Prove-0-1-k-1-1-x-k-dx-4pi-3-23-sinh-23-pi-6-2-cosh-23-pi-3-1-

Question Number 221663 by MrGaster last updated on 09/Jun/25

Prove:∫_0 ^1 Π_(k=1) ^∞ (1−x^k )dx=((4π(√3))/( (√(23))))∙((sinh(((√(23))π)/6))/(2 cosh(((√(23))π)/3)−1))

$$\mathrm{Prove}:\int_{\mathrm{0}} ^{\mathrm{1}} \underset{{k}=\mathrm{1}} {\overset{\infty} {\prod}}\left(\mathrm{1}−{x}^{{k}} \right){dx}=\frac{\mathrm{4}\pi\sqrt{\mathrm{3}}}{\:\sqrt{\mathrm{23}}}\centerdot\frac{\mathrm{sinh}\frac{\sqrt{\mathrm{23}}\pi}{\mathrm{6}}}{\mathrm{2}\:\mathrm{cosh}\frac{\sqrt{\mathrm{23}}\pi}{\mathrm{3}}−\mathrm{1}} \\ $$

Commented by MrGaster last updated on 09/Jun/25

It is difficult for me to give an analytical solution to that integral.

Commented by Tawa11 last updated on 09/Jun/25

$$\mathrm{Sir},\:\mathrm{please}\:\mathrm{any}\:\mathrm{correction}\:\mathrm{on}\:\:\:\mathrm{Q221587}?? \\ $$

Commented by Tawa11 last updated on 10/Jun/25

$$\mathrm{Thanks}\:\mathrm{sir}.\:\mathrm{I}\:\mathrm{appreciate}. \\ $$

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