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

 

Developed with Mathematica -- Download a Free Trial Version
 











Exp




Mathematica Notation

Traditional Notation









Elementary Functions > Exp[z] > Integration > Indefinite integration > Involving functions of the direct function > Involving algebraic functions of the direct function > Other integrals





http://functions.wolfram.com/01.03.21.0524.01









  


  








Input Form





Integrate[E^((3 z)/4)/((-2 + E^((3 z)/4)) Sqrt[-2 + E^((3 z)/4) + E^((3 z)/2)]), z] == (2/3) (Log[-2 + E^((3 z)/4)] - Log[-2 + 5 E^((3 z)/4) + 4 Sqrt[-2 + E^((3 z)/4) + E^((3 z)/2)]])








Standard Form





Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[FractionBox[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["3", " ", "z"]], "/", "4"]]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "2"]], "+", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["3", " ", "z"]], "/", "4"]]]]], ")"]], " ", SqrtBox[RowBox[List[RowBox[List["-", "2"]], "+", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["3", " ", "z"]], "/", "4"]]], "+", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["3", " ", "z"]], "/", "2"]]]]]]]]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[FractionBox["2", "3"], " ", RowBox[List["(", RowBox[List[RowBox[List["Log", "[", RowBox[List[RowBox[List["-", "2"]], "+", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["3", " ", "z"]], "/", "4"]]]]], "]"]], "-", RowBox[List["Log", "[", RowBox[List[RowBox[List["-", "2"]], "+", RowBox[List["5", " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["3", " ", "z"]], "/", "4"]]]]], "+", RowBox[List["4", " ", SqrtBox[RowBox[List[RowBox[List["-", "2"]], "+", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["3", " ", "z"]], "/", "4"]]], "+", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["3", " ", "z"]], "/", "2"]]]]]]]]]], "]"]]]], ")"]]]]]]]]








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> <msup> <mi> &#8519; </mi> <mrow> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> / </mo> <mn> 4 </mn> </mrow> </msup> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> + </mo> <msup> <mi> &#8519; </mi> <mrow> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> / </mo> <mn> 4 </mn> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msqrt> <mrow> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> + </mo> <msup> <mi> &#8519; </mi> <mrow> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> / </mo> <mn> 4 </mn> </mrow> </msup> <mo> + </mo> <msup> <mi> &#8519; </mi> <mrow> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> </mrow> </msqrt> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <mo> &#8518; </mo> <mi> z </mi> </mrow> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <mfrac> <mn> 2 </mn> <mn> 3 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> + </mo> <msup> <mi> &#8519; </mi> <mrow> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> / </mo> <mn> 4 </mn> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <msqrt> <mrow> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> + </mo> <msup> <mi> &#8519; </mi> <mrow> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> / </mo> <mn> 4 </mn> </mrow> </msup> <mo> + </mo> <msup> <mi> &#8519; </mi> <mrow> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> </mrow> </msqrt> </mrow> <mo> + </mo> <mrow> <mn> 5 </mn> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mrow> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> / </mo> <mn> 4 </mn> </mrow> </msup> </mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <int /> <bvar> <ci> z </ci> </bvar> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <apply> <times /> <cn type='integer'> 3 </cn> <ci> z </ci> </apply> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <plus /> <cn type='integer'> -2 </cn> <apply> <power /> <exponentiale /> <apply> <times /> <apply> <times /> <cn type='integer'> 3 </cn> <ci> z </ci> </apply> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> -2 </cn> <apply> <power /> <exponentiale /> <apply> <times /> <apply> <times /> <cn type='integer'> 3 </cn> <ci> z </ci> </apply> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <apply> <times /> <cn type='integer'> 3 </cn> <ci> z </ci> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='rational'> 2 <sep /> 3 </cn> <apply> <plus /> <apply> <ln /> <apply> <plus /> <cn type='integer'> -2 </cn> <apply> <power /> <exponentiale /> <apply> <times /> <apply> <times /> <cn type='integer'> 3 </cn> <ci> z </ci> </apply> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ln /> <apply> <plus /> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <apply> <plus /> <cn type='integer'> -2 </cn> <apply> <power /> <exponentiale /> <apply> <times /> <apply> <times /> <cn type='integer'> 3 </cn> <ci> z </ci> </apply> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <apply> <times /> <cn type='integer'> 3 </cn> <ci> z </ci> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 5 </cn> <apply> <power /> <exponentiale /> <apply> <times /> <apply> <times /> <cn type='integer'> 3 </cn> <ci> z </ci> </apply> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <cn type='integer'> -2 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[FractionBox[SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["3", " ", "z_"]], "4"]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "2"]], "+", SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["3", " ", "z_"]], "4"]]]], ")"]], " ", SqrtBox[RowBox[List[RowBox[List["-", "2"]], "+", SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["3", " ", "z_"]], "4"]], "+", SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["3", " ", "z_"]], "2"]]]]]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox["2", "3"], " ", RowBox[List["(", RowBox[List[RowBox[List["Log", "[", RowBox[List[RowBox[List["-", "2"]], "+", SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["3", " ", "z"]], "4"]]]], "]"]], "-", RowBox[List["Log", "[", RowBox[List[RowBox[List["-", "2"]], "+", RowBox[List["5", " ", SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["3", " ", "z"]], "4"]]]], "+", RowBox[List["4", " ", SqrtBox[RowBox[List[RowBox[List["-", "2"]], "+", SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["3", " ", "z"]], "4"]], "+", SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["3", " ", "z"]], "2"]]]]]]]]], "]"]]]], ")"]]]]]]]]


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





2002-12-18