Integrate[ArcSec[t]/t^(3/2), {t, 1, Infinity}] == (Sqrt[Pi] Gamma[3/4])/Gamma[5/4]

Cell[BoxData[RowBox[List[RowBox[List[SubsuperscriptBox["\[Integral]", "1", "\[Infinity]"], RowBox[List[RowBox[List["ArcSec", "[", "t", "]"]], " ", SuperscriptBox["t", RowBox[List[RowBox[List["-", "3"]], "/", "2"]]], RowBox[List["\[DifferentialD]", "t"]]]]]], "\[Equal]", FractionBox[RowBox[List[SqrtBox["\[Pi]"], " ", RowBox[List["Gamma", "[", FractionBox["3", "4"], "]"]]]], RowBox[List["Gamma", "[", FractionBox["5", "4"], "]"]]]]]]]

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <msubsup> <mo> &#8747; </mo> <mn> 1 </mn> <mi> &#8734; </mi> </msubsup> <mrow> <mrow> <msup> <mi> sec </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mi> t </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> t </mi> <mrow> <mrow> <mo> - </mo> <mn> 3 </mn> </mrow> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mo> &#8518; </mo> <mi> t </mi> </mrow> </mrow> </mrow> <mo> &#10869; </mo> <mfrac> <mrow> <msqrt> <mi> &#960; </mi> </msqrt> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mn> 3 </mn> <mn> 4 </mn> </mfrac> <mo> ) </mo> </mrow> </mrow> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mn> 5 </mn> <mn> 4 </mn> </mfrac> <mo> ) </mo> </mrow> </mfrac> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <int /> <bvar> <ci> t </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <arcsec /> <ci> t </ci> </apply> <apply> <power /> <ci> t </ci> <cn type='rational'> -3 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <pi /> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <ci> Gamma </ci> <cn type='rational'> 3 <sep /> 4 </cn> </apply> <apply> <power /> <apply> <ci> Gamma </ci> <cn type='rational'> 5 <sep /> 4 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubsuperscriptBox["\[Integral]", "1", "\[Infinity]"], RowBox[List[RowBox[List[RowBox[List["ArcSec", "[", "t_", "]"]], " ", SuperscriptBox["t_", RowBox[List[RowBox[List["-", "3"]], "/", "2"]]]]], RowBox[List["\[DifferentialD]", "t_"]]]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[SqrtBox["\[Pi]"], " ", RowBox[List["Gamma", "[", FractionBox["3", "4"], "]"]]]], RowBox[List["Gamma", "[", FractionBox["5", "4"], "]"]]]]]]]

2001-10-29