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http://functions.wolfram.com/01.07.21.0023.01
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Integrate[(Cos[n x] - Cos[n \[Alpha]])/(Cos[x] - Cos[\[Alpha]]),
{x, 0, Pi}] == (Pi Sin[Pi \[Alpha]])/Sin[\[Alpha]] /;
Element[n, Integers] && n > 0 && Element[\[Alpha], Reals]
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[SubsuperscriptBox["\[Integral]", "0", "\[Pi]"], RowBox[List[FractionBox[RowBox[List[RowBox[List["Cos", "[", RowBox[List["n", " ", "x"]], "]"]], "-", RowBox[List["Cos", "[", RowBox[List["n", " ", "\[Alpha]"]], "]"]]]], RowBox[List[RowBox[List["Cos", "[", "x", "]"]], "-", RowBox[List["Cos", "[", "\[Alpha]", "]"]]]]], RowBox[List["\[DifferentialD]", "x"]]]]]], "\[Equal]", FractionBox[RowBox[List["\[Pi]", " ", RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", "\[Alpha]"]], "]"]]]], RowBox[List["Sin", "[", "\[Alpha]", "]"]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", ">", "0"]], "\[And]", RowBox[List["Element", "[", RowBox[List["\[Alpha]", ",", "Reals"]], "]"]]]]]]]]
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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> π </mi> </msubsup> <mrow> <mfrac> <mrow> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> n </mi> <mo> ⁢ </mo> <mi> x </mi> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> n </mi> <mo> ⁢ </mo> <mi> α </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> α </mi> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <mi> x </mi> </mrow> </mrow> </mrow> <mo> ⩵ </mo> <mfrac> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mrow> <mi> sin </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mi> α </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mi> sin </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> α </mi> <mo> ) </mo> </mrow> </mfrac> </mrow> <mo> /. </mo> <mtext> ​ </mtext> <mrow> <mrow> <mi> n </mi> <mo> ∈ </mo> <msup> <mi> ℕ </mi> <mo> + </mo> </msup> </mrow> <mo> ∧ </mo> <mrow> <mi> α </mi> <mo> ∈ </mo> <semantics> <mi> ℝ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalR]", Function[Reals]] </annotation> </semantics> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> ReplaceAll </ci> <apply> <eq /> <apply> <int /> <bvar> <ci> x </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <pi /> </uplimit> <apply> <times /> <apply> <plus /> <apply> <cos /> <apply> <times /> <ci> n </ci> <ci> x </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <cos /> <apply> <times /> <ci> n </ci> <ci> α </ci> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <cos /> <ci> x </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <cos /> <ci> α </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <pi /> <apply> <sin /> <apply> <times /> <pi /> <ci> α </ci> </apply> </apply> <apply> <power /> <apply> <sin /> <ci> α </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <and /> <apply> <in /> <ci> n </ci> <apply> <ci> SuperPlus </ci> <ci> ℕ </ci> </apply> </apply> <apply> <in /> <ci> α </ci> <reals /> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubsuperscriptBox["\[Integral]", "0", "\[Pi]"], RowBox[List[FractionBox[RowBox[List[RowBox[List["Cos", "[", RowBox[List["n_", " ", "x_"]], "]"]], "-", RowBox[List["Cos", "[", RowBox[List["n_", " ", "\[Alpha]_"]], "]"]]]], RowBox[List[RowBox[List["Cos", "[", "x_", "]"]], "-", RowBox[List["Cos", "[", "\[Alpha]_", "]"]]]]], RowBox[List["\[DifferentialD]", "x_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List["\[Pi]", " ", RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", "\[Alpha]"]], "]"]]]], RowBox[List["Sin", "[", "\[Alpha]", "]"]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", ">", "0"]], "&&", RowBox[List["\[Alpha]", "\[Element]", "Reals"]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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){kind=small}
