Theorem: Definite Integral from 0 to 1 of Even Powers of Logarithm of 1 - x over x

Let $n \in \Z_{> 0}$ be a (strictly) positive integer.
:$\ds \int_0^1 \map {\ln^{2n} } {\dfrac {1 - x} x} \rd x = \paren {-1}^{n + 1} B_{2 n} \paren {2^{2 n} - 2} \pi^{2 n}$
:where $B_{2 n}$ is the $2 n$th Bernoulli number.

Proof:
let:
{{begin-eqn}}
{{eqn | l = \map \ln {\dfrac {1 - x} x}
      | r = -u
      | c = Integration by Substitution
}}
{{eqn | l = \dfrac {1 - x} x
      | r = e^{-u}
      | c =
}}
{{eqn | l = \dfrac 1 x - 1
      | r = e^{-u}
      | c =
}}
{{eqn | l = \dfrac 1 x
      | r = 1 + e^{-u}
      | c =
}}
{{eqn | l = x
      | r = \dfrac 1 {1 + e^{-u} }
      | c = 
}}
{{eqn | l = \frac {\rd x} {\rd u}
      | r = \dfrac { {e^{-u} } } {\paren { 1 + e^{-u} }^2 }
      | c = Power Rule for Derivatives, Chain Rule for Derivatives and Derivative of Exponential Function: Corollary 1
}}
{{end-eqn}}
and also:
{{begin-eqn}}
{{eqn | l = \lim_{x \mathop \to 0} \map \ln {\dfrac {1 - x} x}
      | r = \infty
      | c = 
}}
{{eqn | l = \lim_{x \mathop \to 1} \map \ln {\dfrac {1 - x} x}
      | r = -\infty
      | c = 
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn	| l = \int_0^1 \map {\ln^{2n} } {\dfrac {1 - x} x} \rd x
	    | r = -\int_{\infty}^{-\infty} \dfrac { u^{2n} e^{-u} \rd u  } {\paren { 1 + e^{-u} }^2 }
	    | c = 
}}
{{eqn	| r = \int_{-\infty}^{\infty} \dfrac { u^{2n} e^{-u} \rd u  } {\paren { 1 + e^{-u} }^2 }
	    | c = 
}}
{{eqn	| r = \int_{-\infty}^0 \dfrac { u^{2n} e^{-u} \rd u  } {\paren { 1 + e^{-u} }^2 } + \int_0^{\infty} \dfrac { u^{2n} e^{-u} \rd u  } {\paren { 1 + e^{-u} }^2 }
	    | c = Sum of Integrals on Adjacent Intervals for Integrable Functions
}}
{{eqn	| r = \int_0^{\infty} \dfrac { u^{2n} e^{u} \rd u  } {\paren { 1 + e^{u} }^2 } + \int_0^{\infty} \dfrac { u^{2n} e^{-u} \rd u  } {\paren { 1 + e^{-u} }^2 }
	    | c = Rewriting the first integral
}}
{{eqn	| r = \int_0^{\infty} \dfrac { u^{2n} e^{u} \rd u  } {\paren { 1 + e^{u} }^2 } \paren {\dfrac{e^{-2u} }{e^{-2u} } } + \int_0^{\infty} \dfrac { u^{2n} e^{-u} \rd u  } {\paren { 1 + e^{-u} }^2 }
	    | c = Multiplying numerator and denominator by $e^{-2u}$
}}
{{eqn	| r = 2\int_0^{\infty} \dfrac { u^{2n} e^{-u} \rd u  } {\paren { 1 + e^{-u} }^2 }
	    | c = 
}}
{{end-eqn}}
From Sum of Infinite Geometric Sequence, we have:
:$\ds \sum_{k \mathop = 0}^\infty \paren {-1}^k \paren {e^{-u} }^k = \dfrac 1 {1 + e^{-u} }$
Taking the derivative of both sides, we have:
{{begin-eqn}}
{{eqn	| l = \sum_{k \mathop = 0}^\infty \paren {-k} \paren {-1}^k e^{-ku} \rd u
        | r = \dfrac { e^{-u} \rd u} {\paren {1 + e^{-u} }^2}
	    | c = 
}}
{{eqn	| l = \sum_{k \mathop = 1}^\infty k \paren {-1}^{k + 1} e^{-ku} \rd u
        | r = \dfrac {  e^{-u} \rd u} {\paren {1 + e^{-u} }^2}
	    | c = simplifying
}}
{{end-eqn}}
Therefore:
{{begin-eqn}}
{{eqn	| l = \int_0^1 \map {\ln^{2n} } {\dfrac {1 - x} x} \rd x
	    | r = 2\int_0^{\infty} u^{2n} \paren { \sum_{k \mathop = 1}^\infty k \paren {-1}^{k + 1} e^{-ku} \rd u}
	    | c = 
}}
{{eqn	| r = 2\sum_{k \mathop = 1}^\infty k \paren {-1}^{k + 1} \int_0^{\infty} u^{2n} e^{-ku} \rd u
	    | c = Fubini's Theorem
}}
{{eqn	| r = 2\sum_{k \mathop = 1}^\infty k \paren {-1}^{k + 1} \bigintlimits {\frac {e^{-k u} } {-k} \paren {u^{2n} - \dfrac {2n u^{2n - 1} } {\paren {-k} } + \dfrac {2n \paren {2n - 1} u^{2n - 2} } { \paren {-k}^2 } - \dfrac {2n \paren {2n - 1} \paren {2n - 2} u^{2n - 3} } { \paren {-k}^3 } + \cdots + \dfrac {\paren {-1}^{2n} \paren {2n}!} { \paren {-k}^{2n} } } } 0 \infty
	    | c = Primitive of Power of x by Exponential of a x
}}
{{eqn	| r = 2\sum_{k \mathop = 1}^\infty k \paren {-1}^{k + 1} \paren{0 + \frac 1 k \paren{0 - 0 + 0 - 0 + \cdots + \dfrac {\paren {-1}^{2n} \paren {2n}!} { \paren {-k}^{2n} } } }
	    | c = 
}}
{{eqn	| r = 2 \paren {2n}! \sum_{k \mathop = 1}^\infty {-1}^{k + 1} \paren{\dfrac 1 { k^{2n} } }
	    | c = 
}}
{{eqn	| r = 2 \paren {2n}! \paren {\paren {-1}^{n + 1} \dfrac {B_{2 n} \paren {2^{2 n - 1} - 1} \pi^{2 n} } {\paren {2 n}!} }
	    | c = Sum of Reciprocals of Even Powers of Integers Alternating in Sign
}}
{{eqn	| r = \paren {-1}^{n + 1} B_{2 n} \paren {2^{2 n} - 2} \pi^{2 n}
	    | c = 
}}
{{end-eqn}}
{{qed}}
Category:Definite Integrals involving Logarithm Function
Category:Bernoulli Numbers
Category:Riemann Zeta Function at Even Integers
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