How can I prove the following: $$ \int _0^1\:\frac{\sin \left(\pi x\right)}{x^x\left(1-x\right)^{1-x}}dx=\frac{\pi }{e} $$ I'd start by taking the imaginary part of: $$ \int _0^1\:\frac{\exp\left(\pi x i\right)}{x^x\left(1-x\right)^{1-x}}dx $$ But I don't know where to go from there? How can I use complex analysis to evaluate this integral?
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i would rather try with substitution first – Mar 21 '20 at 23:29
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1It has a solution here. – NoName Mar 21 '20 at 23:37