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Question Number 128244 by mnjuly1970 last updated on 05/Jan/21
...nicecalculus...provethat::Ω=∫0π4ln(sin(x))d=−π4log(2)−G2log(2sin(x))=∑∞n=1−1ncos(2nx)Ω=∫0π4{−log(2)−∑∞n=1cos(2nx)n}dx=−π4log(2)−∑∞n=1∫0π4cos(2nx)ndx=−π4log(2)−∑∞n=1[12n2sin(2nx)]0π4=−π4log(2)−12∑∞n=1sin(nπ2)n2=−π4log(2)−12{112−132+152−..}=−π4log(2)−12∑∞n=1(−1)n−1(2n−1)2∴Ω=−π4log(2)−G2✓G:=catalanconstant...
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![...nice calculus... prove that ::Ω= ∫_0 ^(π/4) ln(sin(x))d=((−π)/4)log(2)−(G/2) log(2sin(x))=Σ_(n=1) ^∞ ((−1)/n)cos(2nx) Ω= ∫_0 ^( (π/4)) {−log(2)−Σ_(n=1) ^∞ ((cos(2nx))/n)}dx =((−π)/4)log(2)−Σ_(n=1) ^∞ ∫_0 ^( (π/4)) ((cos(2nx))/n)dx =((−π)/4)log(2)−Σ_(n=1) ^∞ [(1/(2n^2 ))sin(2nx)]_0 ^(π/4) =((−π)/4)log(2)−(1/2)Σ_(n=1) ^∞ ((sin(((nπ)/2)))/n^2 ) =((−π)/4)log(2)−(1/2){(1/1^2 )−(1/3^2 ) +(1/5^2 )−..} =((−π)/4)log(2)−(1/2)Σ_(n=1) ^∞ (((−1)^(n−1) )/((2n−1)^2 )) ∴ Ω =((−π)/4)log(2)−(G/2) ✓ G:= catalan constant...](Q128244.png)