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Question Number 34270 by math khazana by abdo last updated on 03/May/18

let give A_n = ∫_0 ^∞     (dx/((1+x^3 )^n ))  1) calculate A_1   2) for n≥2 find a relation between A_(n+1)  and A_n   3) find the value of A_n .

$${let}\:{give}\:{A}_{{n}} =\:\int_{\mathrm{0}} ^{\infty} \:\:\:\:\frac{{dx}}{\left(\mathrm{1}+{x}^{\mathrm{3}} \right)^{{n}} } \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{A}_{\mathrm{1}} \\ $$$$\left.\mathrm{2}\right)\:{for}\:{n}\geqslant\mathrm{2}\:{find}\:{a}\:{relation}\:{between}\:{A}_{{n}+\mathrm{1}} \:{and}\:{A}_{{n}} \\ $$$$\left.\mathrm{3}\right)\:{find}\:{the}\:{value}\:{of}\:{A}_{{n}} . \\ $$

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