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Question Number 152240 by mnjuly1970 last updated on 26/Aug/21

prove that.. csch (x)= (1/x) + Σ_(n=1) ^∞ ((2.(−1)^( n) x)/(n^( 2) π^( 2) + x^( 2) )) then find: Ω := ∫_0 ^( ∞) ((cosh (x )−(1/x))/x) dx=−ln(2)....■

provethat..csch(x)=1x+∑∞n=12.(−1)nxn2π2+x2thenfind:Ω:=∫0∞cosh(x)−1xxdx=−ln(2)....◼

Answered by Kamel last updated on 27/Aug/21

Ω=∫_0 ^(+∞) ((cosech(x)−(1/x))/x)dx=∫_0 ^(+∞) ∫_0 ^(+∞) (((2e^(−x) )/(1−e^(−2x) ))−(1/x))e^(−tx) dxdt =^(IBP) ∫_0 ^(+∞) (Ln(2)+t∫_0 ^(+∞) (Ln(((1−e^(−x) )/x))−Ln(1+e^(−x) ))e^(−xt) dx)dt =∫_0 ^(+∞) (Ln(2)+t(2∫_0 ^1 Ln(1−u)u^(t−1) du−(1/2)∫_0 ^1 Ln(1−u)u^((t/2)−1) du+((γ+Ln(t))/t)))dt =∫_0 ^(+∞) (Ln(2)+t(−(2/t)∫_0 ^1 ((u^t −1)/(u−1))du+(1/t)∫_0 ^1 ((u^(t/2) −1)/(u−1))du+((γ+Ln(t))/t)))dt =∫_0 ^(+∞) (Ln(t)+Ψ((t/2)+1)−2Ψ(t+1)+Ln(2))dt =lim_(t→+∞) (tLn(t)−t+2Ln(Γ((t/2)+1))−2LnΓ(t+1)+tLn(2)) =lim_(t→+∞) Ln(((t^t e^(−t) Γ^2 ((t/2)+1)2^t )/(Γ^2 (t+1))))=lim_(t→+∞) Ln(((t^(2t) πte^(−2t) )/(t^(2t) 2πte^(−2t) )))=−Ln(2) ∴ ∫_0 ^(+∞) ((cosech(x)−(1/x))/x)dx=−Ln(2) KAMEL BENAICHA

Ω=∫0+∞cosech(x)−1xxdx=∫0+∞∫0+∞(2e−x1−e−2x−1x)e−txdxdt=IBP∫0+∞(Ln(2)+t∫0+∞(Ln(1−e−xx)−Ln(1+e−x))e−xtdx)dt=∫0+∞(Ln(2)+t(2∫01Ln(1−u)ut−1du−12∫01Ln(1−u)ut2−1du+γ+Ln(t)t))dt=∫0+∞(Ln(2)+t(−2t∫01ut−1u−1du+1t∫01ut2−1u−1du+γ+Ln(t)t))dt=∫0+∞(Ln(t)+Ψ(t2+1)−2Ψ(t+1)+Ln(2))dt=limt→+∞(tLn(t)−t+2Ln(Γ(t2+1))−2LnΓ(t+1)+tLn(2))=limLnt→+∞(tte−tΓ2(t2+1)2tΓ2(t+1))=limLnt→+∞(t2tπte−2tt2t2πte−2t)=−Ln(2)∴∫0+∞cosech(x)−1xxdx=−Ln(2)KAMELBENAICHA

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