prove-that-0-1-x-2-x-2-1-2-ln-8pi-

Question Number 167048 by mnjuly1970 last updated on 05/Mar/22

p r o v e t h a t

Φ = \int_{0}^{1} x . ψ (2 + x) = 2 - \frac{1}{2} l n (8 π)

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Answered by shikaridwan last updated on 05/Mar/22

ψ (x + 2) = ψ (x + 1) + \frac{1}{x + 1} = ψ (x) + \frac{1}{x} + \frac{1}{x + 1}

\int_{0}^{1} x ψ (2 + x) d x = \int_{0}^{1} 2 - \frac{1}{x + 1} + x ψ (x) d x

= 2 - \log (2) + {[x l o g (Γ (x))]}_{0}^{1} - \int_{0}^{1} l o g (Γ (x)) d x

= 2 - \log (2) - l o g (\sqrt{2 π})

= 2 - \frac{1}{2} \log (8 π)

Commented by shikaridwan last updated on 05/Mar/22

H i s i r! C a n y o u r e c o g n i z e m e ?

Commented by mnjuly1970 last updated on 05/Mar/22

t h s n k y o u s o m u c h

M r . D w a y p a y a n \dots t h a n k s a l o t

a n d W e l c o m e a g a i n \dots .