... advanced math ... two simple and nice integrals: prove that:: Ω_1 =∫_0 ^( ∞) ((sin(e^(−γ) x)ln(x))/x) dx=0 Ω_2 =∫_0 ^( ∞) ((sin(x^((√2)/2) )ln(x))/x)dx=−πγ note


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Question Number 122713 by mnjuly1970 last updated on 19/Nov/20

$. . . a d v a n c e d m a t h . . .$ $t w o s i m p l e a n d n i c e i n t e g r a l s :$ $p r o v e t h a t ::$ $Ω_{1} = \int_{0}^{\infty} \frac{s i n (e^{- γ} x) l n (x)}{x} d x = 0$ $Ω_{2} = \int_{0}^{\infty} \frac{s i n (x^{\frac{\sqrt{2}}{2}}) l n (x)}{x} d x = - π γ$ $n o t e :: γ : E u l e r - m a s c h e r o n i$ $c o n s t a n t .$ $. m . n .$
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