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Question Number 116554 by Bird last updated on 04/Oct/20

study the integral ∫_0 ^∞ ((sin^n x)/x^n )dx n integr and n≥2

$${study}\:{the}\:{integral}\:\int_{\mathrm{0}} ^{\infty} \:\frac{{sin}^{{n}} {x}}{{x}^{{n}} }{dx} \\ $$$${n}\:{integr}\:{and}\:{n}\geqslant\mathrm{2} \\ $$

Answered by Olaf last updated on 05/Oct/20

I_(2n) = ((nπ)/((2n)!))Σ_(k=1) ^n C_(2n) ^(n−k) (−1)^(n−k) k^(2n−1) I_(2n+1) = (π/(2(2n)!))Σ_(k=0) ^n C_(2n+1) ^(n−k) (−1)^(n−k) (k+(1/2))^(2n) The demonstration is very long and not so easy for this kind of apps.

$$\mathrm{I}_{\mathrm{2}{n}} \:=\:\frac{{n}\pi}{\left(\mathrm{2}{n}\right)!}\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}{C}_{\mathrm{2}{n}} ^{{n}−{k}} \left(−\mathrm{1}\right)^{{n}−{k}} {k}^{\mathrm{2}{n}−\mathrm{1}} \\ $$$$\mathrm{I}_{\mathrm{2}{n}+\mathrm{1}} \:=\:\frac{\pi}{\mathrm{2}\left(\mathrm{2}{n}\right)!}\underset{{k}=\mathrm{0}} {\overset{{n}} {\sum}}{C}_{\mathrm{2}{n}+\mathrm{1}} ^{{n}−{k}} \left(−\mathrm{1}\right)^{{n}−{k}} \left({k}+\frac{\mathrm{1}}{\mathrm{2}}\right)^{\mathrm{2}{n}} \\ $$$$ \\ $$$$\mathrm{The}\:\mathrm{demonstration}\:\mathrm{is}\:\mathrm{very}\:\mathrm{long}\:\mathrm{and} \\ $$$$\mathrm{not}\:\mathrm{so}\:\mathrm{easy}\:\mathrm{for}\:\mathrm{this}\:\mathrm{kind}\:\mathrm{of}\:\mathrm{apps}. \\ $$

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