show that ∫_0 ^1 ∫_0 ^1 ∫_0 ^1 ((log(xyz))/((1+x^2 )(1+y^2 )(1+z^2 ))) dx dy dz=((−3π^2 G)/(16))


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Question Number 84680 by M±th+et£s last updated on 15/Mar/20

$s h o w t h a t$ $\int_{0}^{1} \int_{0}^{1} \int_{0}^{1} \frac{l o g (x y z)}{(1 + x^{2}) (1 + y^{2}) (1 + z^{2})} d x d y d z = \frac{- 3 π^{2} G}{16}$
	Answered by mind is power last updated on 15/Mar/20

	$= \int_{0}^{1} \int_{0}^{1} \int_{0}^{1} \frac{l n (x) + l n (y) + l n (z)}{(1 + x^{2}) (1 + y^{2}) (1 + z^{2})} d x d y d z$ $b y S y m e t r i e$ $= 3 \int_{0}^{1} \int_{0}^{1} \int_{0}^{1} \frac{l n (x)}{(1 + x^{2}) (1 + y^{2}) (1 + z^{2})} d x d y d z$ $= 3 \int_{0}^{1} \int_{0}^{1} \frac{d y d z}{(1 + y^{2}) (1 + z^{2})} \int_{0}^{1} \frac{l n (x)}{(1 + x^{2})} d x$ $- G = \int_{0}^{1} \frac{l n (x)}{1 + x^{2}} \Rightarrow$ $= - 3 G \int_{0}^{1} \int_{0}^{1} \frac{d y d z}{(1 + y^{2}) (1 + z^{2})} = - 3 G \int_{0}^{1} \frac{d y}{(1 + y^{2})} \int_{0}^{1} \frac{d z}{1 + z^{2}}$ $= - 3 G . {(\frac{π}{4})}^{2} = \frac{- 3 π^{2} G}{16}$
	Commented by M±th+et£s last updated on 15/Mar/20

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