prove that i:Σ_(n=0) ^∞ (((−1 )^( n) )/((n +(1/2))cosh(n+(1/2))π)) =(π/4) ii: ∫_0 ^( 1) (( sin( π x ))/(x^( x) ( 1−x )^( 1−x) )) (dx/(1+x)) =(π/4) −−−


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Question Number 163002 by mnjuly1970 last updated on 03/Jan/22

$p r o v e t h a t$ $i : \underset{n = 0}{\sum^{\infty}} \frac{{(- 1)}^{n}}{(n + \frac{1}{2}) c o s h (n + \frac{1}{2}) π} = \frac{π}{4}$ $i i : \int_{0}^{1} \frac{s i n (π x)}{x^{x} {(1 - x)}^{1 - x}} \frac{d x}{1 + x} = \frac{π}{4}$ $- - -$
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