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http://functions.wolfram.com/01.03.21.0033.01
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Integrate[t^k/(E^t + 1), {t, 0, Infinity}] ==
((-1 + 2^k) Gamma[k + 1] Zeta[k + 1])/2^k /; Re[k] > -1
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[SubsuperscriptBox["\[Integral]", "0", "\[Infinity]"], RowBox[List[FractionBox[SuperscriptBox["t", "k"], RowBox[List[SuperscriptBox["\[ExponentialE]", "t"], "+", "1"]]], RowBox[List["\[DifferentialD]", "t"]]]]]], "\[Equal]", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", SuperscriptBox["2", "k"]]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List["k", "+", "1"]], "]"]], " ", RowBox[List["Zeta", "[", RowBox[List["k", "+", "1"]], "]"]]]], SuperscriptBox["2", "k"]]]], "/;", RowBox[List[RowBox[List["Re", "[", "k", "]"]], ">", RowBox[List["-", "1"]]]]]]]]
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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> <mrow> <msubsup> <mo> ∫ </mo> <mn> 0 </mn> <mi> ∞ </mi> </msubsup> <mrow> <mfrac> <msup> <mi> t </mi> <mi> k </mi> </msup> <mrow> <msup> <mi> ⅇ </mi> <mi> t </mi> </msup> <mo> + </mo> <mn> 1 </mn> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <mi> t </mi> </mrow> </mrow> </mrow> <mo> ⩵ </mo> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> + </mo> <msup> <mn> 2 </mn> <mi> k </mi> </msup> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> k </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <semantics> <mrow> <mi> ζ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> k </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Zeta]", "(", TagBox[RowBox[List["k", "+", "1"]], Rule[Editable, True]], ")"]], InterpretTemplate[Function[Zeta[Slot[1]]]]] </annotation> </semantics> </mrow> <msup> <mn> 2 </mn> <mi> k </mi> </msup> </mfrac> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> k </mi> <mo> ) </mo> </mrow> <mo> > </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <int /> <bvar> <ci> t </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <power /> <ci> t </ci> <ci> k </ci> </apply> <apply> <power /> <apply> <plus /> <apply> <power /> <exponentiale /> <ci> t </ci> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <plus /> <cn type='integer'> -1 </cn> <apply> <power /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> k </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <ci> Zeta </ci> <apply> <plus /> <ci> k </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <power /> <apply> <power /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <gt /> <apply> <real /> <ci> k </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubsuperscriptBox["\[Integral]", "0", "\[Infinity]"], RowBox[List[FractionBox[SuperscriptBox["t_", "k_"], RowBox[List[SuperscriptBox["\[ExponentialE]", "t_"], "+", "1"]]], RowBox[List["\[DifferentialD]", "t_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", SuperscriptBox["2", "k"]]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List["k", "+", "1"]], "]"]], " ", RowBox[List["Zeta", "[", RowBox[List["k", "+", "1"]], "]"]]]], SuperscriptBox["2", "k"]], "/;", RowBox[List[RowBox[List["Re", "[", "k", "]"]], ">", RowBox[List["-", "1"]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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