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http://functions.wolfram.com/01.04.21.0041.01
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Integrate[Log[t (1 - t)]^4/(1 - t), {t, 0, 1/2}] ==
(16 Log[2]^5)/5 - 4 Pi^2 Zeta[3] + 72 Zeta[5]
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Cell[BoxData[RowBox[List[RowBox[List[SubsuperscriptBox["\[Integral]", "0", FractionBox["1", "2"]], RowBox[List[FractionBox[SuperscriptBox[RowBox[List["Log", "[", RowBox[List["t", " ", RowBox[List["(", RowBox[List["1", "-", "t"]], ")"]]]], "]"]], "4"], RowBox[List["1", "-", "t"]]], RowBox[List["\[DifferentialD]", "t"]]]]]], "\[Equal]", RowBox[List[FractionBox[RowBox[List["16", " ", SuperscriptBox[RowBox[List["Log", "[", "2", "]"]], "5"]]], "5"], "-", RowBox[List["4", " ", SuperscriptBox["\[Pi]", "2"], " ", RowBox[List["Zeta", "[", "3", "]"]]]], "+", RowBox[List["72", " ", RowBox[List["Zeta", "[", "5", "]"]]]]]]]]]]
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<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <msubsup> <mo> ∫ </mo> <mn> 0 </mn> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </msubsup> <mrow> <mfrac> <mrow> <msup> <mi> log </mi> <mn> 4 </mn> </msup> <mo> ( </mo> <mrow> <mi> t </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> t </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> t </mi> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <mi> t </mi> </mrow> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <mfrac> <mrow> <mn> 16 </mn> <mo> ⁢ </mo> <mrow> <msup> <mi> log </mi> <mn> 5 </mn> </msup> <mo> ( </mo> <mn> 2 </mn> <mo> ) </mo> </mrow> </mrow> <mn> 5 </mn> </mfrac> <mo> - </mo> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <msup> <mi> π </mi> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <semantics> <mrow> <mi> ζ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mn> 3 </mn> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Zeta]", "(", TagBox["3", Rule[Editable, True]], ")"]], InterpretTemplate[Function[BoxForm`e$, Zeta[BoxForm`e$]]]] </annotation> </semantics> </mrow> <mo> + </mo> <mrow> <mn> 72 </mn> <mo> ⁢ </mo> <semantics> <mrow> <mi> ζ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mn> 5 </mn> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Zeta]", "(", TagBox["5", Rule[Editable, True]], ")"]], InterpretTemplate[Function[BoxForm`e$, Zeta[BoxForm`e$]]]] </annotation> </semantics> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <int /> <bvar> <ci> t </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <cn type='rational'> 1 <sep /> 2 </cn> </uplimit> <apply> <times /> <apply> <power /> <apply> <ln /> <apply> <times /> <ci> t </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> t </ci> </apply> </apply> </apply> </apply> <cn type='integer'> 4 </cn> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> t </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 16 </cn> <apply> <power /> <apply> <ln /> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 5 </cn> </apply> <apply> <power /> <cn type='integer'> 5 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <pi /> <cn type='integer'> 2 </cn> </apply> <apply> <ci> Zeta </ci> <cn type='integer'> 3 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 72 </cn> <apply> <ci> Zeta </ci> <cn type='integer'> 5 </cn> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubsuperscriptBox["\[Integral]", "0", FractionBox["1", "2"]], RowBox[List[FractionBox[SuperscriptBox[RowBox[List["Log", "[", RowBox[List["t_", " ", RowBox[List["(", RowBox[List["1", "-", "t_"]], ")"]]]], "]"]], "4"], RowBox[List["1", "-", "t_"]]], RowBox[List["\[DifferentialD]", "t_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List["16", " ", SuperscriptBox[RowBox[List["Log", "[", "2", "]"]], "5"]]], "5"], "-", RowBox[List["4", " ", SuperscriptBox["\[Pi]", "2"], " ", RowBox[List["Zeta", "[", "3", "]"]]]], "+", RowBox[List["72", " ", RowBox[List["Zeta", "[", "5", "]"]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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