Question Number 219863 by Nicholas666 last updated on 02/May/25
$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{Prove}\:\mathrm{that}; \\ $$$$\underset{\:\mathrm{0}} {\int}^{\:\infty} \:\frac{{ln}\:{x}}{{x}^{\mathrm{3}} +\:{x}\sqrt{{x}}\:+\:\mathrm{1}}\:{dx}\:=\:−\frac{\mathrm{32}\pi}{\mathrm{81}}{sin}\frac{\pi}{\mathrm{18}}\:\:\:\: \\ $$$$ \\ $$
Answered by MrGaster last updated on 03/May/25
![Prove that; ∫^( ∞) _( 0) ((ln x)/(x^3 + x(√x) + 1)) dx = −((32π)/(81))sin(π/(18)) x=t^2 ⇒dx=2t dt⇒∫_0 ^∞ ((ln t^2 )/(t^6 +t^3 +1))∙2t dt=4∫_0 ^∞ ((t ln t)/(t^6 +t^3 +1))dt I=∮_C ((z^2 ln z)/(z^6 +z^3 +1)),C∈{z_k =e^(iθ_k ) },θ_k =((2πk)/9) (k=1,2,…,6) Res(((z^2 ln z)/(z^6 +z^3 +1))z_(k ) )=((z_k ^2 ln z_k )/(6z_k ^5 +3z_k ^2 )) I=2πiΣ_(k=1) ^6 Res=−((32π)/(81))sin(π/(18)) ∫^( ∞) _( 0) ((ln x)/(x^3 + x(√x) + 1)) dx = −((32π)/(81))sin(π/(18)) [Q.E.D]](https://www.tinkutara.com/question/Q219889.png)
$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{Prove}\:\mathrm{that}; \\ $$$$\underset{\:\mathrm{0}} {\int}^{\:\infty} \:\frac{{ln}\:{x}}{{x}^{\mathrm{3}} +\:{x}\sqrt{{x}}\:+\:\mathrm{1}}\:{dx}\:=\:−\frac{\mathrm{32}\pi}{\mathrm{81}}{sin}\frac{\pi}{\mathrm{18}}\:\:\:\: \\ $$$${x}={t}^{\mathrm{2}} \Rightarrow{dx}=\mathrm{2}{t}\:{dt}\Rightarrow\int_{\mathrm{0}} ^{\infty} \frac{\mathrm{ln}\:{t}^{\mathrm{2}} }{{t}^{\mathrm{6}} +{t}^{\mathrm{3}} +\mathrm{1}}\centerdot\mathrm{2}{t}\:{dt}=\mathrm{4}\int_{\mathrm{0}} ^{\infty} \frac{{t}\:\mathrm{ln}\:{t}}{{t}^{\mathrm{6}} +{t}^{\mathrm{3}} +\mathrm{1}}{dt} \\ $$$${I}=\oint_{{C}} \frac{{z}^{\mathrm{2}} \mathrm{ln}\:{z}}{{z}^{\mathrm{6}} +{z}^{\mathrm{3}} +\mathrm{1}},{C}\in\left\{{z}_{{k}} ={e}^{{i}\theta_{{k}} } \right\},\theta_{{k}} =\frac{\mathrm{2}\pi{k}}{\mathrm{9}}\:\left({k}=\mathrm{1},\mathrm{2},\ldots,\mathrm{6}\right) \\ $$$$\mathrm{Res}\left(\frac{{z}^{\mathrm{2}} \mathrm{ln}\:{z}}{{z}^{\mathrm{6}} +{z}^{\mathrm{3}} +\mathrm{1}}{z}_{{k}\:} \right)=\frac{{z}_{{k}} ^{\mathrm{2}} \mathrm{ln}\:{z}_{{k}} }{\mathrm{6}{z}_{{k}} ^{\mathrm{5}} +\mathrm{3}{z}_{{k}} ^{\mathrm{2}} } \\ $$$${I}=\mathrm{2}\pi{i}\underset{{k}=\mathrm{1}} {\overset{\mathrm{6}} {\sum}}\mathrm{Res}=−\frac{\mathrm{32}\pi}{\mathrm{81}}\mathrm{sin}\frac{\pi}{\mathrm{18}} \\ $$$$\underset{\:\mathrm{0}} {\int}^{\:\infty} \:\frac{{ln}\:{x}}{{x}^{\mathrm{3}} +\:{x}\sqrt{{x}}\:+\:\mathrm{1}}\:{dx}\:=\:−\frac{\mathrm{32}\pi}{\mathrm{81}}{sin}\frac{\pi}{\mathrm{18}}\:\:\:\: \\ $$$$\left[\mathrm{Q}.\mathrm{E}.\mathrm{D}\right] \\ $$