathrm{10}\pi}{\mathrm{6}}} \:=\:{e}^{{i}\frac{\mathrm{5}\pi}{\mathrm{3}}} \left({pole}\:{of}\varphi\right) \\ $$$${z}_{\mathrm{11}} =\:{e}^{{i}\frac{\mathrm{11}\pi}{\mathrm{6}}} \:=−{e}^{−{i}\frac{\pi}{\mathrm{6}}} \:\:\left({pole}\:{of}\:\varphi\right)\:\:{be}\:{continued}... \\ $$$$ \\ $$

Commented by math khazana by abdo last updated on 04/Jul/18

∫_(−∞) ^(+∞)  ϕ(z)dz =2iπ{ Res(ϕ,z_1 ) +Res(ϕ,z_2 ) +Res(ϕ,z_4 )   +Res(ϕ,z_5 ) +Res(ϕ, z_7 ) +Res(ϕ^� ,z_8 ) +Res(ϕ,z_(10) )  +Res(ϕ,z_(11) ) }  Res(ϕ,z_k ) =  (1/(4z_k ^3  + 6z_k ^5 )) ⇒  Res(ϕ,z_1 )  = (1/(4(e^(i(π/6)) )^3  +6(e^((iπ)/6) )^5 ))  = (1/(4i +6 e^((i5π)/6) ))  Res(ϕ,z_2 ) =   (1/(4(e^((iπ)/3) )^3  +6 (e^((iπ)/3) )^5 )) = (1/(−4 +6 e^((i5π)/3) ))  Res(ϕ,z_4 )= (1/(4(e^((i2π)/3) )^3  +6(e^((i2π)/3) )^5 )) =(1/(4 +6 e^((i10π)/3) ))  =....

$$\int_{−\infty} ^{+\infty} \:\varphi\left({z}\right){dz}\:=\mathrm{2}{i}\pi\left\{\:{Res}\left(\varphi,{z}_{\mathrm{1}} \right)\:+{Res}\left(\varphi,{z}_{\mathrm{2}} \right)\:+{Res}\left(\varphi,{z}_{\mathrm{4}} \right)\:\right. \\ $$$$+{Res}\left(\varphi,{z}_{\mathrm{5}} \right)\:+{Res}\left(\varphi,\:{z}_{\mathrm{7}} \right)\:+{Res}\left(\bar {\varphi},{z}_{\mathrm{8}} \right)\:+{Res}\left(\varphi,{z}_{\mathrm{10}} \right) \\ $$$$\left.+{Res}\left(\varphi,{z}_{\mathrm{11}} \right)\:\right\} \\ $$$${Res}\left(\varphi,{z}_{{k}} \right)\:=\:\:\frac{\mathrm{1}}{\mathrm{4}{z}_{{k}} ^{\mathrm{3}} \:+\:\mathrm{6}{z}_{{k}} ^{\mathrm{5}} }\:\Rightarrow \\ $$$${Res}\left(\varphi,{z}_{\mathrm{1}} \right)\:\:=\:\frac{\mathrm{1}}{\mathrm{4}\left({e}^{{i}\frac{\pi}{\mathrm{6}}} \right)^{\mathrm{3}} \:+\mathrm{6}\left({e}^{\frac{{i}\pi}{\mathrm{6}}} \right)^{\mathrm{5}} } \\ $$$$=\:\frac{\mathrm{1}}{\mathrm{4}{i}\:+\mathrm{6}\:{e}^{\frac{{i}\mathrm{5}\pi}{\mathrm{6}}} } \\ $$$${Res}\left(\varphi,{z}_{\mathrm{2}} \right)\:=\:\:\:\frac{\mathrm{1}}{\mathrm{4}\left({e}^{\frac{{i}\pi}{\mathrm{3}}} \right)^{\mathrm{3}} \:+\mathrm{6}\:\left({e}^{\frac{{i}\pi}{\mathrm{3}}} \right)^{\mathrm{5}} }\:=\:\frac{\mathrm{1}}{−\mathrm{4}\:+\mathrm{6}\:{e}^{\frac{{i}\mathrm{5}\pi}{\mathrm{3}}} } \\ $$$${Res}\left(\varphi,{z}_{\mathrm{4}} \right)=\:\frac{\mathrm{1}}{\mathrm{4}\left({e}^{\frac{{i}\mathrm{2}\pi}{\mathrm{3}}} \right)^{\mathrm{3}} \:+\mathrm{6}\left({e}^{\frac{{i}\mathrm{2}\pi}{\mathrm{3}}} \right)^{\mathrm{5}} }\:=\frac{\mathrm{1}}{\mathrm{4}\:+\mathrm{6}\:{e}^{\frac{{i}\mathrm{10}\pi}{\mathrm{3}}} } \\ $$$$=.... \\ $$

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