Integrate[t^k Abs[t], {t, -a, a}] == ((1 + (-1)^k) a^(2 + k))/(2 + k) /; Element[a, Reals] && a > 0 && Re[k] > -2

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[SubsuperscriptBox["\[Integral]", RowBox[List["-", "a"]], "a"], RowBox[List[SuperscriptBox["t", "k"], " ", RowBox[List["Abs", "[", "t", "]"]], RowBox[List["\[DifferentialD]", "t"]]]]]], "\[Equal]", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["1", "+", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"]]], ")"]], " ", SuperscriptBox["a", RowBox[List["2", "+", "k"]]]]], RowBox[List["2", "+", "k"]]]]], "/;", RowBox[List[RowBox[List["a", "\[Element]", TagBox["Reals", Function[Reals]]]], "\[And]", RowBox[List["a", ">", "0"]], "\[And]", RowBox[List[RowBox[List["Re", "[", "k", "]"]], ">", RowBox[List["-", "2"]]]]]]]]]]

<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> &#8747; </mo> <mrow> <mo> - </mo> <mi> a </mi> </mrow> <mi> a </mi> </msubsup> <mrow> <mrow> <msup> <mi> t </mi> <mi> k </mi> </msup> <mo> &#8290; </mo> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mi> t </mi> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> </mrow> <mo> &#8290; </mo> <mrow> <mo> &#8518; </mo> <mi> t </mi> </mrow> </mrow> </mrow> <mo> &#10869; </mo> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msup> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> a </mi> <mrow> <mi> k </mi> <mo> + </mo> <mn> 2 </mn> </mrow> </msup> </mrow> <mrow> <mi> k </mi> <mo> + </mo> <mn> 2 </mn> </mrow> </mfrac> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> a </mi> <mo> &#8712; </mo> <semantics> <mi> &#8477; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalR]&quot;, Function[Reals]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <mi> a </mi> <mo> &gt; </mo> <mn> 0 </mn> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> k </mi> <mo> ) </mo> </mrow> <mo> &gt; </mo> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <int /> <bvar> <ci> t </ci> </bvar> <lowlimit> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> </lowlimit> <uplimit> <ci> a </ci> </uplimit> <apply> <times /> <apply> <power /> <ci> t </ci> <ci> k </ci> </apply> <apply> <abs /> <ci> t </ci> </apply> </apply> </apply> <apply> <times /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <power /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> </apply> <apply> <power /> <ci> a </ci> <apply> <plus /> <ci> k </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> k </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <and /> <apply> <in /> <ci> a </ci> <reals /> </apply> <apply> <gt /> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> <apply> <gt /> <apply> <real /> <ci> k </ci> </apply> <cn type='integer'> -2 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubsuperscriptBox["\[Integral]", RowBox[List["-", "a_"]], "a_"], RowBox[List[RowBox[List[SuperscriptBox["t_", "k"], " ", RowBox[List["Abs", "[", "t_", "]"]]]], RowBox[List["\[DifferentialD]", "t_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List["1", "+", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"]]], ")"]], " ", SuperscriptBox["a", RowBox[List["2", "+", "k"]]]]], RowBox[List["2", "+", "k"]]], "/;", RowBox[List[RowBox[List["a", "\[Element]", "Reals"]], "&&", RowBox[List["a", ">", "0"]], "&&", RowBox[List[RowBox[List["Re", "[", "k", "]"]], ">", RowBox[List["-", "2"]]]]]]]]]]]]

2001-10-29