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Question Number 152494 by mnjuly1970 last updated on 28/Aug/21

prove that :: Ω := ∫_(0 ) ^( ∞) (( e^( −x) .ln ((( 1)/( x)) ) sin ( x ))/(x )) dx = (( π)/( 8)) ( 2 γ +ln (2 ) ) ...■ m.n

provethat::Ω:=∫0∞e−x.ln(1x)sin(x)xdx=π8(2γ+ln(2))...◼m.n

Answered by Olaf_Thorendsen last updated on 31/Aug/21

Ω = ∫_0 ^∞ ((e^(−x) ln((1/x))sinx)/x) dx Ω = −Σ_(n=0) ^∞ (((−1)^n )/((2n+1)!))∫_0 ^∞ ((x^(2n+1) e^(−x) lnx)/x) dx Ω = −Σ_(n=0) ^∞ (((−1)^n )/((2n+1)!))∫_0 ^∞ x^(2n) e^(−x) lnx dx Ω = −Σ_(n=0) ^∞ (((−1)^n )/((2n+1)!))ψ(2n+1)Γ(2n+1) Ω = −Σ_(n=0) ^∞ (((−1)^n )/(2n+1))ψ(2n+1) to be continued...

Ω=∫0∞e−xln(1x)sinxxdxΩ=−∑∞n=0(−1)n(2n+1)!∫0∞x2n+1e−xlnxxdxΩ=−∑∞n=0(−1)n(2n+1)!∫0∞x2ne−xlnxdxΩ=−∑∞n=0(−1)n(2n+1)!ψ(2n+1)Γ(2n+1)Ω=−∑∞n=0(−1)n2n+1ψ(2n+1)tobecontinued...

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