The result $$ \int_{\infty}^\infty\frac{dx}{\left(e^xx+1\right)^2+\pi^2}=\frac{1}{2}$$ holds and can be easily extracted via residues.
Use the above to prove that: $$ \int_0^\infty x^{x} e^{x}\, \Gamma(x) \sin (\pi x)\, dx= \frac {\pi}{ 2}$$ whereas \( \Gamma \) is the Gamma function defined as \(\displaystyle \Gamma(x)=\int_0^\infty t^{x1}e^{t} \, dx \) and \(x^{x} \) is the iterated function.
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Gamma, trigonometric and iterated integral
 Tolaso J Kos
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Gamma, trigonometric and iterated integral
Imagination is much more important than knowledge.

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Re: Gamma, trigonometric and iterated integral
This is very beautiful, can you give me some hint ?
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