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nice-calculus-evaluate-2-1-x-1-2-x-dx-0-1-ln-x-dx-x-fractional-part-




Question Number 124887 by mnjuly1970 last updated on 06/Dec/20
     ...nice  calculus..     evaluate :    2∫_1 ^( ∞) ((({x}−(1/2))/x))dx−∫_0 ^( 1) ln(Γ(x))dx=???  {x}: fractional part...
nicecalculus..evaluate:21({x}12x)dx01ln(Γ(x))dx=???{x}:fractionalpart
Answered by Dwaipayan Shikari last updated on 06/Dec/20
∫_1 ^∞ ((({x}−(1/2))/x))dx =(1/2)log(2π)−1    (Proved previously)  ∫_0 ^1 log(Γ(x))=(1/2)(∫_0 ^1 log(π)−log(sinπx))=(1/2)log(2π)  2∫_1 ^∞ ((({x}−(1/2))/x))dx −∫_0 ^1 log(Γ(x))dx=(1/2)log(2π)−2
1({x}12x)dx=12log(2π)1(Provedpreviously)01log(Γ(x))=12(01log(π)log(sinπx))=12log(2π)21({x}12x)dx01log(Γ(x))dx=12log(2π)2
Commented by mnjuly1970 last updated on 06/Dec/20
excellent...
excellent

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