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Evaluate-I-0-1-x-5-x-4-x-3-x-2-x-1-dx-

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Question Number 193377 by Humble last updated on 12/Jun/23
Evaluate I=∫_0 ^( ∞) (1/(x^5 +x^4 +x^3 +x^2 +x+1))dx
$${Evaluate} \\ $$$${I}=\int_{\mathrm{0}} ^{\:\infty} \frac{\mathrm{1}}{{x}^{\mathrm{5}} +{x}^{\mathrm{4}} +{x}^{\mathrm{3}} +{x}^{\mathrm{2}} +{x}+\mathrm{1}}{dx} \\ $$
Commented by Frix last updated on 12/Jun/23
x^5 +x^4 +x^3 +x^2 +x+1= =(x+1)(x^2 −x+1)(x^2 +x+1) Just decompose and solve it!
$${x}^{\mathrm{5}} +{x}^{\mathrm{4}} +{x}^{\mathrm{3}} +{x}^{\mathrm{2}} +{x}+\mathrm{1}= \\ $$$$=\left({x}+\mathrm{1}\right)\left({x}^{\mathrm{2}} −{x}+\mathrm{1}\right)\left({x}^{\mathrm{2}} +{x}+\mathrm{1}\right) \\ $$$$\mathrm{Just}\:\mathrm{decompose}\:\mathrm{and}\:\mathrm{solve}\:\mathrm{it}! \\ $$
Answered by Mathspace last updated on 16/Jun/23
I=∫_0 ^∞ (dx/(1+x+x^2 +x^3 +x^4 +x^5 )) I=∫_0 ^∞ (dx/((1−x^6 )/(1−x)))=∫_0 ^∞ ((1−x)/(1−x^6 ))dx =∫_0 ^1 ((1−x)/(1−x^6 ))dx +∫_1 ^∞ ((1−x)/(1−x^6 ))dx(→x=(1/t)) =∫_0 ^1 ((1−x)/(1−x^6 ))dx−∫_0 ^1 ((1−(1/t))/(1−(1/t^6 )))×((−dt)/t^2 ) =∫_0 ^1 ((1−x)/(1−x^6 ))dx+∫_0 ^1 (1/t^3 ).((t−1)/(t^6 −1)).t^6 dt =∫_0 ^1 ((1−x)/(1−x^6 ))dx+∫_0 ^1 ((t^4 −t^3 )/(t^6 −1))dt =∫_0 ^1 ((1−x)/(1−x^6 ))dx+∫_0 ^1 ((x^3 −x^4 )/(1−x^6 ))dx =∫_0 ^1 ((1−x+x^3 −x^4 )/(1−x^6 ))dx =∫_0 ^1 (1−x+x^3 −x^4 )Σ_(n=0) ^∞ x^(6n) dx =Σ_(n=0) ^∞ ∫_0 ^1 (x^(6n) −x^(6n+1) +x^(6n+3) −x^(6n+4) )dx =Σ_(n=0) ^∞ ((1/(6n+1))−(1/(6n+2))+(1/(6n+4))−(1/(6n+5))) =Σ_(n=0) ^∞ (1/((6n+1)(6n+2))) +Σ_(n=0) ^∞ (1/((6n+4)(6n+5))) we have Σ_(n=0) ^∞ (1/((6n+1)(6n+2))) =(1/(36))Σ_(n=0) ^∞ (1/((n+(1/6))(n+(1/3)))) =(1/(36)).((Ψ((1/3))−Ψ((1/6)))/((1/3)−(1/6))) =(1/6){Ψ((1/3))−Ψ((1/6))} Σ_(n=0) ^∞ (1/((6n+4)(6n+5))) =(1/(36))Σ_(n=0) ^∞ (1/((n+(2/3))(n+(5/6)))) =(1/(36)).((Ψ((5/6))−Ψ((2/3)))/((5/6)−(2/3))) =(1/6)(Ψ((5/6))−Ψ((2/3))) ⇒I=(1/6){Ψ((1/3))−Ψ((1/6))+Ψ((5/6))−Ψ((2/3))} after use special values of Ψ....
$${I}=\int_{\mathrm{0}} ^{\infty} \:\:\frac{{dx}}{\mathrm{1}+{x}+{x}^{\mathrm{2}} +{x}^{\mathrm{3}} +{x}^{\mathrm{4}} +{x}^{\mathrm{5}} } \\ $$$${I}=\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{dx}}{\frac{\mathrm{1}−{x}^{\mathrm{6}} }{\mathrm{1}−{x}}}=\int_{\mathrm{0}} ^{\infty} \frac{\mathrm{1}−{x}}{\mathrm{1}−{x}^{\mathrm{6}} }{dx} \\ $$$$=\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\mathrm{1}−{x}}{\mathrm{1}−{x}^{\mathrm{6}} }{dx}\:+\int_{\mathrm{1}} ^{\infty} \frac{\mathrm{1}−{x}}{\mathrm{1}−{x}^{\mathrm{6}} }{dx}\left(\rightarrow{x}=\frac{\mathrm{1}}{{t}}\right) \\ $$$$=\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\mathrm{1}−{x}}{\mathrm{1}−{x}^{\mathrm{6}} }{dx}−\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\mathrm{1}−\frac{\mathrm{1}}{{t}}}{\mathrm{1}−\frac{\mathrm{1}}{{t}^{\mathrm{6}} }}×\frac{−{dt}}{{t}^{\mathrm{2}} } \\ $$$$=\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\mathrm{1}−{x}}{\mathrm{1}−{x}^{\mathrm{6}} }{dx}+\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\mathrm{1}}{{t}^{\mathrm{3}} }.\frac{{t}−\mathrm{1}}{{t}^{\mathrm{6}} −\mathrm{1}}.{t}^{\mathrm{6}} {dt} \\ $$$$=\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\mathrm{1}−{x}}{\mathrm{1}−{x}^{\mathrm{6}} }{dx}+\int_{\mathrm{0}} ^{\mathrm{1}} \frac{{t}^{\mathrm{4}} −{t}^{\mathrm{3}} }{{t}^{\mathrm{6}} −\mathrm{1}}{dt} \\ $$$$=\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\mathrm{1}−{x}}{\mathrm{1}−{x}^{\mathrm{6}} }{dx}+\int_{\mathrm{0}} ^{\mathrm{1}} \frac{{x}^{\mathrm{3}} −{x}^{\mathrm{4}} }{\mathrm{1}−{x}^{\mathrm{6}} }{dx} \\ $$$$=\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\mathrm{1}−{x}+{x}^{\mathrm{3}} −{x}^{\mathrm{4}} }{\mathrm{1}−{x}^{\mathrm{6}} }{dx} \\ $$$$=\int_{\mathrm{0}} ^{\mathrm{1}} \left(\mathrm{1}−{x}+{x}^{\mathrm{3}} −{x}^{\mathrm{4}} \right)\sum_{{n}=\mathrm{0}} ^{\infty} {x}^{\mathrm{6}{n}} {dx} \\ $$$$=\sum_{{n}=\mathrm{0}} ^{\infty} \int_{\mathrm{0}} ^{\mathrm{1}} \left({x}^{\mathrm{6}{n}} −{x}^{\mathrm{6}{n}+\mathrm{1}} +{x}^{\mathrm{6}{n}+\mathrm{3}} −{x}^{\mathrm{6}{n}+\mathrm{4}} \right){dx} \\ $$$$=\sum_{{n}=\mathrm{0}} ^{\infty} \left(\frac{\mathrm{1}}{\mathrm{6}{n}+\mathrm{1}}−\frac{\mathrm{1}}{\mathrm{6}{n}+\mathrm{2}}+\frac{\mathrm{1}}{\mathrm{6}{n}+\mathrm{4}}−\frac{\mathrm{1}}{\mathrm{6}{n}+\mathrm{5}}\right) \\ $$$$=\sum_{{n}=\mathrm{0}} ^{\infty} \frac{\mathrm{1}}{\left(\mathrm{6}{n}+\mathrm{1}\right)\left(\mathrm{6}{n}+\mathrm{2}\right)} \\ $$$$+\sum_{{n}=\mathrm{0}} ^{\infty} \frac{\mathrm{1}}{\left(\mathrm{6}{n}+\mathrm{4}\right)\left(\mathrm{6}{n}+\mathrm{5}\right)}\:\:{we}\:{have} \\ $$$$\sum_{{n}=\mathrm{0}} ^{\infty} \frac{\mathrm{1}}{\left(\mathrm{6}{n}+\mathrm{1}\right)\left(\mathrm{6}{n}+\mathrm{2}\right)} \\ $$$$=\frac{\mathrm{1}}{\mathrm{36}}\sum_{{n}=\mathrm{0}} ^{\infty} \frac{\mathrm{1}}{\left({n}+\frac{\mathrm{1}}{\mathrm{6}}\right)\left({n}+\frac{\mathrm{1}}{\mathrm{3}}\right)} \\ $$$$=\frac{\mathrm{1}}{\mathrm{36}}.\frac{\Psi\left(\frac{\mathrm{1}}{\mathrm{3}}\right)−\Psi\left(\frac{\mathrm{1}}{\mathrm{6}}\right)}{\frac{\mathrm{1}}{\mathrm{3}}−\frac{\mathrm{1}}{\mathrm{6}}} \\ $$$$=\frac{\mathrm{1}}{\mathrm{6}}\left\{\Psi\left(\frac{\mathrm{1}}{\mathrm{3}}\right)−\Psi\left(\frac{\mathrm{1}}{\mathrm{6}}\right)\right\} \\ $$$$\sum_{{n}=\mathrm{0}} ^{\infty} \frac{\mathrm{1}}{\left(\mathrm{6}{n}+\mathrm{4}\right)\left(\mathrm{6}{n}+\mathrm{5}\right)} \\ $$$$=\frac{\mathrm{1}}{\mathrm{36}}\sum_{{n}=\mathrm{0}} ^{\infty} \frac{\mathrm{1}}{\left({n}+\frac{\mathrm{2}}{\mathrm{3}}\right)\left({n}+\frac{\mathrm{5}}{\mathrm{6}}\right)} \\ $$$$=\frac{\mathrm{1}}{\mathrm{36}}.\frac{\Psi\left(\frac{\mathrm{5}}{\mathrm{6}}\right)−\Psi\left(\frac{\mathrm{2}}{\mathrm{3}}\right)}{\frac{\mathrm{5}}{\mathrm{6}}−\frac{\mathrm{2}}{\mathrm{3}}} \\ $$$$=\frac{\mathrm{1}}{\mathrm{6}}\left(\Psi\left(\frac{\mathrm{5}}{\mathrm{6}}\right)−\Psi\left(\frac{\mathrm{2}}{\mathrm{3}}\right)\right) \\ $$$$\Rightarrow{I}=\frac{\mathrm{1}}{\mathrm{6}}\left\{\Psi\left(\frac{\mathrm{1}}{\mathrm{3}}\right)−\Psi\left(\frac{\mathrm{1}}{\mathrm{6}}\right)+\Psi\left(\frac{\mathrm{5}}{\mathrm{6}}\right)−\Psi\left(\frac{\mathrm{2}}{\mathrm{3}}\right)\right\} \\ $$$${after}\:{use}\:{special}\:{values}\:{of}\:\Psi…. \\ $$

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