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0-1-1-x-x-2-x-3-x-4-x-5-1-x-7-dx-

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Question Number 219529 by Nicholas666 last updated on 27/Apr/25
∫_( 0) ^( 1) ((1+x−x^2 +x^3 −x^4 −x^5 )/(1−x^7 )) dx
$$ \\ $$$$\:\:\:\:\int_{\:\mathrm{0}} ^{\:\mathrm{1}} \:\:\:\:\frac{\mathrm{1}+{x}−{x}^{\mathrm{2}} +{x}^{\mathrm{3}} −{x}^{\mathrm{4}} −{x}^{\mathrm{5}} }{\mathrm{1}−{x}^{\mathrm{7}} }\:\:\:{dx} \\ $$$$ \\ $$
Commented by SdC355 last updated on 29/Apr/25
Generalize ∫ ((p(z))/(q(z))) dz=Σ_(ζ_h ; q(ζ)=0) ((p(ζ_h )ln(z−ζ_h ))/(q^((1)) (ζ_h )))+C ζ_h is q(ζ)=0 Solution ζ_h , h=1,2,3... and polynomial p(z)=Σ_(h=1) ^m a_h z^h , q(z)=Σ_(h=1) ^n b_h z^h ∫ ((a_1 z+a_2 z^2 +.....a_m z^m )/(b_1 z+b_2 z^2 +......b_n z^n )) dz=Σ_(ζ;q(ζ_h )=0) ((p(ζ_h )ln(z−ζ_h ))/(q^((1)) (ζ_h )))+C
$$\mathrm{Generalize}\:\:\:\int\:\:\:\frac{\mathrm{p}\left({z}\right)}{\mathrm{q}\left({z}\right)}\:\mathrm{d}{z}=\underset{\zeta_{{h}} ;\:{q}\left(\zeta\right)=\mathrm{0}} {\sum}\:\frac{\mathrm{p}\left(\zeta_{{h}} \right)\mathrm{ln}\left({z}−\zeta_{{h}} \right)}{\mathrm{q}^{\left(\mathrm{1}\right)} \left(\zeta_{{h}} \right)}+{C} \\ $$$$\zeta_{{h}} \:\mathrm{is}\:\mathrm{q}\left(\zeta\right)=\mathrm{0}\:\mathrm{Solution}\:\zeta_{{h}} \:,\:{h}=\mathrm{1},\mathrm{2},\mathrm{3}… \\ $$$$\mathrm{and}\:\mathrm{polynomial} \\ $$$$\mathrm{p}\left({z}\right)=\underset{{h}=\mathrm{1}} {\overset{{m}} {\sum}}\:{a}_{{h}} {z}^{{h}} \:,\:\mathrm{q}\left({z}\right)=\underset{{h}=\mathrm{1}} {\overset{{n}} {\sum}}\:{b}_{{h}} {z}^{{h}} \\ $$$$\int\:\:\:\:\frac{{a}_{\mathrm{1}} {z}+{a}_{\mathrm{2}} {z}^{\mathrm{2}} +…..{a}_{{m}} {z}^{{m}} }{{b}_{\mathrm{1}} {z}+{b}_{\mathrm{2}} {z}^{\mathrm{2}} +……{b}_{{n}} {z}^{{n}} }\:\mathrm{d}{z}=\underset{\zeta;{q}\left(\zeta_{{h}} \right)=\mathrm{0}} {\sum}\:\:\:\frac{\mathrm{p}\left(\zeta_{{h}} \right)\mathrm{ln}\left({z}−\zeta_{{h}} \right)}{\mathrm{q}^{\left(\mathrm{1}\right)} \left(\zeta_{{h}} \right)}+{C} \\ $$
Answered by SdC355 last updated on 28/Apr/25
∫ ∙=Σ_(ζ;ζ^6 +ζ^5 +ζ^4 +ζ^3 +ζ^2 +ζ+1=0) (((ζ^4 +2ζ^3 +ζ^2 +2ζ+1)ln(z−ζ))/(6ζ^5 +5ζ^4 +4ζ^3 +3ζ^2 +2ζ+1)) ∫_0 ^( 1) ∙≈ 1.18741......
$$\int\:\centerdot=\underset{\zeta;\zeta^{\mathrm{6}} +\zeta^{\mathrm{5}} +\zeta^{\mathrm{4}} +\zeta^{\mathrm{3}} +\zeta^{\mathrm{2}} +\zeta+\mathrm{1}=\mathrm{0}} {\sum}\frac{\left(\zeta^{\mathrm{4}} +\mathrm{2}\zeta^{\mathrm{3}} +\zeta^{\mathrm{2}} +\mathrm{2}\zeta+\mathrm{1}\right)\mathrm{ln}\left({z}−\zeta\right)}{\mathrm{6}\zeta^{\mathrm{5}} +\mathrm{5}\zeta^{\mathrm{4}} +\mathrm{4}\zeta^{\mathrm{3}} +\mathrm{3}\zeta^{\mathrm{2}} +\mathrm{2}\zeta+\mathrm{1}} \\ $$$$\int_{\mathrm{0}} ^{\:\mathrm{1}} \:\centerdot\approx\:\mathrm{1}.\mathrm{18741}…… \\ $$
Commented by Nicholas666 last updated on 28/Apr/25
thanks
$${thanks} \\ $$

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