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1-x-5-1-dx-




Question Number 211738 by MrGaster last updated on 19/Sep/24
               ∫(1/(x^5 +1 ))dx.
$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\int\frac{\mathrm{1}}{\boldsymbol{{x}}^{\mathrm{5}} +\mathrm{1}\:}\boldsymbol{{dx}}. \\ $$$$ \\ $$
Commented by Frix last updated on 19/Sep/24
=∫(dx/((x+1)(x^2 −((1+(√5))/2)x+1)(x^2 −((1−(√5))/2)x+1)))  Now decompose and you′re done. It′s more  a writing exercise...
$$=\int\frac{{dx}}{\left({x}+\mathrm{1}\right)\left({x}^{\mathrm{2}} −\frac{\mathrm{1}+\sqrt{\mathrm{5}}}{\mathrm{2}}{x}+\mathrm{1}\right)\left({x}^{\mathrm{2}} −\frac{\mathrm{1}−\sqrt{\mathrm{5}}}{\mathrm{2}}{x}+\mathrm{1}\right)} \\ $$$$\mathrm{Now}\:\mathrm{decompose}\:\mathrm{and}\:\mathrm{you}'\mathrm{re}\:\mathrm{done}.\:\mathrm{It}'\mathrm{s}\:\mathrm{more} \\ $$$$\mathrm{a}\:\mathrm{writing}\:\mathrm{exercise}… \\ $$

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