Wolfram Researchfunctions.wolfram.comOther Wolfram Sites
Search Site
Function CategoriesGraphics GalleryNotationsGeneral IdentitiesAbout This Site Email Comments

View Related Information In
The Documentation Center
MathWorld

Download All Formulas For This Function
Mathematica Notebook
PDF File

Download All Introductions For This Function
Mathematica Notebook
PDF File

 

Developed with Mathematica -- Download a Free Trial Version
 











Cot






Mathematica Notation

Traditional Notation









Elementary Functions > Cot[z] > Integration > Indefinite integration > Involving one direct function and elementary functions > Involving trigonometric functions > Involving algebraic functions of sin > Involving (a+b sin(c z))beta





http://functions.wolfram.com/01.09.21.0067.01









  


  










Input Form





Integrate[Cot[c z]/Sqrt[a - a Sin[c z]], z] == ((Log[-1 + Cos[(c z)/2] - Sin[(c z)/2]] - Log[1 + Cos[(c z)/2] - Sin[(c z)/2]]) (Cos[(c z)/2] - Sin[(c z)/2]))/ (c Sqrt[a - a Sin[c z]])










Standard Form





Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[FractionBox[RowBox[List["Cot", "[", RowBox[List["c", " ", "z"]], "]"]], SqrtBox[RowBox[List["a", "-", RowBox[List["a", " ", RowBox[List["Sin", "[", RowBox[List["c", " ", "z"]], "]"]]]]]]]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Log", "[", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["Cos", "[", FractionBox[RowBox[List["c", " ", "z"]], "2"], "]"]], "-", RowBox[List["Sin", "[", FractionBox[RowBox[List["c", " ", "z"]], "2"], "]"]]]], "]"]], "-", RowBox[List["Log", "[", RowBox[List["1", "+", RowBox[List["Cos", "[", FractionBox[RowBox[List["c", " ", "z"]], "2"], "]"]], "-", RowBox[List["Sin", "[", FractionBox[RowBox[List["c", " ", "z"]], "2"], "]"]]]], "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["Cos", "[", FractionBox[RowBox[List["c", " ", "z"]], "2"], "]"]], "-", RowBox[List["Sin", "[", FractionBox[RowBox[List["c", " ", "z"]], "2"], "]"]]]], ")"]]]], ")"]], "/", RowBox[List["(", RowBox[List["c", " ", SqrtBox[RowBox[List["a", "-", RowBox[List["a", " ", RowBox[List["Sin", "[", RowBox[List["c", " ", "z"]], "]"]]]]]]]]], ")"]]]]]]]]










MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mo> &#8747; </mo> <mrow> <mfrac> <mrow> <mi> cot </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> c </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <msqrt> <mrow> <mi> a </mi> <mo> - </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> c </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </msqrt> </mfrac> <mo> &#8290; </mo> <mrow> <mo> &#8518; </mo> <mi> z </mi> </mrow> </mrow> </mrow> <mo> &#10869; </mo> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mi> cos </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mrow> <mi> c </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mrow> <mi> c </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mi> cos </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mrow> <mi> c </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mrow> <mi> c </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> cos </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mrow> <mi> c </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mrow> <mi> c </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mi> c </mi> <mo> &#8290; </mo> <msqrt> <mrow> <mi> a </mi> <mo> - </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> c </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </msqrt> </mrow> </mfrac> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <int /> <bvar> <ci> z </ci> </bvar> <apply> <times /> <apply> <cot /> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> a </ci> <apply> <sin /> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <ln /> <apply> <plus /> <apply> <cos /> <apply> <times /> <ci> c </ci> <ci> z </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <sin /> <apply> <times /> <ci> c </ci> <ci> z </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ln /> <apply> <plus /> <apply> <cos /> <apply> <times /> <ci> c </ci> <ci> z </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <sin /> <apply> <times /> <ci> c </ci> <ci> z </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> <apply> <plus /> <apply> <cos /> <apply> <times /> <ci> c </ci> <ci> z </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <sin /> <apply> <times /> <ci> c </ci> <ci> z </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <ci> c </ci> <apply> <power /> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> a </ci> <apply> <sin /> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[FractionBox[RowBox[List["Cot", "[", RowBox[List["c_", " ", "z_"]], "]"]], SqrtBox[RowBox[List["a_", "-", RowBox[List["a_", " ", RowBox[List["Sin", "[", RowBox[List["c_", " ", "z_"]], "]"]]]]]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Log", "[", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["Cos", "[", FractionBox[RowBox[List["c", " ", "z"]], "2"], "]"]], "-", RowBox[List["Sin", "[", FractionBox[RowBox[List["c", " ", "z"]], "2"], "]"]]]], "]"]], "-", RowBox[List["Log", "[", RowBox[List["1", "+", RowBox[List["Cos", "[", FractionBox[RowBox[List["c", " ", "z"]], "2"], "]"]], "-", RowBox[List["Sin", "[", FractionBox[RowBox[List["c", " ", "z"]], "2"], "]"]]]], "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["Cos", "[", FractionBox[RowBox[List["c", " ", "z"]], "2"], "]"]], "-", RowBox[List["Sin", "[", FractionBox[RowBox[List["c", " ", "z"]], "2"], "]"]]]], ")"]]]], RowBox[List["c", " ", SqrtBox[RowBox[List["a", "-", RowBox[List["a", " ", RowBox[List["Sin", "[", RowBox[List["c", " ", "z"]], "]"]]]]]]]]]]]]]]










Date Added to functions.wolfram.com (modification date)





2002-12-18